Sturm-Liouville operators with singularities and …my.fit.edu/~cfulton/fulton_html/fulton_langer_2010.pdfSturm-Liouville operators with singularities and generalized Nevanlinna functions

Sturm-Liouville operators with singularities andgeneralized Nevanlinna functionsComplex Analysis and Operator Theory 4 (2) (May 2010), pp. 179-243

C. Fulton and H. Langer

Abstract. The Titchmarsh-Weyl function, which was introduced in [10] for the Sturm-Liouville equation with a hydrogen-like potential on (0,∞), is shown to belong to ageneralized Nevanlinna class Nκ. As a consequence, also in the case of two singularendpoints for the Fourier transformation defined by means of Frobenius solutions thereexists a scalar spectral function. This spectral function is given explicitly for potentials

of the formq0

x2+

q1

x, −

1

4≤ q0 < ∞.

Mathematics Subject Classification (2000). Primary 34B20, 47E05;Secondary 34B24, 34B30, 34B40, 34L10, 34L40, 33C15, 81Q10.

Keywords. Sturm-Liouville operator, Titchmarsh-Weyl function, spectral function, sin-gular potential, generalized Nevanlinna function, Whittaker function, Bessel function.

1. Introduction

In the paper [10] a new Titchmarsh-Weyl function was introduced for differential equa-tions which include

−y′′(x) +( q0x2

+q1x

)y(x) = λy(x), x ∈ (0,∞), (1.1)

under the assumptions

−1

4≤ q0 <∞, q1 ∈ R, q20 + q21 6= 0. (1.2)

It was shown that if x = 0 is Limit Circle (LC) for (1.1), that is if − 14 ≤ q0 < 3

4 ,the new Titchmarsh-Weyl functions belong to the class N0 of Nevanlinna functions, andthat if x = 0 is Limit Point (LP) for (1.1), that is if q0 ≥ 3

4 , they do not belong to N0.For the LC case at x = 0 it was shown that the classical Titchmarsh-Kodaira formula,

The work of the first author was partially supported by National Science Foundation Grant DMS-0109022 toFlorida Institute of Technology.

2 C. Fulton and H. Langer

representing the spectral function by means of the boundary values of the Titchmarsh-Weyl function, holds.

In this paper we will show that for the LP case at x = 0 this new Titchmarsh-Weylfunction belongs to a generalized Nevanlinna class Nκ, and we will establish the valueof κ in its dependence on q0. We will also establish that the Titchmarsh-Kodaira formulacarries over to the LP case at x = 0, so that the spectral function can be calculated in theusual manner from the Titchmarsh-Weyl function.

Recall that in the classical approach, the fundamental system relative to which the Titch-marsh-Weyl function is defined, is chosen normalized either in a regular endpoint, or, ifthere is no regular endpoint, in an interior point of the considered interval; the first caseleads to a scalar spectral function, the second case leads to a 2×2 - matrix spectral function.The novelty of the approach in [10] is that the fundamental system is chosen to be the sys-tem {φ(·;λ), θ(·;λ)} of solutions of (1.1) which arise from application of the Frobeniustheory to equation (1.1), see Subsection 2.1 below. This enables, also in the case of twosingular endpoints, the eigenfunction expansion to be formulated with a scalar spectralfunction and a corresponding Fourier transformation which involves only one solution.As a consequence, the differential operators considered in this paper, also in the cases oftwo singular endpoints, have simple spectrum; compare also [14], [9], [12], and the cor-responding remarks in [10]. However, the question concerning what general conditionswill ensure a simple spectrum in the case of two singular endpoints remains, for the mostpart, open. The eigenfunction expansion theory for Sturm-Liouville problems associatedwith (1.1) was carried out in [10] using the approach of Levitan [25] and Levinson [24]by which the spectral function in the singular case is defined as limit of step spectral func-tions over the finite interval (0, b] (the left endpoint x = 0 being a regular singular pointand nonoscillatory for all real λ). In contrast, in the present paper the spectral functionand the Parseval relation are obtained using Krein’s theory of directing functionals cou-pled with a simple application of the Stieltjes-Livsic inversion formula. This approach tothe eigenfunction expansion theory hinges on the fact that the Titchmarsh-Weyl functionwhich is defined relative to a Frobenius fundamental system belongs to a class Nκ ofgeneralized Nevanlinna functions.

For the Bessel differential equation, by an approach which is different from the one in[10], it was shown in [27] and further studied in [7], that there is an associated generalizedNevanlinna function (and this function coincides with the Titchmarsh-Weyl function from[10] or from the present paper). Similar results for the Laguerre differential equation wereobtained in [2] and [8]. In the more abstract setting of point-like singular perturbationssuch problems were studied in [3], [6], [4]. In these papers an essential role is playedby self-adjoint operators in Pontryagin spaces, in fact, with any generalized Nevanlinnafunction such an operator can be associated and one of the problems considered thereis to describe this Pontryagin space and the corresponding self-adjoint operator in termsof the spaces or operators associated with the given differential operators in the classicalapproach.

Sturm-Liouville operators 3

In the present paper Pontryagin spaces will not play a role. The reason is that the general-ized Nevanlinna functions m which arise here are of the special form

m(λ) =(1 + z2

)nm(λ) + q(λ) (1.3)

with a Nevanlinna function m and a real polynomial q. In this case m has no generalizedpoles of nonpositive type on R and no poles in C \ R, and the spectral function ρ is anondecreasing function on the real axis, with the only difference to the classical case thatρ can have polynomial growth at infinity. In the recent paper [22], exploiting the specialform (1.3) of the generalized Nevanlinna functions, instead of Pontryagin spaces, Hilbertspaces of generalized elements (which are in fact functions with a singularity at zero) areused to give an operator theoretic interpretation for the problem (1.1), (1.2) in the contextof supersingular perturbations. We mention, however, that for other cases of Bessel likeequations (with the singularity in the interior of the interval) there appear Titchmarsh-Weyl functions which are generalized Nevanlinna functions not of the special form (1.3)and with poles of nonpositive type in the finite complex plane, see [23]; in this casePontryagin spaces seem to play an essential role.A short outline of the contents of this paper is as follows. In Section 2 we recall someresults from [10] as well as the definition of generalized Nevanlinna functions and some oftheir properties. In Section 3, for the case of a slightly more general potential than in (1.1)it is shown that the Titchmarsh-Weyl function as introduced in [10], which was shown tobe a Nevanlinna function in the LC case at x = 0 in [10], is a generalized Nevanlinnafunction in the LP case. The Fourier transformation with respect to a Frobenius solutionin the LP case and to a linear combination of Frobenius solutions in the LC case and thecorresponding spectral functions are defined and studied in Section 4.Finally, in Section 5 we work out the spectral theory for equation (1.1) for all cases ofq0, q1 satisfying the assumption (1.2) and the Friedrichs boundary condition at x = 0, andgenerate explicit formulas for the eigenfunction expansions. Finding the Titchmarsh-Weylfunction for these cases is a matter of identifying the appropriate Frobenius solutions,and then representing the solution which is square integrable at infinity (for =λ 6= 0),as a linear combination of them. In all cases, we encounter either Whittaker or Besselfunctions. For the calculations it is convenient to consider the nonlog case (Case I), wherethe difference r2−r1 of the indicial roots r1, r2 is not an integer, and the log case (Case II),where r2−r1 is an integer, separately; in Case II we make a further distinction between thehydrogen atom case q0 = `(`+ 1) and the Bessel like case q0 = N2 − 1

4 , N = 0, 1, . . . .

2. Preliminaries

2.1. In this subsection we recall some results from [10] about the differential equation

−y′′(x) +

(q0x2

+q1x

+

∞∑

n=0

qn+2xn

)y(x) = λy(x), x ∈ (0, b). (2.1)

Here 0 < b ≤ ∞, and q0, q1, . . . are real numbers, satisfying (1.2). We assume that theseries in (2.1) converges in [0, b), and that b is a regular endpoint for (2.1) if b < ∞ andthat (2.1) is LP at b if b = ∞.


The corresponding indicial equation

r2 − r − q0 = 0

has the solutions

r1 =1

2+

√1

4+ q0, r2 =

1

2−√

1

4+ q0,

where r1 ≥ r2, r1 + r2 = 1. As in [10, Theorem 2.1], according to the form of theassociated Frobenius solutions we distinguish the nonlog case (Case I) and the log cases(Case II); the Wronskian of two functions ϕ, ψ is denoted by

Wx(ϕ, ψ) :=

∣∣∣∣∣ϕ(x) ψ(x)

ϕ′(x) ψ′(x)

∣∣∣∣∣ .

Case I: −1

4< q0 <∞, q0 6= M2 − 1

4, M = 1, 2, . . . .

This is Case I A in [10]. We put

q0 = ν2 − 1

4with ν ∈ (0,∞), ν 6= N

2, N = 1, 2, . . . , q1 = −a ∈ R,

so the indicial roots are

r1 =1

2+ ν, r2 =

1

2− ν, ν ∈ (0,∞), ν 6= N

2, N = 1, 2, . . . .

Since r1 − r2 = 2ν 6= 1, 2, . . . , the Frobenius solutions have the general form

y1(x;λ) = x12+ν

(1 +

∞∑

n=1

an(λ)xn

), (2.2)

y2(x;λ) = x12−ν

(1 +

∞∑

n=1

bn(λ)xn

), (2.3)

where an(λ), bn(λ) are polynomials in λ of degree[n2

], and

Wx

(y1(·;λ), y2(·;λ)

)= −2ν. (2.4)

Case II A: q0 =M2 − 1

4, M odd: M = 2`+ 1, ` = 0, 1, . . . .

In this case

q0 = `(`+ 1), ` = 0, 1, . . . .

In [10] this is Case IC for M odd and it includes Case II (for ` = 0). The indicial rootsare

r1 = `+ 1, r2 = −`, ` = 0, 1, . . . .


Accordingly, the Frobenius solutions have the general form

y1(x;λ) = x`+1

(1 +

∞∑

n=1

an(λ)xn

), (2.5)

y2(x;λ) = K`(λ) y1(x;λ) ln x+ x−`

(1 +

∞∑

n=1

dn(λ)xn

), (2.6)

where an(λ), dn(λ) are polynomials in λ of degree[n2

],K`(λ) is a polynomial of degree

`, andWx

(y1(·;λ), y2(·;λ)

)= −(2`+ 1).

Case II B: q0 =M2 − 1

4, M even: M = 2N, N = 0, 1, . . . .

In this case

q0 = N2 − 1

4, N = 0, 1, . . . .

In [10] this is Case IC for M even and it includes Case IB (for N = 0). Now the indicialroots are

r1 =1

2+N, r2 =

1

2−N, N = 0, 1, . . . .

Accordingly, the Frobenius solutions have the general form

y1(x;λ) = x12 +N

(1 +

∞∑

n=1

an(λ)xn

), (2.7)

y2(x;λ) = KN (λ) y1(x;λ) lnx + x12−N

(1 +

∞∑

n=1

dn(λ)xn

), (2.8)

where an(λ), dn(λ) are polynomials in λ of degree[n2

], KN (λ) is a polynomial of

degree N, and

Wx

(y1(·;λ), y2(·;λ)

)=

{−2N if N ≥ 1,

1 if N = 0.(2.9)

The above classification covers all cases of q0 satisfying (1.2). In each of these cases wehave the following properties of the Frobenius solutions (see [10, Theorem 2.1]):

(i) y1(x; ·), y2(x; ·) and their derivatives are entire functions for each x ∈ (0, b) andsatisfy for all λ ∈ C, x ∈ (0, b) the relations

yi

(x;λ

)= yi(x;λ), y′i

(x;λ

)= y′i(x;λ), i = 1, 2.

(ii) y1(·;λ) ∈ L2(0, x0) for 0 < x0 < b and for all λ ∈ C.

(iii) Wx

(y1(·;λ), y2(·;λ)

)= C 6= 0 where C ∈ R, independent of λ.

(iv) limx↓0Wx

(y1(·;λ), y2(·;λ′)

)= C for all λ, λ′ ∈ C, where C is the same constant

as in (iii).

(v) limx↓0Wx

(y1(·;λ), y1(·;λ′)

)= 0 for all λ, λ′ ∈ C.


(vi) limx↓0Wx

(y2(·;λ

), y2(·;λ′)

)=

{0 if x = 0 is LC,∞ if x = 0 is LP,

for all λ, λ′ ∈ C, λ 6= λ′.

Recall (see [10]) that the differential expression in (2.1) is LC at x = 0 if −1/4 ≤ q0 <3/4, and LP if q0 ≥ 3/4. As in [10], we choose the following fundamental system φ, θ ofsolutions of (1.1):

If x = 0 is LP, then

φ(x;λ) := y1(x;λ), θ(x;λ) := y2(x;λ)/C, (2.10)

if x = 0 is LC, then, with α ∈ [0, π),

φα(x;λ) := cosα y1(x;λ) − sinα

Cy2(x;λ),

θα(x;λ) := sinα y1(x;λ) +cosα

Cy2(x;λ).

(2.11)

Then, in the LP case, φ(x;λ) satisfies the boundary condition

f(0) = 0,

and, in the LC case, φα(x;λ) satisfies the boundary condition

limx↓0

Wx

(f, φα(·; 0)

)= 0. (2.12)

Limit circle boundary conditions have been parametrized in the form (2.12) by manyauthors, for example [26, (9)], [15, p.141], [11, (2.18)], [30]. For the case when the LCendpoint is also nonoscillatory, F. Rellich [26] was the first to show that in the LC case atx = 0 the Friedrichs extension associated with the LC endpoint can be characterized bythe boundary condition

limx↓0

Wx

(f, yp(·;λ0)

)= 0

for any real λ0, where yp(·;λ0) is the principal solution at x = 0. Under the assumptionof the present paper, the left endpoint is nonoscillatory, and the first Frobenius solutiony1(·;λ) is the principal solution for all real λ, so it follows that in the LC case the boundarycondition (2.12) with α = 0 gives the classical Friedrichs extension. It is this extensionwhich will be considered in Section 5.As is well known, in (2.12) zero in φα(·; 0) can be replaced by any real λ0, in fact for fin the domain of the maximal operator associated with the left hand side of (2.1), we have

limx↓0

Wx

(f, φα(·;λ0)

)= lim

x↓0Wx

(f, φα(·; 0)

)

for any λ0 ∈ (−∞,∞). This follows by making use of the Plucker identity

Wx(f, g) =

∣∣∣∣Wx(f, v0) Wx(g, v0)−Wx(f, u0) −Wx(g, u0)

∣∣∣∣ ,

where u0, v0 are any two solutions of (2.1) with λ = 0 satisfying Wx(u0, v0) = 1.Namely, take u0(x) = y1(x; 0), v0(x) = 1

C y2(x; 0) and apply the above with x → 0using g(x) = y1(x;λ0) and then g(x) = y2(x;λ0) for any real λ0.In the sequel, when we introduce the Fourier transformation using a single solution ofequation (1.1) it will be helpful to have a common symbol for the solution to be used; so,


henceforth we will take φ(x;λ) = y1(x;λ) when x = 0 is LP and φ(x;λ) = φα(x;λ)when x = 0 is LC; correspondingly, we also write θ instead of θα.In both LC and LP cases, φ and θ have the following properties corresponding to theabove properties of y1, y2:

(a) φ(x; ·), θ(x; ·) and their derivatives φ′(x; ·) =∂φ(x; ·)∂x

, θ′(x; ·) =∂θ(x; ·)∂x

are

entire functions for each x ∈ (0, b), and satisfy for all λ ∈ C, x ∈ (0, b) therelations

φ(x;λ)= φ(x;λ), φ′

(x;λ)= φ′(x;λ), θ

(x;λ)= θ(x;λ), θ′

(x;λ)= θ′(x;λ).

(b) φ(·;λ) ∈ L2(0, x0) for 0 < x0 <∞ and for all λ ∈ C.

(c) Wx

(φ(·;λ), θ(·;λ)

)= 1 for all λ ∈ C.

(d) limx↓0Wx

(φ(·;λ), θ(·;λ′)

)= 1 for all λ, λ′ ∈ C.

(e) limx↓0Wx

(φ(·;λ), φ(·;λ′)

)= 0 for all λ, λ′ ∈ C.

(f) limx↓0Wx

(θ(·;λ), θ(·;λ′)

)=

{0 if x = 0 is LC,∞ if x = 0 is LP,

for all λ, λ′∈ C, λ 6= λ′.

