ON SPECTRAL ASYMPTOTICS OF QUASI-EXACTLY SOLVABLE …staff.math.su.se/shapiro/Articles/QESquartic.pdf · n4=3. (ii) If lim n!1 an n2=3 = 0, the sequence f (an) n gof root-counting

ON SPECTRAL ASYMPTOTICS OF QUASI-EXACTLY

SOLVABLE QUARTIC POTENTIAL

BORIS SHAPIRO AND MILOS TATER

Abstract. Motivated by the earlier results of [19] and [26], we discuss belowthe asymptotics of the solvable part of the spectrum for the quasi-exactlysolvable quartic oscillator. In particular, we formulate a conjecture on thecoincidence of the asymptotic shape of the configuration of the level crossingsof the latter oscillator with the asymptotic shape of zeros of the Yablonskii-Vorob’ev polynomials. In the final section we present a numerical study of thespectral monodromy.

1. Introduction

A quasi-exactly solvable quartic oscillator was introduced by C. M. Bender andS. Boettcher in [3] and (in its restricted form) is a Schrodinger-type eigenvalueproblem of the form

LJ(w) = w00 � (x4/4� ax2/2� Jx)w = �w (1.1)

with the boundary conditions w(te±⇡i/3) ! 0 as t ! +1, where a 2 C and Jare parameters of the spectral problem. With these boundary conditions, real aand J , (1.1) is not Hermitian but is PT -symmetric, see [24], [19] and [13]. Incase when J = n + 1 is a positive integer, then Ln+1(w) maps the linear space

of quasi-polynomials of the form {pe�x3/6+ax/2 : deg p n} to itself where pruns over the linear space of polynomials of degree at most n. The restriction ofLn+1 to the latter space is a finite-dimensional linear operator whose spectrumand eigenfunctions can be found explicitly using linear algebra. This part of thespectrum and eigenfunctions of (1.1) is usually referred to as solvable.

One can easily show that polynomial factors p in the quasi-exactly solutionsw = peh of (1.1) coincide with the polynomial solutions of the degenerate Heunequation

y00 � (x2 � a)y0 + (↵x� �)y = 0, (1.2)

where a 2 C has the same meaning as above and (↵,�) are the spectral variables.Obviously, if equation (1.2) has a polynomial solution of degree n, then ↵ = n. Fur-thermore, to get a polynomial solution of (1.2) of degree n, the remaining spectralvariable � should be chosen as an eigenvalue of the operator

Tn(y) = y00 � (x2 � a)y0 + nxy

when it acts on the linear space of polynomials of degree at most n.

Date: February 24, 2021.2010 Mathematics Subject Classification. 34L20 (Primary); 30C15, 33E05 (Secondary).Key words and phrases. Heun equation, spectral polynomials.

1

2 B. SHAPIRO AND M. TATER

In the standard monomial basis {1, x, x2, . . . , xn}, this action of the operator Tn

is given by the 4-diagonal (n+ 1)⇥ (n+ 1)-matrix M (a)n of the form

M (a)n :=

0

BBBBBBBBB@

0 a 2 0 0 · · · 0n 0 2a 6 0 · · · 00 n� 1 0 3a 12 · · · 0...

.... . .

. . .. . .

. . ....

0 0 · · · 3 0 (n� 1)a n(n� 1)0 0 · · · 0 2 0 na0 0 · · · 0 0 1 0

1

CCCCCCCCCA

. (1.3)

We will call the bivariate polynomial Spn(a,�) := det(M (a)n ��I) the n-th spec-

tral polynomial of (1.2). The degree of polynomial Spn(a,�) equals n+ 1 which isalso its degree with respect to the variable �. (The maximal degree of the variablea in Spn(a,�) equals

⇥n+12

⇤). Additiobally observe that if a is a real number then

Spn(a,�) is a real polynomial in � and therefore the spectrum of M (a)n is symmetric

with respect to the real axis.

The main goal of this paper is to study the asymptotics of the spectrum of M (an)n

in two di↵erent regimes. In the first regime we require that limn!1an

n2/3 = 0 whilein the second regime we require that limn!1

an

n2/3 = A 6= 0. (Our numericalexperiments indicate that if limn!1

an

n2/3 does not exist one can not expect anyinteresting limiting behavior of the spectrum).

Remark. In case of a non-degenerate Heun equation detailed study of similar asymp-totics was carried out in [23] and [26].

Our first result is as follows.

Theorem 1. (i) If limn!1an

n2/3 = 0, the maximal absolute value rn(an) of the

eigenvalues of M (an)n grows as 3

4n4/3.

(ii) If limn!1an

n2/3 = 0, the sequence {µ(an)n } of root-counting measures for the

rescaled spectra of {M (an)n } where each eigenvalue of M (an)

n is divided by n4/3,weakly converges to the measure ⌫0 supported on the union of three straight inter-vals connecting the origin with three cubic roots of 27

64 , see Figure 1.

More information about ⌫0 can be found in § 2.

Our next result is obtained on the physics level of rigor, i.e., modulo three con-vergence assumptions which are explicitly formulated in the proof of Proposition 2,see § 3 below. To formulate it, take the family of equations

C2 � (x2 �A)C + (x� �) = 0 (1.4)

which we consider as quadratic equations in the variable C depending on the spacevariable x and parameters A and �.

Proposition 2. If limn!1an

n2/3 = A 6= 0, then (under three additional convergence

assumptions) the sequence {µ(an)n } of root-counting measures for the rescaled spec-

tra {Spn(an,�n4/3)} of M (an)n weakly converges to a special compactly supported

probability measure ⌫A. In particular, the support of ⌫A consists of all values of

SPECTRAL ASYMPTOTICS OF QES-QUARTIC POTENTIAL 3

Figure 1. Distributions of the eigenvalues of M (0)n scaled by n4/3

for n=200.

the spectral parameter � for which there exists a compactly supported probabil-ity measure in the x-plane whose Cauchy transform C(x) satisfies (1.4) almosteverywhere.

Remark 1. To clarify the above formulation, let us recall that for a complex-valuedmeasure µ compactly supported in C, its logarithmic potential is defined as

uµ(x) :=

Z

Cln |x� ⇠|dµ(⇠)

and its Cauchy transform is defined as

Cµ(x) :=Z

C

dµ(⇠)

x� ⇠=

@uµ(x)

@x.

Remark 2. The existence of a signed (not necessarily positive) measure µ whoseCauchy transform Cµ(x) satisfies (1.4) almost everywhere in C is closely related tothe properties of the family

A,� = �((x2 �A)2 � 4(x� �))dx2 (1.5)

of quadratic di↵erentials depending on parameters A and �. We will present someinformation about this connection in § 3. (For an accessible introduction to qua-dratic di↵erentials, see e.g. [25]).