2.2. In this subsection we recall the definition of Nevanlinna and generalized Nevanlinnafunctions and formulate some of their properties which will be used in this paper. Let DK

be a nonempty open subset of the complex plane, and let κ be a nonnegative integer. Thecomplex valued function or ‘kernel’K(z, ζ), z, ζ ∈ DK , which depends holomorphicallyon z and ζ, is said to have κ negative squares, if the following holds:

(1) K(z, ζ) = K(ζ, z), z, ζ ∈ DK .(2) For any m ∈ N, z1, z2, . . . , zm ∈ DK , zi 6= zj , i, j = 1, 2, . . . ,m, the matrix

(K(zi, zj)

)mi,j=1

has ≤ κ negative eigenvalues and for at least one choice of m, z1, . . . , zm it has κnegative eigenvalues.

If κ = 0 the kernel K(z, ζ) is said to be positive definite. The number of positive squaresof a kernel K(z, ζ) is defined accordingly.A Nevanlinna function is by definition a function F which is defined and holomorphic (atleast) in C+ ∪ C− and has the properties

(i) F (z) = F (z) for all z ∈ hol (F ), the domain of holomorphy of F .

(ii)=F (z)

= z ≥ 0, = z 6= 0.

It is well known (see [20, p.187]) that a function F which is holomorphic in C+ ∪ C

− isa Nevanlinna function if and only if it has the property (i) and the kernel

NF (z, ζ) :=F (z) − F (ζ)

z − ζ, z, ζ ∈ C

+ ∪ C−, z 6= ζ, (2.13)

is positive definite. We denote the set of all Nevanlinna functions by N0 and recall somerelated statements (see, for example, [16]). A Nevanlinna function admits an integral


representation: F ∈ N0 if and only if there exist numbers α ∈ R, β ≥ 0, and a measure

σ on R with the property∫ ∞

−∞

1

1 + t2dσ(t) <∞, such that

F (z) = α+ β z +

∫ ∞

−∞

(1

t− z− t

1 + t2

)dσ(t). (2.14)

Here α, β, and the measure σ are uniquely determined. The termt

1 + t2in (2.14) is

needed only in order regularize the integrand at ∞ so that the integral converges. If thesupport of the measure σ is semibounded from below by a ∈ R then we can choose anarbitrary a′ > a and write (2.14) in the form

F (z) = α′ + β z +

∫ a′

a

1

t− zdσ(t) +

∫ ∞

a′

(1

t− z− t

1 + t2

)dσ(t). (2.15)

Observe that β and σ are the same in both relations, only the constant terms can be dif-ferent. In the examples in Section 5 below, associated with the equation (1.1), σ will bediscrete on the negative real axis with masses σn, n = 1, 2, . . . , at points λ1 < λ2 <· · · < 0; so in these cases we shall choose a′ = 0 and the formula (2.15) then becomes

F (z) = α′ + β z +∞∑

n=1

σn

λn − z+

∫ ∞

0

(1

t− z− t

1 + t2

)dσ(t). (2.16)

A function F ∈ N0 has no nonreal poles. The representation (2.14) implies that for allreal points t0 ∈ R we have

0 ≤ limε↓0

(−iε)F (t0 + iε) <∞,

in fact this limit equals σ({t0}), and that

0 ≤ limy↑∞

F (t0 + iy)

iy<∞,

and this limit equals β in (2.14).The measure σ in the representation (2.14) is available from the boundary values of thefunction F on the real axis by the Stieltjes inversion formula: If σ has no concentratedmass at the points a, b ∈ R, a < b, then

σ([a, b]) = − 1

2πi

∮ ′

Γ[a,b]

F (z) dz = limε↓0

1

π

∫ b

a

=F (s+ iε) ds, (2.17)

where for the real interval [a, b] we denote by Γ[a,b] a rectangular contour intersecting thereal axis in a and b with orientation such that the enclosed interval (a, b) is to the left, andthe ′ at the integral denotes the Cauchy principal value at a and b. The second equalitysign in (2.17) follows easily from Cauchy’s theorem and a parametrization of the contourΓ[a,b]. Clearly, the first equality in (2.17) holds also if the mesure σ in (2.14) is complexvalued.


A generalization of (2.17) is the Stieltjes-Livsic inversion formula: Let F ∈ N0 withintegral representation (2.14) and functions ϕ, ψ, which are holomorphic on the interval[a, b], be given. Consider the function

G := ψ + ϕF.

If σ has no concentrated mass at the points a, b, and Γ[a,b] as above is chosen such thatwith its interior it is in the domain of holomorphy of ϕ and ψ, then the following relationholds:

− 1

2πi

∮ ′

Γ[a,b]

G(z) dz =

∫ b

a

ϕ(t) d σ(t), (2.18)

If, additionally, the functions ϕ and ψ are real on [a, b], then we have also

− 1

2πi

∮ ′

Γ[a,b]

G(z) dz = limε ↓ 0

1

π

∫ b

a

=G(t+ iε) dt =

∫ b

a

ϕ(t) d σ(t). (2.19)

If the measure σ has a concentrated mass e.g. at x = a then on the right hand side of

(2.18) and (2.19) the valueϕ(a)σ({a})

2has to be added.

2.3. If κ is a non-negative integer, by Nκ we denote the set of all functions Q which arelocally meromorphic in C+ ∪ C−, satisfy the symmetry condition (i) of Subsection 2.2,and for which the kernel NQ(z, ζ) from (2.13) has κ negative squares on the domain DQ

of holomorphy of Q. The functions of the classes Nκ are called generalized Nevanlinnafunctions.Let Q ∈ Nκ, κ ≥ 1. The point t0 ∈ R is said to be a generalized pole of non-positivetype of Q if either

lim supε↓0

ε |Q(t0 + iε)| = ∞,

or the limitlimε↓0

(−iε)Q(t0 + iε)

exists and is finite and negative; ∞ is said to be a generalized pole of nonpositive type ofQ if either

lim supy↑∞

|Q(iy)|y

= ∞

or the limit

limy↑∞

Q(iy)

iy

exists and is finite and negative.If t0 ∈ R is a simple pole of the function Q ∈ Nκ, then, according to the representation(2.14), it is a generalized pole of nonpositive type in the sense of the above definition ifits residuum is positive; recall that for a Nevanlinna function the residuum at a pole isnegative.

Lemma 2.1. If Q ∈ Nκ for some κ ≥ 1, then Q has at least one non-real pole orgeneralized pole of nonpositive type (the latter can be ∞).


The statement of this lemma can be made more precise if the notion of degree of nonpos-itivity of a generalized pole of nonpositive type is used, see [5]. For the proof of Lemma2.1 we refer to the papers [20], [21], [23]; for t0 ∈ R the characterization of a generalizedpole of nonpositive type used here as a definition follows from [23, Remark 2.1].The functions Q ∈ Nκ we deal with in this paper have no non-real poles and their onlygeneralized pole of nonpositive type is ∞. In this case the function Q admits the repre-sentation

Q(z) =(1 + z2

)n∫ ∞

−∞

(1

t− z− t

1 + t2

)dσ(t) +

m∑

j=0

αj zj (2.20)

with n,m ∈ N0, αj ∈ R, j = 0, 1, . . . ,m, αm 6= 0 if m > 0, and a measure σ on R

such that∫ ∞

−∞

dσ(t)

1 + t2<∞. Here, if n ≥ 1, the measure σ can be chosen such that

∫ ∞

−∞dσ(t) = ∞.

To see this we observe that in case∫∞−∞ dσ(t) <∞ the Stieltjes-Livsic inversion formula

(2.18) implies

(1+z2)

∫ ∞

−∞

(1

t− z− t

1 + t2

)dσ(t) =

∫ ∞

−∞

(1

t− z− t

1 + t2

)(1+t2

)dσ(t)+· · · ,

where · · · denotes a quadratic polynomial, and thus in the representation (2.20) the ex-ponent n can be replaced by n− 1; in doing this the polynomial

∑mj=0 αjz

j on the righthand side of (2.20) will change.If in (2.20) we have n = 0 or, if n > 0,

∫∞−∞ dσ(t) = ∞, then the representation

(2.20) is called irreducible. For an irreducible representation (2.20), the measure σ andthe polynomial

∑mj=0 αj z

j are uniquely determined, and for the index κ the followingrelation holds (see [20, Lemma 3.3]):

κ =

n if m ≤ 2n,[m2

]if m ≥ 2n+ 1,m even, orm odd and αm > 0,

m+ 1

2if m ≥ 2n+ 1,m odd and αm < 0.

Analogously to (2.16), if the support of σ is bounded from below and σ is discrete onthe negative real axis with masses σn, n = 1, 2, . . . , at points λ1 < λ2 < · · · < 0, therepresentation (2.20) can be written, as

Q(z) =

∞∑

n=0

rnλn − z

+(1 + z2

)n∫ ∞

0

(1

t− z− t

1 + t2

)dσ(t) + · · · ,

where rn = (1 + λ2n)n σn and · · · denotes again a polynomial in z.

We also make use of the following lemma.


Lemma 2.2. If the functions Qν ∈ Nκ, ν = 1, 2, . . . , are holomorphic on some non-empty open set D ⊂ C+, and converge for ν → ∞ locally uniformly on D to the functionQ, then Q ∈ Nκ′ for some κ′ ≤ κ.

Proof. We have to show that the number of negative eigenvalues of all matrices(NQ(zi, zj)

)mi,j=1

, zi, zj ∈ DQ, zi 6= zj , i, j = 1, 2, . . . ,m, (2.21)

is ≤ κ. This is the case because it is true by assumption for the matrices(NQν

(zi, zj))m

i,j=1, ν = 1, 2, . . . ,

since these matrices converge for ν → ∞ elementwise to the matrix in (2.21), and sincefor convergent matrices the number of negative (and of positive) eigenvalues cannot in-crease in the limit. �

3. The Titchmarsh-Weyl function related to theFrobenius solutions

Now we return to equation (2.1):

−y′′(x) +

(q0x2

+q1x

+

∞∑

n=0

qn+2xn

)y(x) = λy(x), x ∈ (0, b), (3.1)

where −1

4≤ q0 < ∞, q1 ∈ R, q20 + q21 6= 0. As above, to express the dependence on λ

we often write y(x;λ) instead of y(x). At x = 0, in the LC case we impose a boundarycondition (2.12) with some α ∈ [0, π); again, instead of φα, θα as in (2.11) we write justφ, θ.At the endpoint b, in case b <∞, since b was then supposed to be regular, we impose theboundary condition

y(b) cosβ + y′(b) sinβ = 0 for some β ∈ [0, π), (3.2)

and in case b = ∞, since we have supposed that the LP case prevails at ∞, no boundarycondition is needed at ∞.First we suppose that b < ∞. We define as in [10, (3.5)] the Titchmarsh-Weyl functionlβb (λ) such that the function

ψb(x;λ) := θ(x;λ) − lβb (λ)φ(x;λ)

satisfies the boundary condition (3.2) at x = b. We have

lβb (λ) =θ(b;λ) cosβ + θ′(b;λ) sinβ

φ(b;λ) cosβ + φ′(b;λ) sinβ=θ(b;λ) cot β + θ′(b;λ)

φ(b;λ) cot β + φ′(b;λ). (3.3)

From the Lagrange identity, for arbitrary x1, x2 ∈ (0, b) it follows that

(λ − µ)

∫ x2

x1

ψ(x;λ)ψ(x;µ) dx = Wx2

(ψ(·;λ), ψ(·;µ)

)−Wx1

(ψ(·;λ), ψ(·;µ)

),


and hence

0 =Wb

(θ(·;λ) − lβb (λ)φ(·;λ), θ(·;µ) − lβb (µ)φ(·;µ)

)

λ− µ

=Wε

(θ(·;λ) − lβb (λ)φ(·;λ), θ(·;µ) − lβb (µ)φ(·;µ)

)

λ− µ

+

∫ b

ε

(θ(x;λ) − lβb (λ)φ(x;λ)

)(θ(x;µ) − lβb (µ)φ(x;µ)

)dx

which yields

0 =Wε

(θ(·;λ), θ(·;µ)

)

λ− µ+ lβb (λ)

Wε

(φ(·;λ), φ(·;µ)

)

λ− µlβb (µ)

− lβb (λ)Wε

(φ(·;λ), θ(·;µ)

)+ lβb (µ)Wε

(θ(·;λ), φ(·;µ)

)

λ− µ

+

∫ b

ε

(θ(x;λ) − lβb (λ)φ(x;λ)

)(θ(x;µ) − lβb (µ)φ(x;µ)

)dx,

(3.4)

for all ε > 0.

Lemma 3.1. Let α0, α1, . . . , αn be complex valued functions which are defined and lin-early independent on some domain D ⊂ C, which is symmetric with respect to the realaxis, and let also the functions α` be symmetric with respect to the real axis: α`(λ) =

α`

(λ), λ ∈ D, ` = 0, 1, . . . , n. Then the kernel

K(λ, µ) =

n∑

`=0

(λ` α`(µ) + α`(λ)µ

`)

has n+ 1 positive and n+ 1 negative squares.

Proof. We havem∑

i,j=1

K(λi, λj)ξiξj =

m∑

i,j=1

n∑

`=0

(λ`

i α`(λj) + α`(λi)λj`)ξiξj

=

n∑

`=0

(m∑

i=1

λ`i ξi

m∑

j=1

α`(λj) ξj +

m∑

i=1

α`(λi) ξi

m∑

j=1

λ`j ξj

)

=

((0n+1 In+1

In+1 0n+1

)(a

b

),

(a

b

))

with

a =

∑mi=1 ξi∑m

i=1 λiξi...∑m

i=1 λni ξi

, b =

∑mi=1 α0(λi)ξi∑mi=1 α1(λi)ξi

...∑mi=1 αn(λi)ξi

.


The matrix

(0n+1 In+1

In+1 0n+1

)has the eigenvalues ±1, each of multiplicity n+ 1. �

Lemma 3.2. If − 14 ≤ q0 <

34 then lβb is a Nevanlinna function, and if q0 ≥ 3/4 then lβb

is a generalized Nevanlinna function.

Proof. The first claim was proved in [10, Theorem 3.2 (Case A)] for general LC boundaryconditions at x = 0. Suppose now q0 ≥ 3

4 . From the relation (3.4), letting ε ↓ 0 and using

the statements (c), (d) and (e), we obtain (we write l instead of lβb )

l(λ) − l(µ)

λ− µ= limε↓0

{Wε

(θ(·;λ), θ(·;µ)

)

λ− µ

+

∫ b

ε

(θ(x;λ) − l(λ)φ(x;λ)

)(θ(x;µ) − l(µ)φ(x;µ)

)dx

}.

(3.5)

Suppose first that q0 6= N2 − 1

4, N = 2, 3, . . . , hence we are in Case I. The first quotient

on the right hand side of (3.5) can be written as an expansion in ε2r2−1+` with r2 =12 −

√14 + q0 and ` = 0, 1, . . . . Since r2 is not a multiple of 1

2 , the exponents are not

integers, therefore with ε ↓ 0 a finite number of the terms tends to ∞, the other termstend to zero. The right hand side of (3.5) is bounded, therefore if ε ↓ 0, the terms of thefirst quotient which go to ∞, that is, for which the exponent 2r2 − 1 + ` is negative,must cancel against such terms in the integral on the right hand side of (3.5). To study thisintegral, we decompose the function θ(x;λ) as

θ(x;λ) =∑

0≤`≤− 12−r2

b`(λ)xr2+` +

∑

`>− 12−r2

b`(λ)xr2+` =: θs(x;λ) + θr(x;λ). (3.6)

Here the first term contains the powers xδ of θ(x;λ) with δ ≤ − 12 , the second term

contains the powers xδ with δ > − 12 . The integral in (3.5) becomes

∫ b

ε

θs(x;λ)θs(x;µ)dx+

∫ b

ε

(θr(x;λ) − l(λ)φ(x;λ)

)(θr(x;µ) − l(µ)φ(x;µ)

)dx

+

∫ b

ε

θs(x;λ)(θr(x;µ) − l(µ)φ(x;µ)

)dx (3.7)

+

∫ b

ε


)θs(x;µ)dx.

The function θs(x;λ)θs(x;µ) in the first integral contains only powers xδ with δ ≤ −1,therefore after integration the first term in (3.7) consists of terms tending to ∞ if ε ↓ 0,which must be cancelled against singular terms in the first quotient on the right handside of (3.5), and from the upper bound b, polynomials in λ and µ of degree at mostdeg b[|r2|− 1

2 ] arise. The smallest exponent of x in(θr(x;λ) − l(λ)φ(x;λ)

)(θr(x;µ) − l(µ)φ(x;µ)

)


is ≥ 2r2 + 2[|r2| − 1

2

]+ 2, therefore the second integral in (3.7) remains finite if ε ↓ 0,

and in the limit it defines a positive definite kernel. It remains to consider the sum

∫ b

ε

θs(x;λ)(θr(x;µ) − l(µ)φ(x;µ)

)dx+

∫ b

ε


)θs(x;µ) dx.

The exponents of x in the product θs(x;λ)φ(x;µ) are ≥ 2r2 + 2[|r2| − 1

2

]+ 1 > −1,

and therefore the corresponding integral is finite in the limit ε ↓ 0 and of the form

b0(λ)a0(µ) + b1(λ)a1(µ) + · · · b|r2|− 12(λ)a|r2|− 1

2(µ)

with analytic functions aj . The product θs(x;λ)θr(x;µ) is(b0(λ)x

r2 + b1(λ)xr2+1 + · · · + b[|r2|− 1

2 ](λ)xr2+[|r2|− 1

2

]) ∑

`≥[|r2|]+1

b`(µ)xr2+`.

If we multiply the sum on the right with the summands in the left bracket, after integrationthe terms which in the limit ε ↓ 0 tend to ∞ must again be canceled, at b these terms givepolynomials in λ of degree at most deg b[|r2|− 1

2 ]; for the following terms, which give alsoa polynomial in λ of degree at most deg b[|r2|− 1

2 ], the integrals are finite:

b0(λ)∑

`≥[|r2|]+1,

2r2+`>−1

b`(µ)x2r2+` + b1(λ)∑

`≥[|r2|]+1,

2r2+`+1>−1

b`(µ)x2r2+`+1

+ · · · + b[|r2|− 12 ](λ)

∑

`≥[|r2|]+1,

2r2+`+[|r2|− 12 ]>−1

b`(µ)x2r2+`+[|r2|− 12 ].

Therefore the kernel defined by the finite terms of the sum of the last two integrals in(3.7) is of the form of the kernel K(λ, µ) in Lemma 3.1, where n is at most the degree

of b[|r2|− 12 ](λ), that is n ≤

[[|r2| − 1

2 ]

2

]. It follows from Lemma 3.1 that the number of

negative squares of the kernel on the left hand side of (3.5) is

≤[[|r2| − 1

2 ]

2

]+ 1 =

[√14 + q0 − 1

]

2+ 1

=

√14 + q0 + 1

2

.

Now consider the log case: q0 =N2 − 1

4, N = 2, 3, . . . . Then the second Frobenius

solution is logarithmic and, since the difference of the indicial roots is an integer, the firstquotient on the right hand side of (3.5) can contain terms with ε0 which in the limit ε ↓ 0give a polynomial in λ and µ. The rest of the proof remains intact if in the decomposition(3.6) of the second solution the logarithmic term is included into θr. �


Remark 3.3. In the proof of Lemma 3.2 we have actually shown that in the nonlog Case Ithe index κ of the generalized Nevanlinna class Nκ to which lβb belongs satisfies

κ ≤

√14 + q0 + 1

2

=

[ν + 1

2

],

(compare [7]). Also in the log case, this index κ is determined by the highest index of thecoefficients b` which appear in θs, that is by the indicial roots, and hence it depends onlyon q0 and is independent of the right endpoint b and the boundary condition at b. We alsonote that in the recent paper [22] of P. Kurasov and A. Luger a different proof is givenwhich establishes κ =

[ν+12

].

In Lemma 3.2 for b any x with 0 < x < b can be chosen, hence the lemma implies thatalso for these x the quotients

θ(x; ·)φ(x; ·) ,

θ′(x; ·)φ′(x; ·)

are Nevanlinna functions if − 14 ≤q0< 3

4 , and generalized Nevanlinna functions if q0 ≥ 34 .

However, the quotientsφ(x; ·)φ′(x; ·) are always Nevanlinna functions, as follows from the next

lemma.

Lemma 3.4. Suppose b <∞. Then the functionφ(b;λ)

φ′(b;λ)is a Nevanlinna function.

Proof. We have

(λ−µ)

∫ b

0

φ(x;λ)φ(x;µ)dx = limε↓0

(∫ b

ε

(Lφ)(x;λ)φ(x;µ)dx−∫ b

ε

φ(x;λ)(Lφ)(x;µ)dx

)

= limε↓0

(−∫ b

ε

φ′′(x;λ)φ(x;µ)dx+

∫ b

ε

φ(x;λ)φ′′(x;µ)dx

)

= limε↓0

(− φ′(x;λ)φ(x;µ)

∣∣∣b

ε+ φ(x;λ)φ′(x;µ)

∣∣∣b

ε

)

= − φ′(b;λ)φ(b;µ) + φ(b;λ)φ′(b;µ),

where for the last equality relation (v) from Subsection 2.1 was used. It follows that

φ(b;λ)

φ′(b;λ)− φ(b;µ)

φ′(b;µ)

λ− µ=

1

φ′(b;λ)φ′(b;µ)

∫ b

0

φ(x;λ)φ(x;µ)dx.

�

Now we consider the case b = ∞. Then, according to [10, Theorem 3.2 (v)], we have forall 0 ≤ β < π,

limb′→∞

lβb′ −→ m, locally uniformly in (C \ R) ∩ hol(m), (3.8)


with some functionm, which according to Lemma 3.2 and Lemma 2.2 is a Nevanlinna orgeneralized Nevanlinna function. For the convenience of the reader we repeat the proofof (3.8). We choose any point b0, 0 < b0, and consider b′ > b0. Denote by ϕ, ψ thefundamental system of solutions of the differential equation (3.1) on [b0,∞) satisfyingthe initial conditions

ϕ(b0;λ) = 1, ϕ′(b0;λ) = 0, ψ(b0;λ) = 0, ψ′(b0;λ) = 1.

Then, for x ≥ b0, we have

θ(x;λ) = θ(b0;λ)ϕ(x;λ) + θ′(b0;λ)ψ(x;λ),

φ(x;λ) = φ(b0;λ)ϕ(x;λ) + φ′(b0;λ)ψ(x;λ).

We choose x = b′ and insert these representations into (3.3) to obtain

lβb′(λ) =θ(b′;λ) cotβ + θ′(b′;λ)

φ(b′;λ) cotβ + φ′(b′;λ)=

θ(b0;λ) + θ′(b0;λ)ψ(b′;λ) cotβ + ψ′(b′;λ)

ϕ(b′;λ) cotβ + ϕ′(b′;λ)

φ(b0;λ) + φ′(b0;λ)ψ(b′;λ) cotβ + ψ′(b′;λ)


.

For the classical singular problem on [b0,∞) with boundary condition y′(b0) = 0 theTitchmarsh-Weyl function is the Nevanlinna function

m0(λ) = limb′→∞

ψ(b′;λ) cotβ + ψ′(b′;λ)

ϕ(b′;λ) cotβ + ϕ′(b′;λ),

where the limit exists and is independent of β. Therefore also the limit

m(λ) : = limb′→∞

lβb′(λ)

= limb′→∞

θ(b0;λ) + θ′(b0;λ)ψ(b′;λ) cotβ + ψ′(b′;λ)


φ(b0;λ) + φ′(b0;λ)ψ(b′;λ) cotβ + ψ′(b′;λ)


=θ(b0;λ) + θ′(b0;λ)m0(λ)

φ(b0;λ) + φ′(b0;λ)m0(λ)

exists and is independent of β. The denominator in the last quotient can be written as

φ′(b0;λ)

(φ(b0;λ)

φ′(b0;λ)+m0(λ)

).

Since both summands in the brackets are nontrivial Nevanlinna functions the sum doesnot vanish on C \ R.

Theorem 3.5. If −1/4 ≤ q0 < 3/4 then the Titchmarsh-Weyl functions lβb (if b < ∞,0 ≤ β < π) or m = limb′→∞ lβb′ (if b = ∞) belong to the class N0, that is, they areNevanlinna functions; if q0 ≥ 3/4, then lβb (if b < ∞, 0 ≤ β < π) or m = limb′→∞ lβb′(if b = ∞) belong to some class Nκ, that is, they are generalized Nevanlinna functions.


Proof. For b <∞ the claims follow from Lemma 3.2. If b = ∞ we choose a finite b′ > 0and a boundary condition of the form (3.2) at b′ with some β′ ∈ [0, π). The corresponding

function lβ′

b′ belongs to some class Nκ where κ depends only on q0 and is independent of

b′ and β′, see Remark 3.3. Since the functions lβ′

b′ for b′ → ∞ converge locally uniformlyin (C \ R) ∩ hol (m) to m, it remains to apply Lemma 2.2. �

4. Fourier transformation and spectral function

In this section we treat the eigenfunction expansion theory for the equation (3.1) for bothb < ∞ and b = ∞ in all LP and LC cases at x = 0. In the LP case at x = 0 (that is ifq0 ≥ 3

4 ), we let A denote the self-adjoint operator associated with the left hand side of(2.1) in L2(0, b) and the regular boundary condition (3.2) at b if b <∞. In the LC case atx = 0 (that is if −1/4 ≤ q0 < 3/4), we let A denote the self-adjoint operator associatedwith the left hand side of (2.1) in L2(0, b), with the boundary condition (2.12) at x = 0,and the regular boundary condition (3.2) at b if b < ∞. The fundamental system φ, θis given by (2.10) in the LP case and by (2.11) in the LC case with boundary condition(2.12). For the Titchmarsh-Weyl function lβb or m from Theorem 3.5 we write here in allcases m.Let L2

0(0, b) denote the set of all functions f ∈ L2(0, b) which vanish identically near b.With

ψ(x;λ) := θ(x;λ) −m(λ)φ(x;λ) (4.1)

the resolvent of A, applied to an element f ∈ L20(0, b), is given by

((A− λ)−1f

)(x) = −ψ(x;λ)

∫ x

0

φ(ξ;λ)f(ξ)dξ − φ(x;λ)

∫ b

x

ψ(ξ;λ)f(ξ)dξ

= m(λ)φ(x;λ)

∫ b

0

φ(ξ;λ)f(ξ)dξ

−θ(x;λ)∫ x

0

φ(ξ;λ)f(ξ)dξ − φ(x;λ)

∫ b

x

θ(ξ;λ)f(ξ)dξ,

see [10, (4.18), (4.33), (5.8), (5.40)]. Taking the inner product with some g ∈ L20(0, b) and

collecting the terms which contain m we find

((A− λ)−1f, g

)= m(λ)

∫ b

0

φ(x;λ)f(x)dx

∫ b

0

φ(x;λ)g(x)dx+ · · · , (4.2)

where · · · denotes an entire function in λ.In the following the Fourier transformation F , associated with the solution φ(x;λ) of thedifferential equation (2.1), plays an important role:

Ff (λ) :=

∫ b

0

φ(x;λ)f(x)dx, λ ∈ C, f ∈ L20(0, b). (4.3)

Recall that here φ = φα given by (2.11) if x = 0 is in LC case, and φ = y1, the firstFrobenius solution, if x = 0 is in LP case.


Clearly, Ff (λ) is an entire function of λ. The Fourier transformation has the followingproperty:

FAf (λ) = λFf (λ) λ ∈ C, f ∈ L20(0, b) ∩ domA. (4.4)

To see this, for given f ∈ L20(0, b) ∩ domA choose b′ < b such that f vanishes in [b′, b).

We choose a function φ(x;λ) ∈ L20(0, b)∩domA which coincides on (0, b′] with φ(x;λ)

and vanishes in a neighbourhood of b. Then we have Aφ(x;λ) = λφ(x;λ), x ∈ (0, b′],and obtain

FAf (λ) =

∫ b

0

φ(x;λ)(Af)(x)dx =

∫ b

0

φ(x;λ)(Af)(x)dx

=

∫ b

0

(Aφ)(x;λ)f(x)dx = λ

∫ b

0

φ(x;λ)f(x)dx = λFf (λ).

The (operator valued) spectral family of the self-adjoint operatorA in L2(0, b) is denotedby E.

Lemma 4.1. The Titchmarsh-Weyl function m from Theorem 3.5 has no poles in C \ R

and no generalized poles of non-positive type in R.

Proof. Since in the LC case at x = 0 the function m belongs to the class N0 only the LPcase needs to be considered. We use (4.2) with f = g. Since the term on the left hand sidein (4.2) is holomorphic in C \ R and for each λ0 ∈ C the functions f, g in (4.2) can bechosen such that their Fourier transformations do not vanish in λ = λ0, the function mcannot have nonreal poles. If t0 is real, for the self-adjoint operatorA in (4.2) we have

limε↓0

(−iε)((A− t0 − iε)−1f, f

)=(E({t0})f, f

), (4.5)

and this number is always nonnegative. Now from relation (4.2) it follows that t0 cannotbe a pole or generalized pole of nonpositive type of m. �

Lemma 4.1 implies that, if m ∈ Nκ with κ ≥ 1, the function m from Theorem 3.5 has ageneralized pole of nonpositive type (of degree of nonpositivity κ) at ∞. Therefore (see[20, p. 211]) it admits a representation

m(z) = (1 + z2)n

∫ ∞

−∞

(1

t− z− t

1 + t2

)dσ(t) +

µ∑

j=0

αj zj (4.6)

with n,m ∈ N0, αj ∈ R, j = 0, 1, . . . ,m, αµ 6= 0 if µ > 0, and a measure σ on R such

that∫ ∞

−∞

dσ(t)

1 + t2< ∞. The representation (2.20) with n = 0 and the sum replaced by

α+ βz holds also for κ = 0, see (2.14).

Lemma 4.2. If f ∈ L20(0, b) and λ0 ∈ C such that

Ff (λ0) = 0, (4.7)

then there exists an element g ∈ domA such that (A− λ0)g = f and hence

Fg(λ) =Ff (λ)

λ− λ0. (4.8)


Proof. We show that the element

g(x) = −θ(x;λ0)

∫ x

0

φ(ξ;λ0)f(ξ)dξ − φ(x;λ0)

∫ b

x

θ(ξ;λ0)f(ξ)dξ, (4.9)

has the desired properties.Case A: x = 0 is LP, or x = 0 is LC with α = 0 in (2.11), (2.12).The integrals exist because of the behavior of φ(x;λ0) at x = 0 and since f vanishes nearb. The latter fact and assumption (4.7) imply that also g vanishes identically near b. Wenow show that in all cases the forms of the Frobenius solutions, see [10, Theorem 2.1] or(2.2), (2.3), (2.5), (2.6), and (2.7), (2.8) above, give rise to the result that

limx→0

g(x) = 0; (4.10)

hence, it follows that g ∈ L2(0, b) in all cases. To prove (4.10) we apply the Schwarzinequality to (4.9):

|g(x)| ≤ |θ(x;λ0)| ‖φ(·;λ0)‖[0,x]‖f‖[0,x] + |φ(x;λ0)| ‖θ(·;λ0)‖[x,b′]‖f‖[0,b], (4.11)

where the norms are L2-norms on the intervals given, and b′ = b if b < ∞ and b′ ischosen such that f vanishes on [b′,∞) if b = ∞. In the nonlog case we have

φ(x;λ0) = x12+ν(1 + O(x)

), θ(x;λ0) =

x12−ν

C

(1 + O(x)

), x ↓ 0,

from which it folows that for x ↓ 0

‖φ(·;λ0)‖[0,x] = O(xν+1

),

‖θ(·;λ0)‖[x,b] =

{O(x1−ν

)if ν > 1,

O (1) if 0 < ν < 1.

Putting these results in (4.11) gives, as x ↓ 0,

g(x) =

{O(x

32

)if ν > 1,

O(x

32

)+ O

(xν+ 1

2

)= O

(xν+ 1

2

)if 0 < ν < 1.