To finish the introduction, let us formulate a rather unexpected conjecture sup-ported by extensive numerical experiments. For a given positive integer n, de-note by ⌃n the set of all branching points of the projection of the algebraic curve�n(a) : {Spn(a,�) = 0} to the a-axis. In other words, ⌃n is the set of all values ofthe complex parameter a for which Spn(a,�) has a multiple root, i.e., the matrix

M (a)n has a multiple eigenvalue. In physics literature such points are called level

crossings. Obviously, one can describe ⌃n as the zero locus of the univariate dis-criminant polynomial Dn(a) := Discriminant(Spn(a,�)) which is the resultant of

Spn(a,�) with @Spn(a,�)@� . One can show that the degree of the latter polynomial

equals�n+1

2

�.


On the other hand, Yablonskii-Vorob’ev polynomials are defined as follows, seee.g. [28, 29]. Set Y V0 = 1, Y V1 = t, and for n � 1, define

Y Vn+1 =t · Y V 2

n � 4(Y Vn · Y V 00n � (Y V 0

n)2)

Y Vn�1.

Although the latter expression apriory determines a rational function, Y Vn is infact a polynomial of degree

�n+12

�, see e.g. [27]. The importance of Yablonskii-

Vorob’ev polynomials is explained by the fact that all rational solutions of thesecond Painleve equation

utt = tu+ 2u3 + ↵,↵ 2 C,are presented in the form

u(t) = u(t;n) =d

dt

⇢ln

Y Vn�1(t)

Y Vn(t)

��, u(t, 0) = 0, u(t;�n) := �u(t;n).

Denote by Zn the zero locus of Y Vn. (Good exposition of the properties of Y Vn

together with several pictures of Zn can be found in [11]).

Remark 3. One can show that the maximal absolute value of points in ⌃n grows

as 33p4

n2/3 while the maximal absolute value of points in Zn grows as�92

� 23 n2/3,

see § 5.

Conjecture 1. (i) After division by 33p4

n2/3 and�92

� 23 n2/3 respectively, the se-

quence {⌃n} tends to the sequence { bZn}, where bZn is the zeros locus of Y Vn(�t).In particular, the limiting triangular domains covered by {⌃n} and { bZn} whenn!1 coincide after the latter scaling. We denote the latter domain by F.

(ii) The interior of F consists of all values of A, for which the support of the abovemeasure ⌫A has a tritronquee form, i.e., looks like a tripod. The complement ofthe closure of F consists of all values of A, for which the support of measure ⌫A isa single smooth arc, see Fig. 7.

For more details on Conjecture 1 see § 4 and for its illustration, see Fig. 2. Wehope that considerations similar to that in [17, 18] can help to settle Conjecture 1.

Acknowledgements. The authors are sincerely grateful to M. Duits, A. Kuijlaars,and A. Gabrielov for discussions. The first author wants to thank F. Stampach forhis help with the complex version of the main result of [16].

2. Case limn!1an

n2/3 = 0. Proof of Theorem 1

Let us start our considerations of the spectral asymptotics of {M (a)n } with the

simplest case an = 0, n = 1, 2, 3, . . . .

Proposition 3. (i) The spectrum of M (0)n , i.e., the zero locus of Spn(0,�), is

invariant under the rotation by 2⇡/3 around the origin.

(ii) The sequence Spn(0,�) splits into the following three subsequences:

(1) for n + 1 = 3k, the polynomial Spn(0,�) contains only the powers of �

divisible by 3; i.e., we have Spn(0,�) = q(0)k (�3);

(2) for n+ 1 = 3k + 1, one has Spn(0,�) = �q(1)k (�3);


Figure 2. The set ⌃40 of the branching points for Sp40(a,�) to-gether with the zero locus bZ40 after scaling sending the three cor-ners to the cubic roots of 1. The blue dots are the roots of bZ40

while the red ones are the branching points of Sp40(a,�). (Ob-serve the surprising closeness of the scaled ⌃40 and bZ40. Theirdistinction is hardly visible by a naked eye!)

(3) for n+ 1 = 3k + 2, one has Spn(0,�) = �2q(2)k (�3).

(iii) Introducing ⇠ = �3, we get that all three polynomials q(0)k (⇠), q(1)k (⇠), q(2)k (⇠)have the same degree k; all their roots are negative and simple which implies that,

for any positive integer n, the spectrum of M (0)n is located on the union of the three

rays through the origin as illustrated in Figure 1.

To prove Proposition 3 following the ideas of [16], we need to introduce a dou-ble indexed polynomial sequence containing our original sequence of characteristic


polynomials {Spn(0,�)}. The most natural way to do this is to consider the prin-

cipal minors of M (0)n � �I given by:

M (0)n � �I :=

0

BBBBBBBBB@

�� 0 2 0 0 · · · 0n �� 0 6 0 · · · 00 n� 1 �� 0 12 · · · 0...

.... . .

. . .. . .

. . ....

0 0 · · · 3 �� 0 n(n� 1)0 0 · · · 0 2 �� 00 0 · · · 0 0 1 ��

1

CCCCCCCCCA

. (2.1)

Namely, denote by �(k)n (�) the k-th principal minor of (2.1). Obviously,

Spn(0,�) = �(n+1)n (�) = det(M (0)

n � �I).

Since the latter matrix is 4-diagonal with one subdiagonal and two superdiago-nals, then, by a general result of [21], its principal minors satisfy a 4-term linearrecurrence relation. Simple explicit calculation gives

�(k)n (�) = ��(k�1)

n (�) + (n� k + 2)(n� k + 3)(k � 1)(k � 2)�(k�3)n (�), (2.2)

where k runs from 1 to n+ 1, with the standard initial conditions

�(�2)n (�) = �(�1)

n (�) = 0, �(0)n (�) = 1.

Recurrence (2.2) has variable coe�cients depending both on k and n.

Another form of (2.2) which looks more promising is as follows. To simplify

the signs, let us instead of M (0)n � �I consider the principal minors of M (0)

n + �I.

Introducing r(k)n (�) := �(k)

n (��), we get the recurrence:

r(k)n (�) = �r(k�1)

n (�) + (n� k + 2)(n� k + 3)(k � 1)(k � 2)r(k�3)n (�). (2.3)

Proof of Proposition 3. Items (i) and (ii) follow immediately from the next state-ment.

Lemma 4. Denote r(3l)n (�) = Pl(⇠), r(3l+1)

n (�) = �Ql(⇠), r(3l+2)n (�) = �2Rl(⇠),

where ⇠ = �3 and where Pl, Ql, Rl are monic polynomials of degree l.

Then in terms of Pl, Ql, Rl recurrence (2.3) has a form of the system:8><

>:

Pl(⇠) = ⇠Rl�1(⇠) + (n� 3l + 2)(n� 3l + 3)(3l � 2)(3l � 1)Pl�1(⇠)

Ql(⇠) = Pl(⇠) + (n� 3l + 1)(n� 3l + 2)(3l � 1)3lQl�1(⇠)

Rl(⇠) = Ql(⇠) + (n� 3l)(n� 3l + 1)3l(3l + 1)Rl�1(⇠)

(2.4)

with the initial conditions P0(⇠) = Q0(⇠) = R0(⇠) = 1. Here l runs from 1 to [n/3].