In the log cases with q0 =M2 − 1

4, M = 0, 1, . . . , we have

φ(·;λ0) = x1+M

2

(1 + O(x)

), x ↓ 0,

and

θ(·;λ0) =

x12 lnx

(1 + O(x)

)if M = 0,

−1 − q1 x lnx+ O(x) if M = 1,

− 1M x

1−M2

(1 + O(x)

)if M = 2, 3, . . . ,

from which it follows that

‖φ(·;λ0)‖[0,x] =

{O(x) if M = 0, 1,

O(x1+ M

2

)if M = 2, 3, . . . ,


and

‖θ(·;λ0)‖[x,b′] =

O(1) if M = 0, 1,

O(| ln x|) if M = 2,

O(x1− M

2

)if M = 3, 4, . . . .

From (4.11) we obtain again (4.10).Case B: x = 0 is LC with α ∈ (0, π) in (2.11), (2.12). In this case we do not needto consider each of the Frobenius solutions separately. The cases which arise are CaseI with ν ∈

(0, 1

2

)∪(

12 , 1), Case II A with ` = 0, and Case II B with M = 0. From

the assumption (4.7) and the fact that g satisfies Ag − λ0g = f , we have, using Green’sformula,

0 = Ff (λ0) =

∫ ∞

0

((Ag)(x)φ(x;λ0) − g(x)λ0φ(x;λ0)

)dx

=

∫ ∞

0

((Ag)(x)φ(x;λ0) − g(x)(Aφ(·;λ))(x)

)dx

= − limx→0

Wx

(φ(·;λ0), g(·)

).

By Remark 2.1, g satisfies the boundary condition (2.12), and hence g ∈ domA. Nowin both Case A and Case B the operator A can be applied to the function g and straight-forward calculation shows that Ag − λ0g = f . If we apply the Fourier transformation tothis relation and observe (4.4), we obtain λFg(λ)− λ0Fg(λ) = Ff (λ), and the lemma isproved. �

Corollary 4.3. For each bounded real interval ∆ there exists an f ∈L20(0, b) for which

the Fourier transformation Ff has no zeros on ∆.

Indeed, if Ff has a zero at λ0, according to (4.8) the order of the zero λ0 of the functionFg with the element g from Lemma 4.2 is reduced by one. Repeated application of thisargument yields the claim.We are now in position to establish the Parseval relation for f, g ∈ L2

0(0, b). First wedefine the spectral function which is associated with the operators arising from the lefthand side of (3.1).

Definition 4.4. For the equation (3.1), in all cases of the self-adjoint operator A as de-scribed in the first paragraph of this section, and with the Titchmarsh-Weyl functionm asin (4.6) we put

ρ(t) :=

∫ t

0

(1 + s2

)nd σ(s), t ∈ (−∞,∞). (4.12)

Clearly, if κ = 0, that is, if m is a Nevanlinna function, then n = 0 in (4.12). Thenonnegative measure defined on R by ρ is called the spectral measure, corresponding tothe Fourier transformation (4.3), of the operator A or of the corresponding eigenvalueproblem; the function ρ or any function on R which generates the same measure is calleda spectral function.

The main result of this section is the following theorem.


Theorem 4.5. Under the assumption (1.2), the following statements hold:

(1) The measure ρ in (4.12) on the real axis is of polynomial growth, that is∫ ∞

−∞

dρ(t)

(1 + t2)n0<∞ for some n0 ∈ N. (4.13)

(2) The Fourier transformation

f 7→ Ff : Ff (t) =

∫ b

0

φ(x; t)f(x)dx, t ∈ R, (4.14)

defined for all f ∈ L20(0, b), satisfies the Parseval relation

(f, g) =

∫

R

Ff (t)Fg(t) dρ(t), f, g ∈ L20(0, b); (4.15)

it can be extended by continuity to a unitary mapping from L2(0, b) onto L2ρ(R).

(3) The inverse mapping to F in (4.14) is given by

f(x) =

∫ ∞

−∞Ff (t)φ(x; t) dρ(t), x ∈ [0, b), (4.16)

where the improper integral converges in the norm of the space L2(0, b).(4) In the correspondence (4.14) and (4.16) between L2(0, b) and L2

ρ(R), the operatorA in L2(0, b) becomes the operator of multiplication by the independent variable inL2

ρ(R). For f ∈ domA, the expansion (4.16) is uniformly and absolutely convergenton compact subsets of (0, b) if x = 0 is LC and α ∈ (0, π) in (2.11), (2.12), anduniformly and absolutely convergent on compact subsets of [0, b) if x = 0 is LP orLC with α = 0 in (2.11), (2.12).

Remark 4.6. The mapping f 7→ Ff is a directing functional in the sense of M.G.Krein(see [19], [1, Appendix 2]) for the symmetric operator S which is the restriction of A todomA∩L2

0(0, b). An application of Krein’s theory on symmetric operators with directingfunctionals yields all the claims of Theorem 4.5 with the exception of formula (4.13). Withthe method of directing functionals, the spectral functions are obtained from the operatorspectral families of the self-adjoint extensions of the operator S. Here we did not use thisapproach, but derived the spectral function from the Titchmarsh-Weyl function instead.However, the proof of the surjectivity of the Fourier transformation f 7→ Ff given belowas well as the use of Lemma 4.2 follow Krein’s reasoning.

Proof of Theorem 4.5. (1) If x = 0 is LC, (4.13) follows from (4.12) with n0 = 1because of the representation (2.14) for m = F . If x = 0 is LP, (4.13) with n0 = nfollows from (4.12) and (4.6) by application of the Stieltjes-Livsic inversion formula.(2) First suppose that x = 0 is LP, in which case m has the representation (4.6). Let∆ = [µ1, µ2] be a real interval, µ1 < µ2, and denote by Γ∆ the rectangular contourintersecting the real axis in µ1 and µ2 with orientation such that the interval (µ1, µ2) isto the left. We integrate both sides of (4.2) along Γ∆. Suppose first that µ1 and µ2 belongto the set R of points of R which are points of continuity of the spectral family E and ofmeasure zero for σ in (4.6). For functions f, g ∈ L2

0(0, b), the left hand side of (4.2) gives


(E(∆)f, g

); in the right hand side of (4.2) we insert for m(λ) the integral representation

(4.6) and apply the Stieltjes-Livsic inversion formula to get∫

∆

Ff (t)Fg(t)(1 + t2

)ndσ(t),

hence

(E(∆)f, g) =

∫

∆

Ff (t)Fg(t)(1 + t2

)ndσ(t).

Since the set R is dense in R this relation extends to all bounded intervals ∆ and also toR, that is, we obtain

(f, g) =

∫ ∞

−∞Ff (t)Fg(t)

(1 + t2

)ndσ(t), f, g ∈ L2

0(0, b). (4.17)

Therefore the Fourier transformation is an isometric mapping from L20(0, b) into L2

ρ(R).The case when x = 0 is LC is established in the same way using instead the integral rep-resentation (2.14) for the Nevanlinna functionm and the corresponding inversion formula(2.17). This yields (4.17) with n = 0. SinceL2

0(0, b) is dense inL2(0, b) the Fourier trans-formation can be extended by continuity to all of L2(0, b) such that the Parseval relation(4.15) holds even for all f, g ∈ L2(0, b).It remains to show that the mapping f 7→ Ff from L2(0, b) to L2

ρ(R) is onto. We will dothis, following the lines of [19, Section 4]; this argument applies irrespective of whetherx = 0 is LP or LC. Denote by L0 the linear span of all Ff , f ∈ L2

0(0, b), and by Lits closure in L2

ρ(R). The Fourier transformation establishes an isomorphism betweenL2(0, b) and L, under which the self-adjoint operator A in L2(0, b) corresponds to theoperator Λ of multiplication by the independent variable in L, which is self-adjoint. Theoperator of multiplication by the independent variable in L2

ρ(R) is denoted by Λ. If E

denotes the spectral family of Λ, and E denotes the spectral family of Λ in L, then wehave for ϕ ∈ L2

ρ(R)

∫ λ

0

|ϕ(t)|2 dρ(t) =

∫ λ

0

d(E(t)ϕ, ϕ

), −∞ < λ <∞,

and for ϕ ∈ L∫ λ

0

|ϕ(t)|2 dρ(t) =

∫ λ

0

d(E(t)ϕ, ϕ

), −∞ < λ <∞, (4.18)

and hence E(t) = P E(t)∣∣L, −∞ < t <∞, where P denotes the orthogonal projection

of L2ρ(R) onto L. We claim that ϕ ∈ L0 for nonreal z implies

ϕ

· − z∈ L.

Denote by Rz, =z 6= 0, the resolvent of the operator Λ in L. Then

Rzϕ =

∫ ∞

−∞

dE(t)ϕ

t− z, ϕ ∈ L,


and (4.18) implies for ϕ, ψ ∈ L

(Rzϕ, ψ) =

∫ ∞

−∞

d(E(t)ϕ, ψ

)

t− z=

∫ ∞

−∞

ϕ(t)ψ(t)

t− zdρ(t),

(Rzϕ,Rzϕ) =1

z − z

((Rzϕ, ϕ) − (Rzϕ, ϕ)

)=

∫ ∞

−∞

|ϕ(t)|2|t− z|2 dρ(t),

and hence

‖Rzϕ− ψ‖2 = (Rzϕ− ψ,Rzϕ− ψ) =

∫ ∞

−∞

∣∣∣∣ϕ(t)

t− z− ψ(t)

∣∣∣∣2

dρ(t).

Since Rzϕ ∈ L, for arbitrary ε > 0 there exists ψ ∈ L0 such that ‖Rzϕ−ψ‖ < ε, hence∫ ∞

−∞

∣∣∣∣ϕ(t)

t− z− ψ(t)

∣∣∣∣2

dρ(t) < ε2,

andϕ

· − z∈ L, which proves the claim.

Now, if h ∈ L2ρ(R) is orthogonal to L, then for arbitrary ϕ ∈ L0 we have

∫ ∞

−∞

ϕ(t)h(t)

t− zdρ(t) = 0, =z 6= 0.

The inversion formula (2.17) yields for arbitrary λ′ < λ′′

∫ λ′′

λ′ϕ(t)h(t) dρ(t) = 0,

hence ∫ λ′′

λ′

∣∣ϕ(t)h(t)∣∣2 dρ(t) = 0.

Since, according to Corollary 4.3, ϕ = Ff can be chosen such that it does not vanish onan arbitrarily given bounded interval it follows that

∫ ∞

−∞|h(t)|2 dρ(t) = 0,

and therefore L = L2ρ(R).

(3) According to part (2) of this proof, in the space L2ρ(R) the operator Λ is the operator

of multiplication by the independent variable t. Therefore the spectral projection E(∆)

of Λ for an arbitrary bounded interval ∆ of R is the multiplication by the characteristicfunction χ∆, that is,

E(∆)Ff = χ∆ Ff .

Hence (E(∆)f, g

)=

∫

∆

Ff (t)Fg(t) dρ(t), f, g ∈ L2(0, b),

and (E(∆)f

)(x) =

∫

∆

Ff (t)φ(x; t) dρ(t).


It follows that

f(x) = lim∆↑R

(E(∆)f

)(x) = lim

∆↑R

∫

∆

Ff (t)φ(x; t) dρ(t),

where the limit is to be understood in the norm of L2(0, b).(4) The first claim was proved in (1). For the absolute and uniform convergence on com-pact subsets of (0, b), the cases when x = 0 is LC are covered by [10, Theorem 4.3], andthe cases when x = 0 is LP by [10, Theorem 5.4]. �

The eigenvalues of the operatorA, introduced at the beginning of this section in any of theLC and LP cases, are the points λ0 ∈ R with ρ0 := ρ({λ0}) > 0 and with correspondingeigenfunction φ(·;λ0). The relations (4.2), (4.5) and the definition of ρ imply that

ρ0 =1

∫ b

0

|φ(x;λ0)|2 dx.

It follows that if the spectrum in the closed interval [c, d] is discrete then

ρ([c, d]) =∑

λj∈[c,d]∩σ(A)

1∫ b

0

|φ(x;λj)|2 dx. (4.19)

In general, the spectral measure ρ in (4.12) can be obtained directly from the Titchmarsh-Weyl function m by the Stieltjes inversion formula, which is sometimes called the Titch-marsh-Kodaira formula.

Theorem 4.7. In any of the LP or LC problems described at the beginning of this section,whenever c, d are points of ρ-measure zero the spectral measure ρ in (4.12) is given bythe formula

ρ([c, d]) = − 1

2πi

∮ ′

Γ[c,d]

m(z) dz = limε↓0

1

π

∫ d

c

=m(t+ iε) dt. (4.20)

Proof. For the LC cases (n = 0 in (4.12)) (4.20) reduces to the known result (2.17). Forthe LP cases (n ≥ 1), apply the Stieltjes-Livsic inversion formula (2.19) to the represen-tation (4.6), with ϕ(z) = (1 + z2)n. �

Similarly to the classical LP problem on [0,∞) with x = 0 a regular endpoint, the spectralmeasure ρ for the operatorsA on (0,∞) in Theorem 4.5 can be obtained as the limit of thespectral measures ρb′ of the corresponding problems on (0, b′], 0 < b′ < ∞, by lettingb′ → ∞. For the cases when x = 0 is LC this was proved in [10, Theorem 4.1(v-vi)]. Forthe cases when x = 0 is LP we have the following analogous result.

Theorem 4.8. If x = 0 is LP and b = ∞, the spectral measure ρ of the problem (3.1) isthe weak limit of the spectral measures ρb′ of problem (3.1), (3.2) on [0, b′], 0 < b′ <∞,with any regular boundary condition (3.2) at b′, that is,

ρ(∆) = limb′→∞

ρb′(∆) = limb′→∞

∑

λj∈∆T

σ(Ab′ )

1∫ b′

0

|φ(x;λj )|2dx(4.21)


for all intervals ∆ with endpoints of ρ-measure zero, that is, with endpoints not in σp(A).

Proof. Denote by Ab′ the self-adjoint operator in L2(0, b′), corresponding to the (0, b′]problem as described in Theorem 4.5. The relations (4.2) and (3.8) imply for all f ∈L2

0(0,∞) and nonreal λ((Ab′ − λ)−1f |(0,b′], f |(0,b′]

)→((A− λ)−1f, f

), b′ → ∞.

Setting Ab′ := Ab′ ⊕ 0 in L2(0,∞) = L2(0, b′) ⊕ L2(b′,∞), it follows that((Ab′ − λ

)−1f, f

)−→

((A− λ)−1f, f

), b′ → ∞.

This relation implies

∥∥(Ab′ − λ)−1

f∥∥2

=

((Ab′ − λ

)−1f, f

)−((Ab′ − λ

)−1f, f

)

λ− λ

−→((A− λ)−1f, f

)−((A− λ)−1f, f

)

λ− λ

=∥∥(A− λ)−1f

∥∥2, b′ → ∞.

Since L20(0,∞) is dense in L2(0,∞) and for fixed nonreal λ the resolvents (Ab′ − λ)−1

are uniformly bounded: ‖(Ab′ − λ)−1‖ ≤ 1

=λ , these resolvents converge strongly to the

resolvent ofA. Therefore, according to a theorem of F. Rellich ([17, Theorem VIII.1.15]),for the spectral families Eb′ andE of the operators Ab′ andA, respectively, it follows thatEb′(∆) → E(∆), strongly, if b′ → ∞ for all intervals ∆ such that the endpoints of ∆

are not eigenvalues of A. Hence(Eb′(∆)f, f

)→(E(∆)f, f

), b → ∞, f ∈ L2

0(0,∞).According to Theorem 4.5, (3), this yields

limb′→∞

∫

∆

|Ff (t)|2 dρb′(t) =

∫

∆

|Ff (t)|2 dρ(t).

Since this relation holds also for any subinterval of ∆ with endpoints not being eigenval-ues of A, and since f can be chosen such that Ff does not vanish on ∆ (see Corollary4.3), we obtain

limb′→∞

ρb′(∆) = ρ(∆),

for all intervals ∆ with endpoints not being eigenvalues of A. The last equality in (4.21)follows by utilizing (4.19). �

Remark 4.9. Of particular interest in the approach utilized in this section for definingthe spectral functions associated with the self-adjoint operators in Theorem 4.5 is thefact that reliance on Helly’s theorems, common in treatments of the spectral theory forthe second order Sturm-Liouville equation, and also employed in [10] for the class ofproblems treated here, is completely avoided. Instead, the Parseval relation and meansquare convergence results for the eigenfunction expansions are obtained via the Stieltjes-Livsic inversion formula. In Theorem 4.8, when both endpoints x = 0 and x = ∞ are LP,the equivalence of the measures, obtained by the two methods and given by the formulas(4.20) and (4.21) is shown (comp. also [30, Theorems 14.12 and 14.13]). It is noteworthy


that in the development of the SLEDGE software package [12, Example 3] the Besselequation on (0,∞) with the two LP endpoints was used as a test problem, and that thetheoretical formulas for ρ(λ) and f(λ) = ρ′(λ) (see (5.15) and (5.19), (5.26) below) wereobtained using the Titchmarsh-Kodaira formula (4.20), while the numerical computationsfor these test problems followed the Levitan-Levinson formula (4.21).

5. Examples

In this section, under the asumptions (1.2), we give explicit formulas for the spectralfunction of the problem (1.1) on (0,∞) and with the Friedrichs boundary condition atx = 0 in the LC case (which corresponds to the choice α = 0 in the bounday condition(2.12)). Thus, in all the LP or LC cases, the corresponding Fourier transformation (4.3)is with respect to the first Frobenius solution φ(x;λ) = y1(x;λ). According to the abovewe have to find the Titchmarsh-Weyl function m in (4.1), that is, we have to find thesolutionψ(x;λ) of (1.1), which is square integrable at ∞, and the two Frobenius solutionsy1(x;λ), y2(x;λ) or φ(x;λ), θ(x;λ), and to express ψ(x;λ) as a linear combination ofthe latter as in (4.1); such relations of linear dependence are called connection formulasin the literature on special functions. The key to this detailed analysis is the identification,in each case, of the solutions φ(x;λ) and θ(x;λ) and of the square integrable solutionψ(x;λ) of (1.1) in terms of Whittaker or Bessel functions. For convenience, we considerthe Cases I, IIA, and II B of Section 2 separately.

Example 1 (Case I). Consider for ν ∈ (0,∞), ν 6= N

2for N = 1, 2, . . . , and a ∈ R the

equation

−y′′(x) +

[−ax

+ν2 − 1

4

x2

]y(x) = λy(x), 0 < x <∞, (5.1)

where for ν < 1 the boundary condition (2.12) with α = 0 is imposed at x = 0. Theequation (5.1) converts under the change of variables

t = −2ix√λ, W(t) = y(x),

to the Whittaker equation

W ′′(t) +

(−1

4+

β

t+

14 − ν2

t2

)W(t) = 0, β := β(λ) :=

ia

2√λ, (5.2)

and under the further change of variable

W(t) = tν+ 12 e−

12 tu(t)

to the confluent hypergeometric equation

tu′′(t) + (1 + 2ν − t)u′(t) −(ν +

1

2− β

)u(t) = 0. (5.3)

The latter equation has a two-term recurrence relation in the Frobenius theory at t = 0with indicial roots r1 = 0, r2 = −2ν. The Frobenius solutions of (5.3) are therefore


available in closed form, and are readily found to be

u1(t; β) := M

(ν +

1

2− β(λ), 1 + 2ν, t

)=

∞∑

n=0

(ν + 1

2 − β(λ))

n

n! (1 + 2ν)ntn

and

u2(t; β) := t−2ν M

(−ν +

1

2− β(λ), 1 − 2ν, t

)=

∞∑

n=0

(−ν + 1

2 − β(λ))

n

n! (1 − 2ν)ntn−2ν ,

whereM(a, b, t) is the confluent hypergeometric function of first kind. The correspondingfundamental system of solutions of (5.2) is given by Mβ,ν(t) and Mβ,−ν(t), where

Mβ,ν(t) := tν+ 12 e−

12 tM

(ν +

1

2− β, 2ν + 1, t

)

is the Whittaker function of first kind ([28, (1.6.4)]). When we change back from t to thevariable x these Whittaker functions fail to be entire in λ; however, we can express theFrobenius solutions (2.2), (2.3) of (5.1) in terms of them as

y1(x;λ) = xν+ 12

[1 +

∑∞n=1 an(λ)xn

]

= xν+ 12 eix

√λM

(ν + 1

2 − β(λ), 1 + 2ν,−2ix√λ)

=1

(−2i

√λ)ν+ 1

2

Mβ,ν

(−2ix

√λ),

(5.4)

and

y2(x;λ) = x−ν+ 12

[1 +

∑∞n=1 bn(λ)xn

]

= x−ν+ 12 eix

√λ M

(− ν + 1

2 − β(λ), 1 − 2ν,−2ix√λ)

=1

(−2i

√λ) 1

2−νMβ,−ν

(−2ix

√λ),

(5.5)

where the polynomials an(λ) and bn(λ) of degree[

n2

]arise on performing the Cauchy

products of the two series in the second lines. From (2.4) or [10, Theorem 2.1, Case IA]we have

Wx

(y1(x;λ), y2(x;λ)

)= −2ν for all λ ∈ C,

so for (5.1) with α = 0 in the boundary condition (2.12) we need to take

φ(x;λ) = y1(x;λ), θ(x;λ) = − 1

2νy2(x;λ) (5.6)

as the fundamental system {φ(·;λ), θ(·;λ)} of (5.1) which satisfies all the properties of[10, Corollary 2.4].


For all ν ∈ (0,∞), ν 6= N

2, N = 1, 2, . . . , the Whittaker function of second kind is (see

[28, (1.7.1)])

Wβ,ν

(− 2ix

√λ): =

π

sin(2πν) Γ(1+2ν) Γ(−ν+ 12−β(λ))

Mβ,ν

(− 2ix

√λ)

+π

sin(2πν)Γ(1−2ν)Γ(ν+ 12−β(λ))

Mβ,−ν

(− 2ix

√λ).

(5.7)

Taking the square root of λ to be defined with branch cut on the positive real λ-axis, i.e.

λ12 = |λ| 12 ei arg λ

2 , 0 ≤ argλ < 2π,

it follows from the asymptotic expansion of Wβ,ν(t) (see [28, p. 61]), that for all ν ∈(0,∞), ν 6= N

2 , N = 1, 2, . . . , and 0 < argλ < 2π we have∣∣∣Wβ,ν

(−2ix

√λ)∣∣∣ 5 D exp

(−(sin

argλ

2

)[x|λ| 12 − a

2|λ| 12ln(2x|λ| 12 )

])(1+O

(1

x

))

for some D > 0, as x → ∞. Since sin arg λ2 > 0 for argλ ∈ (0, 2π), it follows that

Wβ,ν

(− 2ix

√λ)∈ L2(x0,∞) for all λ ∈ C \ [0,∞).

With the Frobenius solutions y1, y2 in (5.4) and (5.5), which are entire in λ, (5.7) can bewritten as

Wβ,ν

(− 2ix

√λ)

= α1(λ) y1(x;λ) + α2(λ) y2(x;λ), (5.8)

where the coefficients α1, α2 are

α1(λ) := −(− 2i

√λ) 1

2+ν π

sin(2πν) Γ(1 + 2ν) Γ(− ν + 1

2 − β(λ)) , (5.9)

andα2(λ) :=

(− 2i

√λ) 1

2−ν π

sin(2πν) Γ(1 − 2ν) Γ(ν + 1

2 − β(λ)) , (5.10)

with branch cut on the positive real λ-axis.If in (5.8) the functions y1, y2 are replaced by φ, θ according to (5.6), we obtain theconnection formula

θ(x;λ) − α1(λ)

2ν α2(λ)φ(x;λ) = − 1

2ν α2(λ)Wβ,ν

(− 2ix

√λ), =λ 6= 0,

and for the Titchmarsh-Weyl functionmν it follows that

mν(λ) =α1(λ)

2ν α2(λ)

= −(− 2i

√λ)2ν

2ν· Γ(1 − 2ν) Γ

(12 +ν−β(λ)

)

Γ(1 + 2ν) Γ(

12−ν−β(λ)

)

= − Γ(1−2ν)

2ν Γ(1+ 2ν)

Γ(

12 +ν−β(λ)

)

Γ(

12−ν−β(λ)

) 22ν [cos(νπ)−i sin(νπ)]λν

= − π 22νe−iνπ

Γ2(1+ 2ν) sin(2νπ)

Γ(

12 +ν−β(λ)

)

Γ(

12−ν−β(λ)

) λν .

(5.11)


This holds for all ν > 0, ν 6= N2 , N = 1, 2, . . . , and all real a; the branch cut of λν and

λ1/2, arising in β(λ), is taken on the positive real λ-axis, that is, we define

λν = |λ|ν eiν·argλ, 0 ≤ argλ < 2π. (5.12)

It follows that

arg(λ)

= 2π − argλ for all λ ∈ C, λ 6= 0,(λ)ν

= e2πiν(λν),(λ)1/2

= −(λ1/2

),

β(λ) =ia

2(λ)1/2

= β(λ),

and these formulas can be utilized in (5.11) (4th and 5th lines) to verify that mν(λ) =

mν(λ), λ ∈ C \ R, in accordance with [10, Theorem 3.2 (v)].In the particular case a = 0 the solutions y1(x;λ) and y2(x;λ) reduce to Bessel functions.Indeed, using the formulas (see [28, (1.8.13),(1.8.15)])

M0,ν

(− 2ix

√λ)

= Γ(1 + ν) e−πi2 ( 1

2+ν) 22ν+ 12λ

14√x Jν

(x√λ)

and

W0,ν

(− 2ix

√λ)

=

√π

2e

πi2 ( 1

2+ν)λ14√xH(1)

ν

(x√λ),

the Frobenius solutions y1, y2 in (5.4), (5.5) become

y1(x;λ) = 2νΓ(ν + 1)λ−ν2 x

12Jν

(x√λ)

= x12+ν

(1 +

∞∑

j=1

(−1)jλj

j! (ν + 1)j 22jx2j

)(5.13)

and

y2(x;λ)= 2−νΓ(1 − ν)λν2 x

12J−ν

(x√λ)= x

12−ν

(1 +

∞∑

j=1

(−1)jλj

j! (1−ν)j 22jx2j

), (5.14)

which have the required form (2.2), (2.3). The connection formula (5.8) becomes√π

2e

iπ2 (ν+ 1

2 )λ14√x[Jν

(x√λ)+ iYν

(x√λ)]

=α1(λ) y1(x;λ)+α2(λ) y2(x;λ),

where y1, y2 are the Bessel functions from (5.13) and (5.14), andα1, α2 follow from (5.9),(5.10) for a = 0:

α1(λ) = −(− 2i

√λ)ν+ 1

2π

sin(2πν)Γ(1 + 2ν)Γ(−ν + 1

2

) ,

α2(λ) =(− 2i

√λ)−ν+ 1

2π

sin(2πν)Γ(1 − 2ν)Γ(ν + 1

2

) .

Using the relations

Γ(x)Γ(1 − x) =−π

sin(π x), Γ

(x+ 1

2

)Γ(x

2

)=

√π

2x−1Γ(x),


from the third line of (5.11) we find for the Titchmarsh-Weyl function mν easily

mν(λ) = − π

22ν+1 sin(νπ)Γ2(ν + 1)e−iνπ λν , (5.15)

where the branch cut is again taken on the positive λ-axis. This formula was also obtainedin [10, (6.5)].

The following lemma enables the general case ν ∈ (0,∞), ν 6= N

2for N = 1, 2, . . . ,

to be reduced to the case ν ∈(0, 1

2

)∪(

12 , 1)

where we have LC case at zero and thecorrespondingm-function is a Nevanlinna function. It makes use of the representation ofmν(λ) in the last line of (5.11).

Lemma 5.1. If ν ∈ (0,∞), ν 6= N

2for N = 1, 2, . . . , is written as

ν = M + ν′, M := [ν], ν′ ∈(

0,1

2

)∪(

1

2, 1

),

thenmν(λ) = kν′,M (λ)mν′ (λ) (5.16)

with kν′,0(λ) = 1 and

kν′,M (λ) =1

[(1 + 2ν′)2M ]2

M∏

j=1

(4λ(ν′ − 1

2+ j)2

+ a2

), M = 1, 2, . . . . (5.17)

Proof. Using the Pochhammer symbol

(α)M = α(α+ 1) · · · (α+M − 1) =Γ(α+M)

Γ(α),

we findΓ(M+ν′+ 1

2−β)

Γ(−M−ν′+ 1

2−β) =

Γ(ν′ + 1

2 − β)

Γ(−ν′ + 1

2 − β)(−M−ν′+ 1

2−β

)

M

(ν′ +

1

2− β

)

M

=Γ(ν′ + 1

2 − β)

Γ(−ν′ + 1

2 − β) (−1)M

M∏

j=1

(j − 1

2+ ν′ + β

)(j − 1

2+ ν′ − β

)

=Γ(ν′ + 1

2 − β(λ))

Γ(−ν′ + 1

2 − β(λ)) (−1)M

M∏

j=1

[(j − 1

2+ ν′

)2

+a2

4λ

].

Now we use the fourth line of (5.11):

mν(λ) = − π22ν e−iνπ

Γ2(1+ 2ν) sin(2νπ)

Γ(ν + 1

2 − β(λ))

Γ(−ν + 1

2 − β(λ)) λν

= − π22M22ν′e−i(M+ν′)π

Γ2(1+ 2ν′ + 2M) sin(2ν′π)

Γ(ν′ +M + 1

2 − β(λ))

Γ(−ν′ −M + 1

2 − β(λ)) λν′

λM

= mν′(λ)1

[(1 + 2ν′)2M ]2

M∏

j=1

(4λ(ν′ − 1

2+ j)2

+ a2

). �


Remark 5.2. In the case a = 0, that is β = 0, in the relation (5.16) we have

mν′(λ) = − π

22ν′+1 sin(ν′π)Γ2(ν′ + 1)e−iν′π λν′

(5.18)

and

kν′,M (λ) =λM

22M [(1 + ν′)M ]2 . (5.19)

Next we suppose ν ∈(0, 1

2

)∪(

12 , 1). Recall that in this case x = 0 is LC and it follows

from [10, Theorem 3.2 (viii), Case A] thatmν ∈ N0 for all real a. In the following lemmawe find the integral representation (2.14) of mν .

Lemma 5.3. For ν ∈(0, 1

2

)∪(

12 , 1), the Titchmarsh-Weyl functions in (5.11) and (5.15)

are of class N0 and admit the following representations (2.14):

a > 0 :

mν(λ) = αν +

∞∑

n=1

rν,n

λν,n − λ+

∫ ∞

0

(1

t− λ− t

1 + t2

)fν(t) dt (5.20)

with, for n = 1, 2, . . . ,

λν,n =− a2

4(ν− 1

2 +n)2 , (5.21)

rν,n =π a2ν+2

2(n−1)!∣∣Γ(1−2ν−n) sin(2νπ)

∣∣Γ2(1+2ν)(ν− 1

2 +n)2ν+3 , (5.22)

and

fν(t) =π 22ν−1 e

aπ

2√

t

Γ2(1+2ν)∣∣Γ(−ν+ 1

2−β(t))∣∣2{cos2(πν) + sinh2

(aπ2√

t

)} tν , t ≥ 0. (5.23)

a < 0 :

mν(λ) = αν +

∫ ∞

0

(1

t− λ− t

1 + t2

)fν(t) dt (5.24)

with fν(t) given by (5.23).

a = 0 :

mν(λ) = αν +

∫ ∞

0

(1

t− λ− t

1 + t2

)fν(t) dt (5.25)

with

fν(t) :=tν

22ν+1 Γ2(ν + 1), t ≥ 0. (5.26)

Proof. It follows from (5.11) that

β = limy↑∞

mν(iy)

iy= lim

y↑∞

(iy)ν

iy= lim

y↑∞

|y|ν eπ2 iν

iy= 0,


since ν < 1. If t > 0, we put λ = t + iε in the last line of (5.11), and make use of thecontinuity of the Gamma functions as ε ↓ 0 to compute the limit of mν(t + iε) as ε ↓ 0(branch cut of λν and λ1/2 on the positive λ-axis as in (5.12)):

mν(t) = limε↓0 mν(t+ iε)

= − π22νe−iνπ

Γ2(2ν+1) sin(2νπ)

Γ(ν+ 1

2−β(t))Γ(−ν+ 1

2 +β(t))

∣∣Γ(−ν+ 1

2−β(t))∣∣2 tν

= − π2 22νe−iνπ cos[π(ν+β(t))]

Γ2(2ν+1) sin(2νπ)∣∣Γ(−ν+ 1

2−β(t))∣∣2 ∣∣ cos[π(ν−β(t))]

∣∣2 tν ,

where

β(t) = limε↓0

β(t+ iε) = limε↓0

ia

2√t+ iε

=ia

2√t,

and hence β(t) = −β(t) for t ∈ (0,∞)). It follows that

fν(t) = limε↓01

π=mν(t+ iε)

=1

π= [ limε↓0mν(t+ iε) ]

=π 22ν [− sin(πν) cos(πν)] e

aπ

2√

t

Γ2(2ν+1) sin(2νπ)∣∣Γ(−ν+ 1

2−β(t))∣∣2 ∣∣ cos[π(−ν+β(t))]

∣∣2 tν

=π 22ν−1 e

aπ

2√

t

Γ2(2ν + 1)∣∣Γ(−ν + 1

2 − β(t))∣∣2{cos2(πν) + sinh2

(aπ2√

t

)} tν .