Proof. Simple algebra. ⇤To prove item (iii) of Proposition 3, we will use notation of Lemma 4. Hence Pk,

Qk, Rk correspond to q(0)k , q(1)k and q(2)k respectively. Our first step is to prove thateach polynomial in the recurrences has positive coe�cients and real roots. Moreoverthe roots of the polynomials appearing in the same recurrence are interlacing. Notethat, in our case, positive coe�cients imply that all roots are negative.


We will use induction on l. As the base of induction we check the case l = 2since l = 1 is trivial. Elementary calculations give ⇠R1 = ⇠2 + (84 � 8n + 20n2)⇠and P1 = ⇠ � 2n + 2n2, which implies that the roots of each of them are negativeand are interlacing for all positive integers n � 3. Hence, since P2 = ⇠R1 + 20(n�4)(n� 3)P1, then using Lemma 1.10 of [14] or conducting elementary calculations,we can conclude that P2 has real roots. Using similar elementary calculations weobtain similar results for the pairs P2, Q1 and Q2, R1.

Assume now that our hypothesis holds for a given positive integer l. Note that,in recurrence for l, degrees of the polynomials di↵er by one. This indicates thatthe largest root belongs to the polynomial with the larger degree. Furthermore byCorollary 1.30 of [14], we can derive the following results:

⇠Rl�1 Pl Pl�1 (2.5)

Pl Ql Ql�1 (2.6)

Ql Rl Rl�1. (2.7)

Here the arrow ” “ indicates that the corresponding pair of polynomials haveinterlacing roots with the largest root belonging to the polynomial at which thearrow points.

(a) Consider the recurrence

Pl+1 = ⇠Rl + (n� 3l � 1)(n� 3l)(3l + 1)(3l + 2)Pl.

We can rewrite ⇠Rl as ⇠Ql + (n � 3l)(n � 3l + 1)3l(3l + 1)⇠Rl�1. Observethat ⇠Rl�1 Pl by induction hypothesis and Corollary 1.30 in [14]. Using(2.2), we can conclude that ⇠Ql Pl since all roots of Pl are negative.Hence ⇠Rl Pl, by Lemma 1.31 in [14]. Therefore Pl+1 has real roots, byLemma 1.10 in [14]. Furthermore, since Pl and Rl have positive coe�cients,so does Pl+1, hence it has negative roots.

(b) Consider the recurrence

Ql+1 = Pl+1 + (n� 3l � 2)(n� 3l � 1)(3l + 2)(3l + 3)Ql.

One can rewrite Pl+1 as in part (a). By (2.2), Pl Ql and by (2.3),⇠Rl Ql since Rl and Ql only have negative roots. Proof is concluded bythe same argument as in part (a).

(c) Consider the recurrence

Rl+1 = Ql+1 + (n� 3l � 3)(n� 3l � 2)(3l + 3)(3l + 4)Rl.

One can rewrite Ql+1 as in part (b). By (2.3), Ql Rl. Pl+1 Rl bypart (a), Corollary 1.30 in [14] and the fact that both Pl+1 and Rl haveonly negative roots. Proof is concluded by the same argument as in parts(a) and (b).

The second step is to prove that all roots of the considered polynomials are simple.Assume that Pl has a double root p. Then by Corollary 1.30 in [14], p should alsobe a root of ⇠Rl�1 and Pl�1. Hence when we conduct our induction on l and assumethat ⇠Rl�1 and Pl�1 have only simple roots, we get a contradiction since Pl is alinear combination of those polynomials. Note that, for l = 2, we have only simplezeros from the latter formulas. The same argument applies to Ql and Rl. ⇤


Proof of Theorem 1 in case an = 0; n = 1, 2, 3, . . . . In order to apply the approachof [16], we need the variable recurrence coe�cients to stabilize when k

n ! ⌧ , forany fixed ⌧ 2 [0, 1]. This stabilization would indeed hold if instead of (2.2) oneconsiders the scaled recurrence

e�(k)n (�) = �� e�(k�1)

n (�) +(n� k + 2)(n� k + 3)(k � 1)(k � 2)

n4e�(k�3)n (�), (2.8)

which is obtained from (2.2) by using the scaling � = n4/3�. Then when kn ! ⌧,

the latter recurrence tends to following relation with constant coe�cients:

e�(k)⌧ (�) = �� e�(k�1)

⌧ (�) + (1� ⌧)2⌧2 e�(k�3)⌧ (�). (2.9)

Observe that (2.8) is the recurrence for the principal minors of the rescaled

matrix 1n4/3M

(0)n � �I given by:

1

n4/3M (0)

n � �I :=

0

BBBBBBBBB@

�� 0 2n4/3 0 0 · · · 0

nn4/3 �� 0 6

n4/3 0 · · · 00 n�1

n4/3 �� 0 12n4/3 · · · 0

......

. . .. . .

. . .. . .

...

0 0 · · · 3n4/3 �� 0 n(n�1)

n4/3

0 0 · · · 0 2n4/3 �� 0

0 0 · · · 0 0 1n4/3 ��

1

CCCCCCCCCA

. (2.10)

In other words, (2.8) is satisfied by the characteristic polynomials of the principal

minors of 1n4/3M

(0)n . Following [16], we conclude that the Cauchy transform of the

asymptotic root-counting measure for the polynomial sequencenSpn(0,�n

4/3)o=nn

4(n+1)3 · e�(n+1)

n (�)o

is obtained by averaging the Cauchy transforms of the asymptotic distributions of(2.9) over ⌧ 2 [0, 1].

In fact, in the case under consideration even the density of the former distributioncan be obtained by averaging the densities of the latter family of distributions ifwe argue as follows.

Recurrence (2.9) is similar to the one considered in the last section of [4] andhas very nice asymptotic distribution of its roots, see Figure 3. Observe that forany ⌧ 2 [0, 1], the initial conditions for (2.9) are taken as

e�(�2)⌧ (�) = e�(�1)

⌧ (�) = 0, e�(0)⌧ (�) = 1.

Since (2.9) has constant coe�cients, the support of the asymptotic root-countingmeasure of its solution is given by the well-known result of Beraha-Cahane-Weiss[2]. Namely, it coincides with the set of all � 2 C such that the characteristicequation

3 + � 2 � (1� ⌧)2⌧2 = 0 (2.11)

has two solutions with respect to which have the same modulus and, additionally,this modulus is the largest among all three solutions of (2.11)). From considerationsof [4] one can easily derive that, for any fixed ⌧ 2 [0, 1], this support is the union ofthree intervals starting at the origin and ending at the branching points of (2.11),


Figure 3. Root distributions of the solution to the recurrencerelation (2.9) for ⌧ = 1/4, ⌧ = 1/2, ⌧ = 3/4 and n = 150.

i.e., those values of � and ⌧ for which (2.11) has a multiple root with respect to ;the latter branching points are given by the equation:

�3 =27

4(1� ⌧)2⌧2. (2.12)

These branching points are located as shown in Figure 3. Observe that if for⌧ 2 [0, 1], �+(⌧) denotes the branching point lying on the positive half-axis, thenit attains its maximum at ⌧ = 1/2 and this maximum equals 3

4 .As we mentioned before, in the case under consideration all supports of (2.9)

lie on three fixed rays through the origin. Therefore, the density of the asymptoticroot-counting measure of the polynomial sequence

nSpn(0,�n

4/3)o=nn

4(n+1)3 e�(n+1)

n (�)o

is obtained by averaging the densities of (2.9) over ⌧ 2 [0, 1], comp. [12] and [16].Therefore, the support of the asymptotic root distribution for {Spn(0,�n4/3)} inthe �-plane is the union of three intervals of length 3

4 . ⇤

Proof of Theorem 1 in case when limn!1an

n2/3 = 0, We have to show that Propo-sition 3 holds asymptotically for any sequence {an} satisfying the condition

limn!1

ann2/3

= 0.