(5.27)

This proves (5.23) for the cases a > 0 and a < 0. Putting a = 0 in (5.27) or using theTitchmarsh-Kodaira formula (2.17) for the m-function (5.18) will prove (5.26).For a > 0 we now consider t < 0. It follows from the last line in (5.11) that mν has polesat the points where Γ

(ν + 1

2 − β(λ))

has poles, and is continuous and real between thesepoles. Therefore the only contribution to σ in (2.14) for t < 0 is at these poles λν,n andequals −rν,n, the negative residue of mν at λν,n. To find the λν,n we have to solve

ν +1

2− β(λν,n) = ν +

1

2− ia

2√λν,n

= −(n− 1), n = 1, 2, . . . ,

which yields

λν,n = − a2

4(ν + 1

2 + (n− 1))2 , n = 1, 2, . . . .

For the case a ≤ 0 the above equation has no solution, so Γ(ν + 1

2 − β(t))

has no polesfor t < 0. To calculate the residue of mν at the poles for a > 0 we make use of

Res Γ(z)∣∣z=−(n−1)

=(−1)n−1

(n− 1)!, n = 1, 2, . . . .


Employing L’Hospital’s rule we find (writing λn instead of λν,n)

Res Γ

(ν+

1

2− β(λ)

) ∣∣∣∣∣λ=λn

= limλ→λn

(−1)n−1

(n− 1)!

(λ−λn

ν + 12 − β(λ) + (n− 1)

)

= limλ→λn

(−1)n−1

(n− 1)!

(4

ia λ−32

)

=(−1)n−1

(n− 1)!

(− a2

2(ν − 1

2 + n)3

).

Evaluating the remaining factors of mν in the last line of (5.11) at λ = λn gives

− π 22ν

Γ2(2ν+1)

1

Γ

(12−ν−

ia

2√λn

) e−iνπ

sin(2νπ)λν

=−π a2ν

Γ2(2ν+1)Γ(−2ν−(n−1)

)sin(2νπ)

(ν− 1

2 +n)2ν ,

so it follows that

rν,n = −Res∣∣λ=λn

mν(λ)

= − (−1)n−1

2(n− 1)!

πa2ν+2

Γ(−2ν−(n− 1)

)sin(2νπ) Γ2(1+2ν)

(ν − 1

2 + n)2ν+3

=πa2ν+2

2(n− 1)!∣∣Γ(−2ν−(n− 1)

)sin(2νπ)

∣∣ Γ2(1 + 2ν)(ν − 1

2 + n)2ν+3 .

Here we have used that Γ(−2ν−(n − 1)) sin(2νπ) has sign (−1)n for all n = 1, 2, . . . ,and for all ν ∈

(0, 1

2

)∪(

12 , 1). �

Remark 5.4. According to Lemma 5.3, in the case a > 0 the spectral function σν of mν

in the representation (2.16) is given by

σν(t) =

∑

n:λν,n≤t

rν,n, t < 0,

∞∑

n=1

rν,n +

∫ t

0

fν(u) du, t ≥ 0,(5.28)

with λν,n, rν,n, fν as in (5.21), (5.22), (5.23). For a < 0 and a = 0 the spectral measureσν is supported on the nonnegative half axis and absolutely continuous with density fν

given by (5.23) and (5.26), respectively.

Remark 5.5. The constants αν in (5.20) and (5.24) can be found by using the asymptoticsofmν(λ) with λ = −u2, u→ ∞, obtained from the last line of (5.11) and from the righthand sides of (5.20) and (5.24), and then matching them. To this end it becomes necessary


to know the asymptotic behavior of fν(t) in (5.23) as t→ ∞. It follows from (5.23) that

fν(t) =tν

22ν+1Γ2(ν + 1)

{1 +

aπ

2√t

+c

t+ O

(1

t3/2

)}, t→ ∞,

where

c :=a2

4

[π2 − sec2(πν) −

[Γ′( 1

2 − ν)]2 − Γ( 1

2 − ν)Γ′′( 12 − ν)

Γ2( 12 − ν)

].

Making use of this asymptotic for fν(t) in the integrals on the right hand side of (5.20)and (5.24), a lengthy calculation (which we omit since these constants will not play a rolein the sequel) gives for ν ∈

(0, 1

2

)

αν = − π cos(

νπ2

)

22ν+1Γ2(ν+1) sin(νπ)−∫ ∞

0

t

1+t2

[tν

22ν+1Γ2(ν+1)− fν(t)

]dt,

and for ν ∈(

12 , 1)

αν = − π cos(

νπ2

)

22ν+1Γ2(ν+1) sin(νπ)−∫ ∞

0

t

1 + t2

[tν

22ν+1Γ2(ν+1)

(1+

aπ

2√t

)−fν(t)

]dt

− aπ2[sin(

νπ2

)+ cos

(νπ2

)]

22ν+2Γ2(ν+1)√

2 cos(νπ).

For the case a = 0 the αν in (5.25) is obtained by putting a = 0 in the above formulaswhich gives

αν = − π cos(

νπ2

)

22ν+1Γ2(ν + 1) sin(νπ).

Now we are ready to describe the spectral function ρ of the problem (5.1) for ν > 0,ν 6= N

2 , N = 1, 2, . . . , and with the Friedrichs boundary condition in the LC case.

Theorem 5.6. If we set

ν = M + ν′ with M = [ν], ν′ ∈(

0,1

2

)∪(

1

2, 1

), (5.29)

then the spectral function ρν of the problem (5.1), for the boundary condition (2.12) withα = 0 in the LC case M = 0, corresponding to the Fourier transformation (4.14), is asfollows:

a > 0 :

ρν(t) =

∑

λν′,M+1≤λν′,n≤t

kν′,M (λν′,n) rν,n, t < 0,

∞∑

n=M+1

kν′,M (λν′,n) rν′ ,n +

∫ t

0

kν′,M (u)fν′(u) du, t ≥ 0;(5.30)


a < 0 :

ρν(t) =

0, t < 0,∫ t

0

kν′,M (u) fν′(u) du, t ≥ 0;

a = 0 :

ρν(t) =

0, t < 0,∫ t

0

kν′,M (u)uν′

22ν′+1Γ2(ν′ + 1)du, t ≥ 0;

here kν′,M , λν′,n, rν′,n, and fν′ are given by (5.17), (5.21), (5.22), and (5.23), respec-tively, for a 6= 0, and by (5.19), (5.26) for a = 0.

Proof. First we observe that for a > 0 the function kν′,M in (5.17) has simple zeros atthe points λν′,1, λν′,2, . . . , λν′,M , therefore according to (5.16) the function mν does nothave poles at these points. Now making use of the Titchmarsh-Kodaira formula (4.20), therepresentation (5.16) for mν , and the Stieltjes-Livsic inversion formula (2.19) we have,since kν′,M (z) is real on the real z-axis, that

ρν([c, d]) = limε↓0

1

π

∫ d

c

=mν(t+ iε) dt

= − 1

2πi

∮ ′

Γ[c,d]

mν(z) dz

= − 1

2πi

∮ ′

Γ[c,d]

kν′,M (z)mν′(z) dz

=

∫ d

c

kν′,M (t) d σν′(t),

whenever c, d are points with no concentrated σν′ -mass, where σν′ is given in (5.28). Theproof for a ≤ 0 is similar. �

The expansion formula or inverse Fourier transformation (4.16) for a > 0 takes the fol-lowing form:

f(x) =

∑∞n=M+1 Ff (λν,n)φ(x;λν,n) kν′,M (λν′,n) rν′,n

+

∫ ∞

0

Ff (λ)φ(x;λ) kν′ ,M (λ)fν′(λ) dλ if a > 0,

∫ ∞

o

Ff (λ)φ(x;λ)kν′ ,Mfν′(λ)dλ if a < 0,

∫ ∞

0

Ff (λ)φ(x;λ)λν

22ν+1Γ2(ν + 1)dλ if a = 0.


Here Ff is the Fourier transformation (4.3), where φ(x;λ) is given by (5.4) for a 6= 0 and(5.13) for a = 0.

Remark 5.7. It will be observed that the decomposition (5.16) ofmν enables the residuesof mν at its poles λν′,n, n ≥ M + 1, to be calculated in terms of the residues of mν′ in(5.22): namely, for ν = M + ν ′, M ≥ 1, we have for all n ≥M + 1

rν,n := Res∣∣λ=λν′,n

mν(λ) = Res∣∣λ=λν′,n

kν′,M (λ)mν′ (λ) = kν′,M (λν′,n)rν′,n,

where rν′ ,n is given in (5.22).

In the following theorem we give the representation (2.20) of the m-function mν . It isobtained from Lemma 5.1 and Lemma 5.3, and for a = 0 from (5.18), (5.19).

Theorem 5.8. The function mν in (5.16) with ν = M + ν ′, M = [ν], as in (5.29) is ofclass Nκ with κ :=

[ν+12

]and admits the following representation:

a > 0 :

mν(λ) =∞∑

n=M+1

kν′,M (λν′ ,n)rν′,n

λν′,n − λ

+ (1 + λ2)κ

∫ ∞

0

(1

t−λ−t

1+ t2

)kν′,M (t)fν′(t)

(1+ t2)κdt+ qM (λ),

where qM (λ) is a polynomial of degree M and kν′,M , λν′,M , rν′,M , and fν′ are givenby (5.17), (5.21),(5.22), (5.23), respectively.

a < 0 :

mν(λ) =(1 + λ2

)κ∫ ∞

0

(1

t− λ− t

1 + t2

)kν′,M (t) fν′(t)

(1 + t2)κdt+ qM (λ),

where qM (λ) is a polynomial of degree M and kν′,M and fν′ are given by (5.17), (5.23),respectively.

a = 0 :

mν(λ)=(1+λ2

)κ∫ ∞

0

(1

t−λ − t

1+ t2

)kν′,M (t) tν

′

22ν′+1Γ2(ν′+1)(1+ t2)κdt+ qM (λ),

where qM (λ) is a polynomial of degree M and kν′,M is given by (5.19).

Proof. We consider the case a > 0. Making use of the relation (5.16) and the representa-tion (5.20) of mν′(λ) for ν′ ∈

(0, 1

2

)∪(

12 , 1)

with σν′ from (5.28) we find

mν(λ) = kν′,M (λ)

[αν′ +

∞∑

n=1

rν′,n

λν′,n−λ+

∫ ∞

0

(1

t−λ − t

1+t2

)fν′(t) dt

], (5.31)

where kν′,M (λ) is the polynomial of degree M given in (5.17). Since kν′,M has simplezeros at the first M poles λν′,1, λν′,2, . . . , λν′,M of mν(λ), these poles in the sum in


(5.31) cancel and the corresponding terms contribute to the right hand side of (5.31) as apolynomial of degree ≤M − 1: namely, we have for 1 ≤ n ≤M

kν′,M (λ)1

λν′ ,n − λ= −4

(ν′ − 1

2+ n

)2 M∏

j=1j 6=n

4

(ν′ − 1

2+ j

)2

(λ− λν′,j) ,

which is a polynomial of degree M − 1.Now we use the relation (with κ =

[ν+12

])

(1 + λ2

)κ(

1

t− λ− t

1 + t2

)k(t)

(1 + t2)κ− k(λ)

(1

t− λ− t

1 + t2

)

=1 + λt

(1 + t2)κ+1

k(t)(1 + λ2

)κk(λ)

(1 + t2

)κ

t− λ.

The right hand side is a polynomial in λ of degree ≤ 2κ, and it is O(

1t2

)for t → ∞. It

follows that

mν(λ)=

∞∑

n=M+1

kν′,M (λν′,n)rν′,n

λν′,n − λ+(1+λ2

)κ∫ ∞

0

(1

t−λ−t

1+ t2

)kν′,M (t)

(1+ t2)κfν′(t) dt+· · · ,

where · · · denotes a polynomial of degree ≤ 2κ.The proof in the cases a > 0 and a = 0 is similar. �

Example 2 (Case IIA). Consider for ` = 0, 1, . . . and a ∈ R the equation

−y′′(x) +

(−ax

+`(`+ 1)

x2

)y(x) = λ y(x), x ∈ (0,∞), (5.32)

where in case a 6= 0 and ` = 0 the boundary condition (2.12) with α = 0 is imposed atx = 0. The equation (5.32) converts under the change of variables

t = −2i√λ, W(t) = y(x),


W ′′(t) +

(−1

4+

β

t+

14 −

(`+ 1

2

)2

t2

)W(t) = 0, β := β(λ) :=

ia

2√λ,

and with the further change of variable W(t) = t`+1e−12 tu(t) to the confluent hypergeo-

metric equation

tu′′(t) + (2`+ 2 − t)u′(t) − (`+ 1 − β)u(t) = 0.

The latter equation has a two-term recurrence relation in the Frobenius theory at t = 0,with indicial roots r1 = 0 and r2 = −(2`+ 1), and the two Frobenius solutions u1(t; β)and u2(t; β) were obtained in [10, (7.20)-(7.23)] for all β. The corresponding solutionsW1(−2ix

√λ) and W2(−2ix

√λ) of equation (5.32) under the above changes of variable


were also shown in [10, (7.30), (7.39)] to be connected to the Frobenius solutions (2.5)-(2.6) by the relations of linear dependence,

y1(x;λ) =1

(−2i√λ)`+1

W1

(− 2ix

√λ)=

1

(−2i√λ)`+1

Mβ,`+ 12(−2ix

√λ),

y2(x;λ) = (−2i√λ)` W2

(− 2ix

√λ)−A(λ)y1(x;λ).

(5.33)

where A(λ) is given by [10, (7.38)]. Here y1(x;λ) and y2(x;λ) are entire in λ, whileW1

(−2ix

√λ), W2

(−2ix

√λ)

and A(λ) have branch cuts on the positive real λ-axis.Since

W (y1, y2) = −(2`+ 1),

we have for the functions φ, θ from (2.10),

φ(·;λ) = y1(·;λ), θ(·;λ) = − 1

2`+ 1y2(·;λ). (5.34)

It was also shown in [10, (7.45)] that with the standard Whittaker function of second kind

Wβ,`+ 12(t) := t`+1e−

12 tU(`+ 1 − β, 2`+ 2, t),

where U is the confluent hypergeometric function of second kind [28, (1.5.24)], the solu-tion

ψ(x;λ) := Wβ,`+ 12

(− 2ix

√λ)

of (5.32) is square integrable at ∞ for all λ ∈ C\[0,∞). To see this, observe that itfollows from the asymptotic expansion of Wβ,`+ 1

2(t) (see [28, p. 61]), that for all N ≥ 0

and 0 < argλ < 2π we have∣∣∣Wβ,`+ 1

2

(−2ix

√λ)∣∣∣ 5 D exp

(−(sin

argλ

2

)[x|λ| 12 − a

2|λ| 12ln(2x|λ| 12 )

])(1+O

(1

x

))

for someD > 0, as x→ ∞1. Also, from [10, (8.4), (7.60), (8.8), (8.13)-(8.17)] it followsthat Wβ,`+ 1

2

(− 2ix

√λ)

is connected to the Frobenius solutions φ(x;λ), θ(x;λ) for β 6=`+ 1 +m, m = 0, 1, 2, ..., by

Wβ,`+ 12

(− 2ix

√λ)

= −(2`+ 1)α2(λ)[θ(x;λ) −m`(λ)φ(x;λ)

],

where α2(λ) :=(2`)!