The principal minors of Mn(an)� �I given by:

M (an)n � �I :=

0

BBBBBBBBB@

�� an 2 0 0 · · · 0n �� 2an 6 0 · · · 00 n� 1 �� 3an 12 · · · 0...

.... . .

. . .. . .

. . ....

0 0 · · · 3 �� (n� 1)an n(n� 1)0 0 · · · 0 2 �� nan0 0 · · · 0 0 1 ��

1

CCCCCCCCCA

, (2.13)


satisfy the recurrence

�(k)n (an,�) = ��(k�1)

n (an,�) + (k � 1)(n� k + 2)an�(k�2)n (an,�)

+(n� k + 2)(n� k + 3)(k � 1)(k � 2)�(k�3)n (an,�), (2.14)

where k runs from 1 to n+ 1, with the initial conditions

�(�2)n (an,�) = �

(�1)n (an,�) = 0, �(0)

n (an,�) = 1.

To obtain a converging sequence of root-counting measures one has to consider

e�(k)n (an,�) = �� e�(k�1)

n (an,�)�(k � 1)(n� k + 2)an

n8/3e�(k�2)n (an,�)

+(n� k + 2)(n� k + 3)(k � 1)(k � 2)

n4e�(k�3)n (an,�), (2.15)

which is the recurrence for the principal minors of 1n4/3M

(an)n � �I given by:

0

BBBBBBBBB@

�� an

n4/32

n4/3 0 0 · · · 0n

n4/3 �� 2an

n4/36

n4/3 0 · · · 00 n�1

n4/3 �� 3an

n4/312

n4/3 · · · 0...

.... . .

. . .. . .

. . ....

0 0 · · · 3n4/3 �� (n�1)an

n4/3

n(n�1)n4/3

0 0 · · · 0 2n4/3 �� nan

n4/3

0 0 · · · 0 0 1n4/3 ��

1

CCCCCCCCCA

. (2.16)

When kn ! ⌧, the family (2.15) of recurrence relations converges to the earlier

family (2.9) corrsponding to the case an = 0 due to the fact that limn!1an

n2/3 =0. Therefore the measure obtained by averaging of the family of root-countingmeasures for recurrence relations with constant coe�cient is exactly the same asin case an = 0, i.e., it coincides with ⌫0. Additionally observe that by a generalresult of [16], the support of the asymptotic root-counting measure of the sequence

{e�(k)n (an,�)} can only be smaller than that of ⌫0 and their Cauchy transforms

coincide outside the support of ⌫0. Since the support of ⌫0 is the union of threestraight intervals through the origin the resulting asymptotic root-counting measurefor the sequence {Spn(an,�n4/3)} in case limn!1

an

n2/3 = 0 coincides with ⌫0. ⇤

Remark 4. Since the support of the limiting measure ⌫0 consists of three segmentsthrough the origin it is in principle possible to find integral formulas for the densityand the Cauchy transform of ⌫0 similar to those presented in [16], [12] and [23]. Inparticular, in the complement to the support of ⌫0 its Cauchy transform is givenby

C⌫0(�) =

Z 1

0

@ @� d⌧

,

where is the unique solution of (2.11) satisfying lim�!1 � = �1 and 0

� isits partial derivative with respect to �. However it seems di�cult to find eithera somewhat explicit expression for C⌫0(�) or a linear di↵erential operator withpolynomial coe�cients annihilating C⌫0(�). (Observe that such an operator alwaysexists since C⌫0(�) belongs to the Nilsson class, see [20]).


3. Case limn!1an

n2/3 = A 6= 0. “Proof” of Proposition 2

Similarly to the previous section, introduce the double-indexed sequence {�(k)n (a,�)}

of the characteristic polynomials of the principal minors of M (a)n , see (1.3). For any

fixed A, we obtain the corresponding recurrence relation of length 4:

�(k)n (An2/3,�) = ��(k�1)

n (An2/3,�)� (k � 1)(n� k + 2)An2/3�(k�2)n (An2/3,�)

+(n� k + 2)(n� k + 3)(k � 1)(k � 2)�(k�3)n (An2/3,�).

The same scaling by n4/3 of the eigenvalues of M (An2/3)n leads to

e�(k)n (An2/3,�) = �� e�(k�1)

n (An2/3,�)� (k � 1)(n� k + 2)A

n2e�(k�2)n (An2/3,�)

+(n� k + 2)(n� k + 3)(k � 1)(k � 2)

n4e�(k�3)n (An2/3,�), (3.1)

where � = �n�4/3. Observe that (3.1) is the recurrence relation for the sequence

of characteristic polynomials of the principal minors of 1n4/3M

(An2/3)n ��I given by

1

n4/3M (An2/3)

n ��I :=

0

BBBBBBBBBB@

�� An2/3

2n4/3 0 0 · · · 0

nn4/3 �� 2A

n2/36

n4/3 0 · · · 00 n�1

n4/3 �� 3An2/3

12n4/3 · · · 0

......

. . .. . .

. . .. . .

...

0 0 · · · 3n4/3 �� (n�1)A

n2/3

n(n�1)n4/3

0 0 · · · 0 2n4/3 �� nA

n2/3

0 0 · · · 0 0 1n4/3 ��

1

CCCCCCCCCCA

. (3.2)

Assuming that kn ! ⌧, we obtain that (3.1) tends to the following relation with

constant coe�cients:

e�(k)⌧ (�) = �� e�(k�1)

⌧ (�)� a⌧(1� ⌧)e�(k�2)⌧ + (1� ⌧)2⌧2 e�(k�3)

⌧ (�), (3.3)

whose characteristic equation is given by

3 + � 2 +A⌧(1� ⌧) � (1� ⌧)2⌧2 = 0. (3.4)

Typically the union over ⌧ 2 [0, 1] of the supports of the asymptotic root-counting

measures µ(⌧)A for the polynomial sequences (3.3) depending on ⌧ 2 [0, 1] is larger

than that of ⌫A, see Fig. 4. By the complex version of the main result of [16]corresponded to the first author by A. Kuijlaars in such case we can only concludethat the corresponding Cauchy transforms coincide in the complement to the largersupport. This complex version is worked out in details in § 6 below. The lattercircumstance implies that the measure

MA =

Z 1

⌧=0µ(⌧)A d⌧

is obtained as the balayage of measure ⌫A.

However, in case A � 33p4

(which fits the situation covered by the main result

of [12],) these supports coincide and one can obtain the density of ⌫A by averagingthe densities of (3.3).


Figure 4. The (blue) domain is (an approximation to) the unionover ⌧ 2 [0, 1] of the supports for the asymptotic root-counting

measures for the polynomial sequence {e�(k)⌧ (�)} defined by (3.3)

while the red curve is the zero locus of Sp200(An2/3,�n4/3) forA = (1 � i)/2 (upper left), A = 1 + i (upper right), and A = i/2(bottom) resp.

-2 -1 0 1-1.0

-0.5

0.0

0.5

1.0

Figure 5. The union over ⌧ 2 [0, 1] of the supports for {e�(k)⌧ (�)}

in case A = 3 together with 3 branching points given by (3.8).

Lemma 5. If A � 33p4

, then for ⌧ 2 [0, 1], each support of the asymptotic root-

counting measure given by the polynomial sequence defined by (3.3) is a real interval.This interval connects two branching points defined by (3.5).

Proof. Using the standard expression for the discriminant, one can check that allthree branching points of (3.4) with respect to satisfy the equation:

4�3 +A2�2 � 18A�⌧(1� ⌧) + ⌧(1� ⌧)(27⌧2 � 27⌧ � 4A3) = 0. (3.5)

To check that for A � 33p4

, and any ⌧ 2 [0, 1], all three solutions of (3.5) are real,

we calculate the discriminant of (3.5) with respect to �. Again using symbolicmanipulations, we get that this discriminant is given by:

Dsc := 16⌧(1� ⌧)(A3 � 27⌧ + 27⌧2)3.


Figure 6. Plot of A3 � 27⌧ + 27⌧2 = 0 in the real (A, ⌧)-plane.

For ⌧ 2 [0, 1], the graph of Dsc in the real (A, ⌧)-plane is presented in Fig. 6.One can easily check that the maximal value of A on this graph is obtained when⌧ = 1/2 and is equal to 3

3p4. At the same time, for A = 3 one can easily check that

all three roots of (3.5) are real. ⇤

Remark 5. We will show below that for A � 33p4

, the union of the supports=real

intervals from Lemma 5 coincides with the interval bounded by the two rightmostbranching points given by (3.8), see Fig 5.

Remark 6. Similarly to the case A = 0, for any A � 33p4

, one can represent the

Cauchy transform of ⌫A in the complement to its support (which is an intervalexplicitly given in Lemma 5) as

C⌫A(�) =

Z 1

0

@ @� d⌧

,

where is the unique solution of (3.4) satisfying the condition

lim�!1

�= �1.

(In an appropriate domain in C such presentation for the Cauchy transform of ⌫Ais valid for any complex A.)

”Proof“ of Proposition 2 under additional convergence assumptions given below. Toobtain the support of ⌫A we argue as follows. Assume that we have a (sub)sequence

�jn,n of the eigenvalues of fMn(An2/3) (one for each n) converging to some finitelimit denoted by ⇤. Denote by {pn} the corresponding (sub)sequence of eigenpoly-nomials. (These eigenpolynomials are classically referred to as Stieltjes polynomials


Figure 7. Root distributions of Sp200(An2/3,�n4/3) for A = (1�i)/2 (left), A = 4/5� 2i/3 (right) and A = 2/3� i (down) in the�-plane. Larger dots are the endpoints of the support given by(3.8).

and in special cases their properties are well studied in the literature). Each pnsatisfies its own di↵erential equation:

p00n � (x2 �An2/3)p0n + (nx� �jn,nn4/3)pn = 0.

First assumption. We assume that if the subsequence {�jn,n} has a finite limit ⇤,then the sequence {µn} of the root-counting measures of {pn} also weakly converges,after appropriate scaling of x, to some limiting measure a,⇤ whose support consistsof finitely many compact curves and points.

This assumption implies that the sequence of Cauchy transforms of scaled {µn}converges to the Cauchy transform of the limiting measure a,⇤. The appropriatescaling of x can be easy guessed from the latter equation. Namely, substitutingx = ⇥n1/3 and dividing the above equation by n4/3pn, we get the following relation:

d2pn

d2⇥

n2pn� (⇥2 �A)

dpn

d⇥

npn+ (⇥� �jn,n) = 0 (3.6)

with respect to the new independent variable⇥. Observe that the scaled logarithmic

derivativedpnd⇥npn

is the Cauchy transform of the root-counting measure of pn(⇥n1/3)with respect to the variable ⇥.

Second assumption. Assuming that the sequence {µn} of the root-counting mea-sures of {pn(⇥n1/3)} converges to a,⇤, we additionally assume that the sequencesof the root-counting measures of its first and second derivatives converge to thesame measure a,⇤.

(Apparently this assumption can be rigorously proved by using the same argu-ments presented in [4].)


Under the above two main assumptions, using (3.6), we get that the Cauchytransform CA,⇤ of A,⇤ satisfies the quadratic equation:

C2A,⇤ � (⇥2 �A)CA,⇤ + (⇥� ⇤) = 0 (3.7)

almost everywhere in C. Up to a variable change x $ ⇥ and ⇤ $ � the latterequation coincides with (1.4).

Third assumption. So far we presented a physics-style argument showing that

if a (sub)sequence {�jn,n} of the eigenvalues of fM (An2/3)n converges to some limit

⇤, then there exists a probability measure A,⇤ whose Cauchy transform satisfies(3.7) almost everywhere in the ⇥-plane. Our final assumption is that the converseto the latter statement is true as well, i.e., for each ⇤ with the above propertiesthere exists an appropriate subsequence {�jn,n} converging to ⇤. ⇤3.1. Quadratic equations with polynomial coe�cients and quadratic dif-

ferentials. The next result is a special case of Proposition 9 and Theorem 12 in[6].

Proposition 6. There exists a signed measure µA,⇤ whose Cauchy transform sat-isfies (3.7) almost everywhere if and only if the set of critical horizontal trajectoriesof the quadratic di↵erential

�((⇥2 �A)2 � 4(⇥� ⇤))d⇥2

contains all its turning points, i.e., all roots of P (⇥,⇤) = (⇥2 � A)2 � 4(⇥ � ⇤).(Here by a critical horizontal trajectory of a quadratic di↵erential we mean itshorizontal trajectory which starts and ends at the turning points.)

We know that the support of µA,⇤ should include all the branching points of(3.7) and consists of the critical horizontal trajectories of (1.5).

Lemma 7. The set of the critical values of the polynomial P (⇥,⇤) = ((⇥2�A)2�4(⇥�⇤)), i.e., the set of all ⇤ for which P (⇥,⇤) has a double root with respect to⇥ is given by the equation:

4⇤3 +A2⇤2 � 9A⇤/2�A3 � 27/16 = 0. (3.8)

Proof. Straight-forward calculations. ⇤Corollary 1. The endpoints of the support of ⌫A are contained among the threeroots of equation (3.5) when ⌧ = 1/2, see Fig. 7. This equation coincides with (3.8)where ⇤ is substituted by �.

4. On branching points and monodromy of the spectrum

Observe that, for any positive integer n and generic values of parameter a, theroots of Spn(a,�) with respect to � are simple. The latter roots are called thequasy-exactly solvable spectrum of the quartic oscillator under consideration.