(− 2i

√λ)`

Γ(`+ 1 − β). Thus, the Titchmarsh-Weyl function is for

` = 0:

m0(λ) = −a(log(− 2i

√λ)

+ Ψ(1 − β(λ)

)+ 2γ

)+ i

√λ, (5.35)

and for ` ≥ 1:

m`(λ) = k`(λ)m0(λ) + p`(λ), (5.36)

1The factor a2

4|λ|in [10, (7.45)] is incorrect.


where (observing i√

λa = − 1

2β(λ) in [10, (8.13)])

p`(λ) : = aH2` k`(λ) − ak`(λ)

(− 1

2β(λ)+∑

j=1

1

j − β(λ)

)

+(i√λ)2`+1

(2`+ 1)!

2∑

k=0

2k (−`− β(λ))k

k!(2`+ 1 − k)

(5.37)

and

k`(λ) :=1

[(2`+ 1)!]2

∏

j=1

(4λj2 + a2

). (5.38)

Here p`(λ) is a polynomial of degree ` in λwith real coefficients (see Remark 5.10 below).

By passing β → `+ 1 +m(

andλ → λ`m = − a2

4(`+1+m)2

), m = 0, 1, 2, ..., in (5.34) we

obtain (see [10, (7.62)])

W`+1+m,`+ 12

(2x√|λ`

m|)

=( a

`+1+m

)`+1 (−1)m(2`+1+m)!

(2`+ 1)!φ(x;λ`

m),

which shows that for an eigenvalue λ of the problem (5.32) the first Frobenius solution(which satisfies the boundary condition at x = 0) is square integrable at ∞ and hence it isthe corresponding eigenfunction. In this case α2(λ) → 0, and the zero of α2(λ) cancelswith the pole of m`(λ).Summing up we have proved the following analog to Lemma 5.1.

Lemma 5.9. For ` = 1, 2, . . . , the Titchmarsh-Weyl function m` is

m`(λ) = k`(λ)m0(λ) + p`(λ), (5.39)

with the Nevanlinna function

m0(λ) = −a log(− 2i

√λ)− aΨ

(1 − ia

2√λ

)− 2γa+ i

√λ, =λ 6= 0, (5.40)

and p`(λ), k`(λ) being real polynomials given by (5.37) and (5.38).

Remark 5.10. That k`(λ) is a real polynomial is clear from its definition (5.38), forp`(λ) this follows from the fact that m`(λ) and m0(λ) are real for negative λ, which arenot poles. It can also be shown directly from the representation (5.37) as follows: For thefirst summand on the right hand side of (5.37) this is clear. For the other terms on the righthand side of (5.37) it follows if we show that

r`(β) :=

(a

β

)2`+1(−`− β)2`+1

(2`+ 1)!(2`)!

− 1

2β+∑

j=1

1

j − β

− a2`+1

(2`)!(2β)2`+1

2∑

k=0

2k (−`− β)k

k!(2`+ 1 − k)


is an even function: r(β) = r(−β). It is easy to see that this amounts to checking thefollowing equality:

(−`− β)2`+1

(2`+ 1)!

∑

j=−`

1

j − β=

1

22`+1

2∑

k=0

2k (−`− β)k

k!(2`+ 1 − k)+

1

22`+1

2∑

k=0

2k (−`+ β)k

k!(2`+ 1 − k).

Here on both sides are even polynomials of degree 2`, hence it is sufficient to show theequality for the points β = 0, 1, . . . , `. We thank our colleagues Aad Dijksma and BrankoCurgus for having done this, using Mathematica and induction with respect to β. For anexplicit calculation of the polynomials p`(λ) a Mathematica program is given in [18],where also the first polynomials p`(λ) up to ` = 4 are listed; this paper gives two inde-pendent Mathematica programs and the output for r` was found to be in agreement withthe above r` formula up to ` = 30.

Remark 5.11. In the special case a = 0 the functionm0(λ) specializes to

m0(λ) = i√λ, (5.41)

and the relation (5.39) becomes

m`(λ) = k`(λ)i√λ =

22`(`!)2λ`

[(2`+ 1)!]2i√λ.

The Whittaker functions in (5.33) reduce in the case a = 0 to Bessel functions of order`+ 1

2 (see [28, (1.8.13),(1.8.15)]), and the formulas for y1, y2 become

y1(x;λ) = 22`+ 12 Γ(`+

3

2

)λ−( `

2+ 14 )√xJ`+ 1

2(√λx)

= x`+ 12

(1 +

∞∑

n=1

(−1)nλn x2n

22nn!(`+ 3

2

)n

),

y2(x;λ) = 2−(2`+ 12 )Γ(− `+

1

2

)λ

`2+ 1

4√xJ−(`+ 1

2 )(√

λx)

= x−`

(1 +

∞∑

n=1

(−1)nλn x2n

22nn!(

12 − `

)n

),

which have the required normalization (2.5) - (2.6) (with K` = 0 in (2.6)). Since

Wx (y1(·;λ), y2(·;λ)) = −(2`+ 1),

we take

φ(x;λ) = y1(x;λ), θ(x;λ) = − 1

2`+ 1y2(x;λ).

The abovem-function may also be obtained from the requirement that

θ(x;λ) −m`(λ)φ(x;λ) = K√xH

(1)

`+ 12

(√λx)

for some constant K, since√xH

(1)

`+ 12

(√λx)∈ L2(x0,∞) for all λ ∈ C \ [0,∞).

Next we find the integral representation (2.16) of the Nevanlinna functionm0 in (5.40).


Lemma 5.12. The functionm0 in (5.40) is of class N0 and it admits the following repre-sentation:

a > 0:

m0(λ) = α+

∞∑

n=1

rnλn − λ

+

∫ ∞

0

(1

t− λ− t

1 + t2

)a

1 − e− aπ√

t

dt, (5.42)

with

λn = − a2

4n2, rn =

a3

2n3, n = 1, 2, . . . ; (5.43)

hence the function σ = σ0 in the representation (2.14) of m = m0 is

σ0(t) =

∑

n:λn≤t

rn if t < 0,

∞∑

n=1

rn +

∫ t

0

a

1 − e− aπ√

u

du if t ≥ 0.

a < 0:

m0(λ) = α+

∫ ∞

0

(1

t− λ− t

1 + t2

)a

1 − e− aπ√

t

dt; (5.44)

hence the spectral function σ0 of m0 in (2.14) is

σ0(t) =

0 if t ≤ 0,∫ t

0

a

1 − e− aπ√

u

du if t ≥ 0.(5.45)

a = 0:

m0(λ) = α+

∫ ∞

0

(1

t− λ− t

1 + t2

) √t

πdt; (5.46)

hence the spectral function σ0 of m0 in (2.14) is

σ0(t) =

0 if t ≤ 0,

2

3πt

32 if t ≥ 0.

(5.47)

Proof. As was mentioned already, for ` = 0 we have LC case at x = 0, so it follows from[10, Theorem 2(viii), Case A] that m0 ∈ N0.For β in (2.14) we have

β = limy→∞

m0(iy)

iy= 0.

To find the spectral function σ(t) associated with m0(λ) we apply the inversion formulaas in [10, (8.9)]. For λ = t+ iε, t > 0, this gives

limε↓0

−a

π=Ψ

(1 − ia

2√t+ iε

)= −a

π=Ψ

(1 − ia

2√t

)= −

√t

π+a

2coth

(πa

2√t

),

limε↓0

−a

π= log

(− 2i

√t+ iε

)= −a

π= log

(− 2i

√t)

=a

2,


limε↓0

1

π=(i√t+ iε

)=

1

π=(i√t)

=

√t

π,

and thus

limε↓0

1

π=m0

(t+ iε

)=a

2

[1 + coth

(πa

2√t

)]=

a

1 − e−πa√

t

.

For the log term and the term i√λ in (5.40) we have for t < 0:

= log(− 2i

√t)

= 0, =(i√t)

= 0.

Moreover, for an interval [t1, t2] ⊂ (−∞, 0) where the Ψ term does not give a pole wehave

limε↓0

=(−aΨ

(1− ia

2√t+ iε

))= =

(−aΨ

(1− ia

2√t

))= =

(−aΨ

(1− a

2√−t

))= 0,

so σ is constant on [t1, t2]. These calculations apply to both a > 0 and a < 0. For a ≤ 0,m0 does not have any poles. For a > 0 and t < 0, m0 has poles at the points λn, where

1 − ia

2√λn

= −n+ 1, n = 1, 2, . . . , that is, at

λn = − a2

4n2, n = 1, 2, . . . .

The corresponding coefficient rn is the negative residue of m0 at λn. Using

Res∣∣z=−(n−1)

Ψ(z) = −1,

we find with L’Hospital’s rule that

rn = Res∣∣λ=λn

[aΨ(1 − β(λ)

)]

= a limλ→λn

{(λ− λn)

−1(1 − β(λ)

)+ (n− 1)

}=

a3

2n3.

For the case a = 0 we have from (5.41)

β = limy→∞

m0(iy)

iy= 0,

and

σ′0(t) = lim

ε↓0

=m0(t+ iε)

π=

√t

π,

which gives (5.47). �

Remark 5.13. The constants in (5.42) and (5.44) for a 6= 0 can be found by using theasymptotics of m0(λ) with λ = −u2, u → ∞ from (5.35) and from the right hand sidesof (5.42) and (5.44), and then matching them. This yields

α = −aγ − a ln 2 − 1√2

+

∫ ∞

0

1

1 + t2

(a

1 − e− aπ√

t

− a

π−

√t

π

)dt,

and in (5.46), α = − 1√2

.


Theorem 5.14. The spectral function ρ`, corresponding to the Fourier transformation(4.14), is as follows (k`, λn, and rn are given by (5.38) and (5.43)):

a > 0 :

ρ`(t) =

∑

λ`+1≤λn≤t

k`(λn) rn, t < 0,

∞∑

n=`+1

k`(λn) rn +

∫ t

0

f`(u) du, t ≥ 0,

with

f`(t) :=ak`(t)

1 − e− aπ√

t

, t ≥ 0. (5.48)

a < 0 :

ρ`(t) =

0, t < 0,∫ t

0

f`(u) du, t ≥ 0,

with f`(t) given by (5.48).

a = 0 :

ρ`(t) =

0, t < 0,

1

π

∫ t

0

k`(u)u12 du, t ≥ 0.

Proof. The proof is similar to the proof of Theorem 5.6. The poles λ1, λ2, . . . , λ` of m0

in (5.42) are canceled by the factor k` in (5.39). Making use of (4.20) and (5.39), and theStieltjes-Livsic inversion formula (2.19) we have, since k`(z) is real on the real axis, that

ρ`([a, b]) = − 1

2πi

∮ ′

Γ[a,b]

m`(z) dz = − 1

2πi

∮ ′

Γ[a,b]

(k`(z)m0(z) + p`(z)) dz

=

∫ b

a

k`(t)dσ0(t),

whenever a, b ∈ R, a < b are points with σ0-measure zero, and where σ0 is given in(5.45). �

The expansion theorem or inverse Fourier transformation (4.16) takes the following formfor ` ≥ 0:

f(x) =

∑∞n=`+1 Ff (λn)φ(x;λn) k`(λn) rn +

∫ ∞

0

Ff (λ)φ(x;λ) f`(λ) dλ, a > 0,

∫ ∞

0

Ff (λ)φ(x;λ)f`(λ)dλ, a < 0,

∫ ∞

0

Ff (λ)φ(x;λ)22`(`!)2λ`

[(2`+ 1)!]2dλ, a = 0.


Here Ff is again the Fourier transformation (4.3), where φ(x;λ) is given in (5.33) fora 6= 0 and in Remark 5.11 for a = 0 (which yields the Hankel formula of order ` + 1

2 ).The case a > 0 corresponds to the usual eigenfunction expansion for the radial part ofthe separated hydrogen atom and agrees with the formulas in [10, (8.26)] and Titchmarsh[29, p. 100] and Jorgens-Rellich [15, p. 221].The representation (2.20) of the function m` we formulate in the next theorem; its proofis similar to the proof of Theorem 5.8 and therefore omitted.

Theorem 5.15. The function m` in (5.36) is of class Nκ with κ =[

`+12

]. Its representa-

tion (2.20) is as follows (here k` is given by (5.38), q` is a real polynomial of degree `) :

a > 0 :

m`(λ) =

∞∑

n=`+1

k`(λn)rnλn − λ

+(1 + λ2

)κ∫ ∞

0

(1

t− λ− t

1 + t2

)dσ`(t) + q`(λ),

where

σ`(t) :=

∫ t

0

k`(u) a

(1 + u2)κ(1 − e

− aπ√u

) du, t ≥ 0. (5.49)

a < 0 :

m`(λ) =(1 + λ2

)κ∫ ∞

0

(1

t− λ− t

1 + t2

)dσ`(t) + q`(λ),

with σ` as in (5.49).

a = 0 :

m`(λ) =(1 + λ2

)κ∫ ∞

0

(1

t− λ− t

1 + t2

)dσ`(t) + q`(λ),

with

σ`(t) :=

∫ t

0

k`(u)u12

π (1 + u2)κdu, t ≥ 0.

Example 3 (Case IIB). The equation is now

−y′′(x) +

[−ax

+N2 − 1

4

x2

]y(x) = λy(x), 0 < x <∞, (5.50)

for N = 0, 1, . . . , a ∈ R, and with the boundary condition (2.12) with α = 0 for N = 0.The equation (5.50) converts under the change of variables

t = −2ix√λ, W(t) = y(x)


W ′′(t) +

(−1

4+

β

t+

14 −N2

t2

)W(t) = 0, β :=

ia

2√λ, (5.51)


and with the further change of variable W(t) = tN+ 12 e−

12 tu(t) to the confluent hyperge-

ometric equation

tu′′(t) + (2N + 1 − t)u′(t) −(N +

1

2− β

)u(t) = 0. (5.52)

The Frobenius solutions of (5.52) are therefore available in closed form as follows. Thecase N = 0 has equal indicial roots, r1 = r2 = 1

2 , and N ≥ 1 has indicial roots,r1,2 = ±N+ 1

2 , differing by an integer. For this reason, the logarithmic Frobenius solutionforN = 0 does not arise as a special case of the logarithmic solution forN ≥ 1; however,the nonlogarithmic Frobenius solution forN = 0 is of the same form as forN ≥ 1. Moreexplicitly, we obtain

u1(t; β) = M(N +

1

2− β, 1 + 2N, t

):=

∞∑

n=0

(N + 1

2 − β)n

n!(1 + 2N)ntn if N ≥ 0,

and

u2(t; β) =

u1(t; β) log t+ 2βt+

∞∑

n=2

bn(β)tn if N = 0,

−(−N − β + 1

2

)2N

(2N)!(2N − 1)!u1(t; β) log t+

2N−1∑

n=0

(−N−β + 1

2

)n

n!(1−2N)ntn−2N

−(−N−β+ 1

2

)2N

(2N−1)!(2N)!

∞∑

n=0

bn(β)tn if N ≥ 1,

where for n = 0, 1, . . . , and N ≥ 0,

bn(β) =

(N + 12 − β)n

n!(2N + 1)n

n−1∑

j=−2N

1

j + 12 +N − β

+H2N−1 −Hn −H2N+n

if β 6= N + 12 +m, m = 0, 1, 2, . . . ,

(−1)mm!(n−m− 1)!

n!(2N + 1)nif β = N + 1

2 +m, m = 0, 1, . . . , n− 1,

(−m)n

n!(2N + 1)n

n−1∑

j=−2N

1

j −m+H2N−1 −Hn −H2N+n

if β = N + 12 +m, m = n, n+ 1, . . . .

Here, the formulas for β = N +m+ 12 arise by passing to the limit β → N +m+ 1

2 inthe formulas for β 6= N +m+ 1

2 , for N ≥ 0.


Converting these Frobenius solutions via the above changes of variable to solutions of(5.51) and (5.50) we define for N = 0, 1, . . .