Moreover, for any given n, and any su�ciently large positive a, these roots arereal and distinct. The set ⌃n ⇢ C of branching points of Spn(a,�), i.e., the setof all values of a for which two eigenvalues coalesce, has cardinality

�n+12

�. When

plotted these branching points form a regular pattern in the complex plane shownin Figures 2 and 8.

In this subsection we present our (mostly) numerical results and conjecturesabout the monodromy of the roots of Spn(a,�) when a runs along di↵erent closed


-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-2

-1

0

1

2

Figure 8. The triangle of the branching points for Sp10(a,�).

!20 0 20

!20

!10

0

10

20

!a

!20 0 20

!20

!10

0

10

20

!a

Figure 9. Roots of Sp8(a, 84/3�). The left figure shows the sit-uation with a = 500 exp(i'a),'a = 4⇡/5, the right one a =500 exp(i'a),'a = 6⇡/5. In both cases roots are almost uniformlydistributed on the interval [�

pa,pa].

paths in the complement to ⌃n in the a-plane. We start with the following state-ment.

Proposition 8. For any given n, if |a| ! 1 with arg a = � fixed, then the rootsof Spn(a,�) divided by n4/3 will be asymptotically uniformly distributed on thestraight segment [�

pa,+pa], see Fig. 9. In particular, if a runs over the circle

Re2⇡it, t 2 [0, 1] for any su�ciently large R, then the resulting monodromy of rootsof Spn(R,�) (which are all real) is the complete reversing of their order, i.e., theleftmost and the rightmost roots change places, the second from the left and thesecond from the right change places etc.

Proof of Proposition 8. Observe that if a = Ke2⇡i� with K very large then polyno-

mial n� 4(n+1)3 Spn(a,�n4/3), comp. (2.15),ı¿œ is close to cSpn(a,�) where cSpn(a,�)


is the characteristic polynomial of the tridiagonal matrix

cM (a)n :=

0

BBBBBBBBB@

0 a/n 0 0 0 · · · 0n/n 0 2a/n 0 0 · · · 00 (n� 1)/n 0 3a/n 0 · · · 0...

.... . .

. . .. . .

. . ....

0 0 · · · 3/n 0 (n� 1)a/n 00 0 · · · 0 2/n 0 na/n0 0 · · · 0 0 1/n 0

1

CCCCCCCCCA

. (4.1)

To make the situation more transparent, let us consider the sequence of character-

istic polynomials of 1pafM (a)

n and of 1pacM (a)

n . In other words, we are comparing the

roots of cSpn(a,�) divided bypa with that of cSpn(a,�) divided by

pa. The char-

acteristic polynomials of the respective principal minors of 1pafM (a)

n and of 1pacM (a)

n

satisfy the recurrences:

e�(k)n (�) = �� e�(k�1)

n (�)� (k � 1)(n� k + 2)

n8/3e�(k�2)n (�)

+(n� k + 2)(n� k + 3)(k � 1)(k � 2)

a3/2n4e�(k�3)n (�), (4.2)

and

b�(k)n (�) = �� b�(k�1)

n (�)� (k � 1)(n� k + 2)

n8/3b�(k�2)n (�) (4.3)

where � = �/pa and both recurrences have the standard boundary conditions:

e�(�1)n (�) = b�(�1)

n (�) = 0, e�(0)n (�) = b�(0)

n (�) = 1. As before fSpn(a, �) = e�(n)n (�)

and cSpn(a, �) = b�(n)n (�). Observe now that, for any fixed n and any ✏ > 0, one

can choose |a| so large that (due to the presence of a3/2 in the denominator ofthe third term in (4.2)) each equation in (4.2) for k = 1, 2, . . . , n deviated from

the corresponding equation in (4.3) so little that e�(n)n (�) � b�(n)

n (�) can be madearbitrary small coe�cientwise.

Now one can easily check by induction that bivariate polynomial cSpn(a,�) isquasihomogeneous with weight 1 for variable � and weight 2 for variable a. Whena is positive, then cSpn(a,�) is a real-rooted polynomial. Since multiplication of

� by e⇡it and multiplication of a by e2⇡it multiplies the whole cSpn(a,�) by a

constant, the roots of cSpn(a,�) with respect to � lie for any fixed a on the linethrough the origin whose slope is half of the slope of a. Consider the roots ofcSpn(a,�) with respect to �. Observe that if a = 1, then the spectrum of (4.1) is(�1,�1+ 2

n ,�1+4n , . . . , 1�

4n , 1�

2n , 1). The same argument as above gives that for

any a 6= 0 the roots of cSpn(a,�) will be equally spaced on the interval [�pa,pa].

Since choosing |a| su�ciently large, we can achieve that all roots of eSp(n)n (a, �) lie

arbitrarily close to that of bSp(n)n (a, �), and the latter roots are equally distributedon the interval [e�2⇡i�, e�2⇡i�], the result follows. ⇤

To describe our conjecture on the monodromy of the spectrum, let us introducea system of standard paths connecting a base point chosen as a su�ciently largepositive number with every branching point, see Fig. 10. Based on our numericalexperiments, we see that ⌃n has a triangular shape Fn with points regularly ar-ranged into columns and rows in C. There are n columns (enumerated from left to


base point

H1,2L

H1,2L

H1,2L

H1,2L

H2,3L

H2,3L

H2,3L

H3,4L

H3,4LH4,5L B

Figure 10. The system of standard paths and the monodromy(transpositions) of the spectrum which they produce, n = 4.

right) where the j-th column consists of n� j branching points with approximatelythe same real part and there are n rows (enumerated from bottom to top) wherethe i-th row consists of points with approximately the same imaginary part. Wedenote the branching points �i,j 2 ⌃n where i = 1, . . . , 2n � 1 is the row numberand j = 1, . . . , n is the column number.

Fixing a base point B, connect B with every �i,j by a ”vertical hook“ Pi,j , i.e.,move from B vertically to the imaginary part of �i,j , then move horizontally to theleft untill you almost hit �i,j , then circumgo �i,j counterclockwise along a smallcircle centered at �i,j and return back to B along the same path. Conjecturally,along such a path one will never hit any other branching points unless �i,j lies onthe real axis. In other words, the imaginary parts of all branching points except forthe real ones are all distinct. In case when �i,j is real one can slightly deform thesuggested path (which is a real interval) in an arbitrary way to move it away fromthe real axis. The resulting monodromy will (conjecturally) be independent of anysuch small deformation, see below.

Conjecture 2. For any �i,j 2 ⌃n and any su�ciently large positive base point B,the monodromy corresponding to the standard path Pi,j is a simple transposition(j, j + 1) of the roots of Spn(B,�) (which are all real) ordered from left to right.

This conjecture has been numerically checked for all n 10. Observe that sincethe system of standard paths gives a basis of the fundamental group ⇡1(C\⌃n), thenknowing the monodromy for the standard paths, one can calculate the monodromyalong any loop in C \ ⌃n based at B.

4.1. Asymptotic distribution of branching points of Spn(a,�) and Yablonskii-

Vorob’ev polynomials. There are two natural asymptotic regimes for the se-quence {⌃n} when n!1. In the first regime we just consider the sequence {⌃n}and study the stabilization of its portion in any fixed bounded domain of C whenn ! 1. In the second regime we consider the sequence {⌃n/n2/3} and study thelimiting density of zeros in the triangular shape F which is more and more denselyfilled by the roots of ⌃n/n2/3 when n ! 1. Numerical experiments show thatboth limits exist.

Conjecture 3. The sequence {⌃n} splits into three subsequences depending onn mod 3 and each subsequence stabilizes to its own hexagonal lattice in C, seeFig. 11.


Figure 11. Stabilization of the lattice, n = 34 and n = 37.

Conjecture 4. The sequences {⌃n/(274 )1/3n2/3} and { bZn/(

92 )

2/3n2/3} when n!1 asymptotically fills in the curvilinear triangular shape F, see Fig 2. The interiorof F is the set of all values a 2 C for which the support of ⌫a consists of threesmooth segments=legs with a common point, see Fig. 7 (left). The complement ofF consists of all values a 2 C for which the support of ⌫a is a single smooth segment,see Fig. 7 (down). The boundary of F consists of those a 2 C for which the supportof ⌫a is a single curve with a singularity, i.e., a belongs to the boundary betweenthe domain where this support has a tritronquee form and the domain where thesupport is a smooth single curve, see Fig. 7 (right).

Conjecture 4 complements Conjecture 1 from the Introduction. For the sequence{ bZn} parts of the latter conjecture have been settled in [9], see also [3].

5. Appendix I. Estimates for roots of Yablonskii-Vorob’ev

polynomials

We want to find estimates both from above and below as well as the asymptoticbehaviour for roots of maximal modulus of Yablonskii-Vorob’ev polynomials.

The Yablonskii-Vorob’ev polynomials satisfy the di↵erential-di↵erence relation

Qn+1(t) =tQn(t)2 � 4(Qn(t)Q

00

n(t)� (Q0

n(t))2)

Qn�1(t)(5.1)


with Q0(t) = 1 and Q1(t) = t. The roots of Qn cover approximately an equilateraltriangle, the edges, however, are rather curves than straight lines. The wholepattern is invariant under rotation through 2⇡/3 and reflections in the lines arg(t) =n⇡/3, for n = 0,±1,±2. The roots of maximal modulus lie on a circle with centrethe origin. These and other properties are listed in [11].

Our strategy is to connect roots and coe�cients of Yablonskii-Vorob’ev poly-nomials. Similar approach was used in [15]. Our results are sharper and we alsoderive the asymptotic formula. As the first step we form rational functions

pn(t) = (log(Qn(t)))00� t/4.

Another relation between pn and Qn is

pn+1(t) = �Qn(t)Qn+2(t)/(4Qn+1(t)2).

From this and from

Qn(t) =

n(n+1)/2X

k=1

(t� ↵n,k) =

n(n+1)/2X

k=0

cn,ktk

we see that the Laurent expansion at 1 for pn is convergent in |t| > An+1, whereAn := max |↵n,k|1kn(n+1)/2:

pn(t) = �t

4�

1X

j=0

(�1)jpn,jt�(3j+2). (5.2)

A closer inspection reveals thatpn,j

3j + 1= (�1)jsn,3j , (5.3)

where sn,j is the sum of jth powers of roots of Qn, i.e.

sn,j =

n(n+1)/2X

k=1

↵jn,k.

The second step is to find the relation between sn,j and cn,k, i.e.

sn,j = �jcn,n(n+1)/2�j �j�1X

`=1

cn,n(n+1)/2�`sn,j�`. (5.4)

To this end it is suitable to rewrite Qn(t) as

Qn(t) =

[n(n+1)/6]+1X

k=0

qk(n)t(n2+n�6k)/2. (5.5)

We can find explicit form of qn’s:

q0(n) =1

q1(n) =(n+ 2)(n+ 1)n(n� 1)/6

q2(n) =(n+ 5)(n+ 3)(n+ 2)(n+ 1)n(n� 1)(n� 2)(n� 4)/72

q3(n) =(n4 + 2n3 � 57n2 � 58n+ 1120)(n+ 4)(n+ 3)(n+ 2)(n+ 1)

n(n� 1)(n� 2)(n� 3)/1296

· · ·

(5.6)


Figure 12. Dependence of |An| on n (red dots) and the up-per (5.8) (k = 1, 2, 3, 10) (green asterisks) and lower (5.9) (k =1, 2, 4, 15) (blue asterisks) limits.

We get sn,j from (5.4). We see that sn,j 6= 0 only if j ⌘ 0mod 3.

sn,3 =� (n+ 2)(n+ 1)n(n� 1)/2

sn,6 =2(n2 + n� 5)(n+ 2)(n+ 1)n(n� 1)

sn,9 =� 4(n2 + n� 7)(3n2 + 3n� 20)(n+ 2)(n+ 1)n(n� 1)

sn,12 =8(11n6 + 33n5 � 259n4 � 573n3 + 2348n2 + 2640n� 7700)

(n+ 2)(n+ 1)n(n� 1)

· · ·

(5.7)

Now, we can limit |A3kn |. It is clear that sn,3k does not exceed n(n+1)|A3k

n |/2, i.e.

|An| 3ipsn,3k, k = 1, 2, . . . (5.8)

and clearly 3kpsn,3k 3(k�1)

psn,3(k�1). On the other hand, from (5.3)

|An| � 3k

s2pn,k

(3k + 1)n(n+ 1). (5.9)

Again,

3i

s2pn,i

(3k + 1)n(n+ 1)� 3(k�1)

s2pn,k�1

(3k � 2)n(n+ 1),

so that we have an infinite series of both upper and lower bounds on |An|. Thismakes possible to find the asymptotic behaviour of |An| when n!1.

Conserving only the leading term of (�1)3ksn,3k we arrive at a series

n2, 4n4, 24n6, 176n8, 1456n10, 13056n12, 124032n14, 1230592n16, . . .


The general term is

vk =(�1)k2k�(3k + 1)

�(k + 2)�(2k + 2)and

limk!1

3kpvk = 3

p�1 3

3p2n2/3.

Besides, the factor 3p�1 shows the direction in which the three roots of maximal

modulus move when n!1.Numerical results show the rate of convergence to the asymptotic value, cf. table.

n |An| 3n2/3/ 3p2� |An|

10 9.226620959867741 1.8254720 15.85575092198829 1.6883630 21.37667838759338 1.6126040 26.28885026360517 1.5606850 30.79511630027872 1.5214060 35.00327495653731 1.4899470 38.97923077181361 1.4637680 42.76697737664885 1.4414090 46.39772099260680 1.42191100 49.89460772047459 1.40467110 53.27540358774959 1.38922

6. Appendix II. Comments on the main result of [16]

We adopt the notation used in paper [16] by A. Kuijlaars and W. Van Assche:

limn/N!t

Xn,N = X

which denotes that for the doubly indexed sequence it holds that

limj!1

Xnj ,Nj = X,

for any {nj}j2N, {Nj}j2N ⇢ N such that Nj ! 1 and nj/Nj ! t, as j ! 1.Additionally, we occasionally add the meaning of the limit sense. For example,

w-limn/N!t

µn,N = µ

expresses the limit of the double indexed sequence of measures µn,N converging toµ in the weak? topology. Similarly,

s-limn/N!t

An,N = A

stands for the limit of the double indexed sequence of bounded operators An,N

converging to A strongly.Recall some well known facts from the theory of linear operators. First, for any

closed operator T on a Banach space, one has

k(T � z)�1k � 1

dist(z,�(T )), 8z 2 ⇢(T ).