WNj (t) = tN+ 1

2 e−12 tuj(t; β), j = 1, 2,

as solutions of (5.51), and obtain

WNj

(− 2ix

√λ)

=(− 2ix

√λ)N+ 1

2 eix√

λuj

(− 2ix

√λ; β(λ)

), j = 1, 2, (5.53)

as solutions of (5.50). Similarly to the analysis in [10, Section 7], we can express thesolutions in (5.53) in terms of the Frobenius solutions y1(x;λ) and y2(x;λ) of (2.7)-(2.8),and vice versa. This gives

y1(x;λ) =WN

1

(−2ix

√λ)

(−2i

√λ)N+ 1

2

=Mβ,N+ 1

2

(−2ix

√λ)

(−2i

√λ)N+ 1

2

= xN+ 12 eix

√λM(N+ 1

2−β(λ), 2N+1,−2ix√λ), N ≥ 0,

(5.54)

and

y2(x;λ) =

(−2i

√λ)− 1

2W02 (−2ix

√λ) − log(−2i

√λ)y1(x;λ), N = 0,

(−2i

√λ)N− 1

2WN2 (−2ix

√λ) −AN (λ)y1(x;λ), N ≥ 1,

(5.55)

where

AN (λ) := KN (λ)

log

(−2i

√λ)

+

2N−1∑

j=0

1

N − β(λ) − j − 12

+

(i√λ)2N

(2N − 1)!

2N−1∑

k=0

2k(−N − β(λ) + 1

2

)k

k!(2N − k)

and

KN(λ) := − 1

(2N − 1)!(2N)!

N−1∏

j=0

[(j +

1

2

)2

4λ+ a2

]

= −(−N − β(λ) + 1

2

)2N

(− 2i

√λ)2N

(2N − 1)!(2N)!.

Of interest here is that the branch cuts associated with√λ in the solutions WN

1 (−2ix√λ)

and WN2 (−2ix

√λ), and u1(−2ix

√λ) and u2(−2ix

√λ), cancel to make y1(x;λ) and

y2(x;λ) entire in λ.In order to introduce the Titchmarsh-Weyl function for the doubly singular Sturm-Liou-ville problem (5.50), we have to write the solution of (5.50) which is square integrable atx = ∞ as linear combination of the above Frobenius solutions y1(x;λ) and y2(x;λ). Tothis end we make use of U

(N + 1

2 − β(λ), 1 + 2N, t), the standard confluent hypergeo-

metric function of second kind ([28, (1.5.24)]), and the corresponding Whittaker function


of second kind [28, (1.7.18)], which comes from it by the change of variable from (5.51)to (5.52),

Wβ,N(t) = tN+ 12 e−

t2U(N +

1

2− β, 1 + 2N, t

), N ≥ 0. (5.56)

Similarly to [10, (7.48),(7.50)] we can make use of well known properties of the digammafunction to obtain the connection formula,

Wβ,N (t) = ε1(β, N)W1(t) + ε2(β, N)W2(t), (5.57)

where, for N = 0 and β 6= m+ 12 , m = 0, 1, 2, ...,

ε1(β, 0) := −2γ + Ψ( 12 − β)

Γ( 12 − β)

, ε2(β, 0) = − 1

Γ( 12 − β)

,

and for N ≥ 1 and β 6= N + 12 +m, m = 0, 1, 2, ...,

ε1(β, N) :=1

(2N)!Γ(−N−β+ 1

2

)[ −1∑

j=1−2N

1

j − 12 +N−β

+H2N−1

−2γ−Ψ(N+ 1

2−β−1)],

ε2(β, N) :=(2N − 1)!

Γ(N + 12 − β)

.

Putting t = −2ix√λ on the left hand side of (5.57) gives the solution of (5.50) which is

square integrable near x = ∞ for all λ ∈ C \ [0,∞). Taking the square root of λ to bedefined with branch cut on the positive real λ-axis, i.e.

λ12 = |λ| 12 ei arg λ

2 , 0 ≤ argλ < 2π,

it follows from the asymptotic expansion of Wβ,N (t) (see [28, p. 61]), that for all N ≥ 0and 0 < argλ < 2π we have∣∣∣Wβ,N

(−2ix

√λ)∣∣∣ 5 D exp

(−(sin

argλ

2

)[x|λ| 12 − a

2|λ| 12ln(2x|λ| 12

)])(1+O

(1

x

))

for some D > 0, as x → ∞. Since sin arg λ2 > 0 for argλ ∈ (0, 2π), it follows that

Wβ,N

(− 2ix

√λ)∈ L2(x0,∞) for all λ ∈ C \ [0,∞).

It follows from (2.9) that we should take for N = 0

φ(x;λ) = y1(x;λ), θ(x;λ) = y2(x;λ), (5.58)

and for N ≥ 1

φ(x;λ) = y1(x;λ), θ(x;λ) = − 1

2Ny2(x;λ). (5.59)

Solving (5.54) and (5.55) for W1 and W2, putting the results into (5.57), making use of(5.58) and (5.59), and taking t = −2ix

√λ, after a bit of algebra, we find

Wβ,0

(− 2ix

√λ)

= α2(λ, 0)(θ(x;λ) −m0(λ)φ(x;λ)

)


and

Wβ,N

(− 2ix

√λ)

= −2Nα2(λ,N)(θ(x;λ) −mN (λ)φ(x;λ)

), N ≥ 1, (5.60)

where

α2(λ, 0) := −(− 2i

√λ) 1

2

Γ

(1

2− ia

2√λ

) ,

and

m0(λ) := −(

log(− 2i

√λ)

+ Ψ

(1

2− ia

2√λ

)+ 2γ

), (5.61)

and, for N ≥ 1,

α2(λ,N) :=(2N − 1)!

Γ(N + 1

2 − β(λ)) (

− 2i√λ)N− 1

2

,

mN (λ) : =KN (λ)

2N

{log(− 2i

√λ)

+1

N−β(λ)− 12

+ Ψ(N− 1

2 −β(λ))

−H2N−1 + 2γ

}+

(i√λ)2N

(2N)!

2N−1∑

k=0

2k(−N−β(λ)+ 1

2

)k

k! (2N−k) .

(5.62)

If N ≥ 0, these formulas are valid for ia2√

λ6= N + 1

2 + m, m = 0, 1, .... For ia2√

λ=

N + 12 + m, m = 0, 1, . . . , i.e. λ = λm =

−a2

4(N + 1

2 +m)2 , passage to the limit

β(λ) → N + 12 +m gives

WN+m+ 12 ,N

(ax

N + 12 +m

)=

(a

N + 12 +m

)N+ 12 (−1)m+1(m+ 2N)!

(2N)!φ(x;λm),

and also α2 → 0, hence the zero of α2 cancels with the pole of mN .The functionsm0(λ) andmN (λ) in (5.61) and (5.62) are Titchmarsh-Weyl functions, rel-ative to the fundamental system {φ(·;λ), θ(·;λ)} for the doubly singular problem (5.50).For N ≥ 2 we make use of the identity

Ψ

(N − β − 1

2

)= Ψ

(1

2− β

)+

N−2∑

j=0

112 − β + j

,

and of (5.61), to obtain the m-function in the form

mN (λ)= kN (λ)m0(λ) + pN (λ), pN (λ) := kN (λ)H2N−1 + rN (λ), (5.63)

where

kN (λ) := −KN(λ)

2N=

N−1∏

j=0

[4λ

(j +

1

2

)2

+ a2

]

[(2N)!]2 (5.64)


and

rN (λ) := −kN(λ)N−1∑

j=0

1(12−β(λ)+j

)+(−1)NλN

(2N)!

2N−1∑

k=0

2k(−N−β(λ)+ 1

2

)k

k! (2N − k). (5.65)

The case N = 1 is included in (5.63) with p1(λ) = a2/4 (by making use of (5.62)).The expression for rN (λ) can be seen to be a polynomial of degree N in λ with realcoefficients in a manner similar to Remark 5.10.

Remark 5.16. Similarly to [10, equa (8.14)] and [18] it can be shown that

rN (λ) =

N∑

j=1

[ −1

[(2N)!]2C(N, j − 1) +

(−1)N

(2N)!d(N, j)

]λj

where

C(N, j) :=

N−1∑

m=0

4

(m+

1

2

)K(N,m)γj(N,m), 0 ≤ j ≤ N − 1,

K(N,m) :=

∏N−1j=0 (2j + 1)2

4(m+ 1

2

)2 ,

and with k1 =[

k2

]

d(N, j) :=

2k1+1∑

k=0

[(−1)N−j2kα2N−2j(k,N)

(a2

)2N−2j

k!(2N − k)

],

where γj = γj(N,m) and αn = αn(k,N) are defined by the equations

N−1∏

i=0,i6=m

(λ+

a2

4(i+ 12 )2

)=

N−1∑

j=0

γj(N,m)λj

andk−1∏

j=0

(−N +

1

2+ j − t

)=

k∑

n=0

αn(k,N)tn

Summing up, we have proved the following lemma.

Lemma 5.17. For N = 1, 2, 3, . . . , the Titchmarsh-Weyl function mN in (5.62) admitsthe representation

mN (λ) = kN (λ)m0(λ) + pN(λ),

where m0 is the Titchmarsh-Weyl function (5.61) for N = 0, and kN , pN , and rN are thepolynomials of degree N given in (5.63)-(5.65).

Remark 5.18. In the special case a = 0 the functionm0(λ) specializes to

m0(λ) = − log(− 2i

√λ)

+ γ − 2 ln 2, (5.66)


and the relation (5.66) becomes

mN (λ) =λN

4N(N !)2m0(λ) +

HN−1λN

22N+1(N !)2,

for N ≥ 1 (with H0 = 0). The Whittaker functions in (5.55) reduce in the case a = 0to Bessel functions of order N (see [28, (1.8.13), (1.8.15)]) and the formulas for y1, y2become

y1(x;λ) = N !2Nλ−N2√xJN

(√λx)

= xN+ 12

[1 +

∞∑

n=1

((−1)nλnN !

22nn!(N + n)!

)x2n

], N ≥ 0,

(5.67)

and

y2(x;λ) =

y1(x;λ) ln x+ x12

[ ∞∑

n=1

(−1)n+1Hnλnx2n

(n!)222n

], N = 0,

(−2NλN

4N (N !)2

)y1(x;λ) ln x+ x−N+ 1

2

{1+

N−1∑

n=1

((N − n− 1)!λn

(N − 1)!n!22n

)x2n

+

∞∑

n=N

((−1)n+N (Hn+Hn−N −HN−1)λ

n

n!(n−N)!(N − 1)!22n

)x2n

}, N ≥ 1,

which have the required normalization (2.7)-(2.8), and can also be written as

y2(x;λ) =

π

2

√xY0

(√λx)−[γ + log

(√λ

2

)]√xJ0

(√λx), N = 0,

− πλN/2

2N (N − 1)!

√xYN

(√λx)

+

[2NλN

4N(N !)2

(log(√

λ2

)+ γ)− HN−1λ

N

4NN !(N − 1)!

]y1(x;λ), N ≥ 1.

From the Wronskian relation in (2.9) we need to take

φ(x;λ) = y1(x;λ), θ(x;λ) =

{y2(x;λ), N = 0

− 12N y2(x;λ), N ≥ 1.

The above Titchmarsh-Weyl functionmN may also be obtained from the requirement that

θ(x;λ) −mN (λ)φ(x;λ) = K√xH

(1)N (

√λx)

for some constant K, since√xH

(1)N (

√λx) ∈ L2(x0,∞) for all λ ∈ C \ [0,∞), x0 > 0.

Now the integral representation (2.16) of the functionm0 in (5.61) is given in the follow-ing lemma. The proof is similar to Lemmas 5.3 and 5.12.

Lemma 5.19. The function m0 in (5.61) is of class N0 and admits the following repre-sentation:


a > 0 :

m0(λ)= α+

∞∑

n=1

rnλn−λ

+

∫ ∞

0

(1

t−λ − t

1+ t2

)1

1+ e− aπ√

t

dt (5.68)

with

λn = − a2

4(n− 1

2

)2 , rn :=a2

2(n− 1

2

)3 , n = 1, 2, . . . . (5.69)

a ≤ 0 :

m0(λ)= α+

∫ ∞

0

(1

t−λ − t

1+ t2

)1

1+ e|a|π√

t

dt. (5.70)

Remark 5.20. The constants in (5.68) and (5.70) for a 6= 0 can be found by using theasymptotics of m0(λ) with λ = −u2, u→ ∞, from (5.61) and from the right hand sidesof (5.68) and (5.70), and then matching them. This yields,

α = −γ + ln 2 +

∫ ∞

0

t

1 + t2

1

1 + exp(

aπ√t

) − 1

2

dt

for the constant in (5.68); and for (5.70) the sign in front of the integral becomes negativefor a < 0. For a = 0, we get α = −γ + ln 2.

Finally, for the spectral function ρN we have the following theorem. Its proof is similarto the proofs of Theorems 5.6 and 5.14 and therefore omitted.

Theorem 5.21. For N = 0, 1, . . . , the spectral function ρN corresponding to the Fouriertransformation (4.14) is as follows (kN , λn, and rn given by (5.64) and (5.69)) :

a > 0 :

ρN (t) =

∑

λN+1≤λn≤t

kN (λn) rn, t < 0,

∞∑

n=1

kN (λn) rn +

∫ t

0

fN (u) du, t ≥ 0,

with

fN (t) :=kN (t)

1 + e− aπ√

t

, t ≥ 0. (5.71)

a < 0 :

ρN (t) =

0 t < 0,∫ t

0

fN (u) du, t ≥ 0,


with fN(t) given by (5.71).

a = 0 :

ρN (t) =

0 t < 0,

1

2

∫ t

0

uN

4N (N !)2du, t ≥ 0.

The expansion theorem or inverse Fourier transformation (4.16) takes the following formfor N ≥ 0:

f(x) =

∞∑

n=N+1

Ff (λn)φ(x;λn) kN (λn) rn

+

∫ ∞

0

Ff (λ)φ(x;λ) fN (λ) dλ, a > 0,

∫ ∞

0

Ff (λ)φ(x;λ)fN (λ)dλ, a < 0,

∫ ∞

0

Ff (λ)φ(x;λ)1

2

(λN

4N (N !)2

)dλ, a = 0.

Here Ff is again the Fourier transformation (4.3), where φ(x;λ) is given by (5.54) forN = 0 and (5.67) for a = 0 (which yields the Hankel formula of order N ).The representation (2.20) of the function mN for N = 1 is given in the next theorem; itsproof is similar to the proof of Theorem 5.8 and is therefore omitted.

Theorem 5.22. The function mN in (5.62)-(5.63) is of class Nκ with κ :=[

N+12

]. Its

representation (2.20) is as follows, where kN is given by (5.64), qN is a real polynomialof degree N , and λn,rn are given by (5.69).a > 0:

mN (λ) =

∞∑

n=N+1

kN (λn)rnλn − λ

+(1 + λ2

)κ∫ ∞

0

(1

t− λ− t

1+ t2

)dσN (t) + qN (λ)

where

σN (t) :=

∫ t

0

kN (u)

(1 + u2)κ(1 + e− aπ√

u )du, t = 0. (5.72)

a < 0:

mN (λ) =(1 + λ2

)κ∫ ∞

0

(1

t− λ− t

1 + t2

)dσN (t) + qN (λ)

with σN as in (5.72).a = 0 :

mN (λ) =(1 + λ2

)κ∫ ∞

0

(1

t− λ− t

1 + t2

)dσN (t) + qN (λ)

with

σN (t) :=

∫ t

0

1

(1 + u2)k

uN

4N (N !)21

2du, t ≥ 0.


Remark 5.23. Our examples 1-3 exhaust all cases of equation (1.1) under the assumption(1.2), and our analysis of these examples, involving special functions, therefore brings tocompletion the work in [10], [14] and [9]. The cases considered previously were, in [10]:Example 1 with a = 0, Example 2 with a > 0; in [14]: Examples 1, 2, 3 with a = 0; in[9]: Example 1 with a = 0 and ν ∈ [0, 1). Also, for the Bessel differential equation, thesame m-function as in Example 1 was obtained in [27] and [7]. Further work on the topicof Sturm-Liouville problems with two singular endpoints, including numerical algorithmsfor computing the spectral functions for the problems we have considered here, has beendone in [13].

Acknowledgement: Charles Fulton acknowledges support for this work from the Collegeof Science, Florida Institute of Technology, for a sabbatical leave in Spring 2006 whichenabled him to spend three weeks at University of Vienna, Vienna, Austria.

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C. FultonDepartment of Mathematical SciencesFlorida Institute of Technology150 W. University Blvd.Melbourne, FL. 32904, USAe-mail: [email protected]

H. LangerInstitute for Analysis and Scientific ComputingVienna University of TechnologyWiedner Hauptstrasse 8-101040 Vienna, Austriae-mail: [email protected]

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