On the other hand, recall that for a bounded operator B it holds

k(B � z)�1k 1

dist(z,DkBk), 8z, |z| > kBk,


where DkBk = {z 2 C | |z| kBk}. Let us remark that �(B) ⇢ DkBk.Let {an}n�1, {bn}n�0 ⇢ C and an 6= 0, 8n � 1. Further Jn denotes the (n+1)⇥

(n+ 1) Jacobi matrix of the form

Jn =

0

BBBBBBB@

b0 a1a1 b1 a2

a2 b2 a3. . .

. . .. . .

an�1 bn�1 anan bn

1

CCCCCCCA

.

Recall also that the polynomial sequence {pn}n�0 determined by recurrence

anpn�1(x) + bnpn(x) + an+1pn+1(x) = xpn(x), n � 0,

and initial conditions p�1 = 0 and p0 = 1, is related with Jn by formula

pn(x) =

nY

k=1

1

ak

!det (x� Jn�1) , n � 1.

Lemma 9. [16, Lem. 2.2] Let kJnk M , then��

pn(z)

an+1pn+1(z)

�� 1

dist(z,DM )=

1

|z|�M, 8z 2 C, |z| > M.

On the other hand, one has��

pn(z)

an+1pn+1(z)

�� 1

2|z| 8z 2 C, |z| > 3M.

Proof. By simple linear algebra,

pn(z)

an+1pn+1(z)=

det (z � Jn�1)

det (z � Jn)= hen, (z � Jn)

�1eni

(note that en stands for the (n + 1)-th vector of the standard basis of Cn+1).Consequently,��

pn(z)

an+1pn+1(z)

�� = |hen, (z�Jn)�1eni| k(z�Jn)�1k 1

dist(z,DkJnk) 1

dist(z,DM ),

whenever |z| > M .One the other hand,

|hen, (z � Jn)�1eni| =

1

|z|

��1 + hen,1X

k=1

z�kJkneni

�� 1

|z|

✓1� kJnk

|z|� kJnk

◆� 1

2|z| ,

whenever |z| � 3M . ⇤Observe that if Rn 2 Cn+1,n+1 is the permutation matrix determined by equa-

tions Rnek = en�k, 8k 2 {0, 1, . . . , n}, then

RnJnRn =

0

BBBBBBB@

bn anan bn�1 an�1

an�1 bn�2 an�2

. . .. . .

. . .a2 b1 a1

a1 b0

1

CCCCCCCA

,


det(Jn � z) = det(RnJnRn � z), and

hen, (Jn � z)�1eni = he0, (RnJnRn � z)�1e0i, 8z 2 ⇢(Jn).

If convenient, we identify matrix Rn with the operator Rn � 0 acting on `2(N0)In what follows, {an,N | n,N 2 N} ⇢ C, {bn,N | n 2 N0, N 2 N} ⇢ C, and

J(N) =

0

BBB@

b0,N a1,Na1,N b1,N a2,N

a2,N b2,N a3,N. . .

. . .. . .

1

CCCA

stands for the semi-infinite (complex) Jacobi matrix. Finally, Pn 2 B(`2(N0))denotes the orthogonal projection on span{e0, . . . , en}.

Proposition 10. Let

limn/N!t

an,N = A and limn/N!t

bn,N = B. (6.1)

Further assume that there exists ✏ > 0 such that

sup{kPnJ(N)Pnk | |n/N � t| < ✏} <1. (6.2)

Thens-limn/N!t

RnJ(N)Rn = J(A,B)

where J(A,B) 2 B(`2(N0)) stands for the Jacobi matrix with constant diagonal Band constant o↵-diagonal A.

Proof. Recall an easily verifiable statement: Let X be a Banach space, B 2 B(X )and Bn a uniformly bounded sequence of operators acting on X . If Bn'! B', asn!1, for all ' from a dense subset of X , then s-lim n!1 Bn = B.

Take arbitrary {nj}, {Nj} ⇢ N, Nj ! 1 and nj/Nj ! t, for j ! 1. Let us

denote temporarily Jj = RnjJ(Nj)Rnj Assumption (6.2) guarantees the existenceof j0 2 N such that

supj�j0

kPnjJ(Nj)Pnjk = supj�j0

kJjk <1.

Hence, taking into account the above statement, it su�ces to verify that

limj!1

Jjen = J(A,B)en, 8n 2 N0.

Take n 2 N0 and j > n, then

kJjen� J(A,B)enk2 = |anj�n+1,Nj �A|2 + |bnj�n,Nj �B|2 + |anj�n,Nj �A|2 ! 0,

for j !1, by assumption (6.1). If n = 0, set anj+1,Nj = 0 in the above equation.All in all, the claim is verified. ⇤

It is again quite easy to see that for any Bn, B 2 B(X ) such that supn kBnk Mand s-lim n!1 Bn = B, one has

s-limn!1

(Bn � z)�1 = (B � z)�1, 8z /2 DM

and the convergence is local uniform in z. Indeed, since

(B � z)�1 � (Bn � z)�1 = (Bn � z)�1(Bn �B)(B � z)�1


one obtains

k(B � z)�1'� (Bn � z)�1'k 1

|z|�Mk(Bn �B)(B � z)�1'k,

from which the strong convergence of resolvents follows. The local uniformnessfollows, for example, from the Mantel’s theorem.

The next statement is in fact a corollary of Proposition 10, however, it is also acomplex generalization of [16, Thm. 2.1]. Therefore we formulate it as a proposition.

Proposition 11. [16, Thm. 2.1] Let the assumptions (6.1) and (6.2) hold. Denotethe value of the supremum in (6.2) by M . Then

limn/N!t

hen, (z � PnJ(N)Pn)�1 eni =

2

z �B +p

(z �B)2 � 4A2

locally uniformly in C \DM .

Remark 7. Note that by (6.1),

M = sup{kPnJ(N)Pnk | |n/N � t| < ✏} � |B|+ 2|A|.

Proof. It follows from Proposition 10 that

s-limn/N!t

(z �RnJ(N)Rn)�1 = (z � J(A,B))�1

locally uniformly in z /2 DM . Hence,

limn/N!t

hen, (z � PnJ(N)Pn)�1 eni = lim

n/N!the0, (z �RnJ(N)Rn)

�1 e0i

= he0, (z � J(A,B))�1 e0i

locally uniformly in z /2 DM . It is a standard result that

he0, (z � J(A,B))�1 e0i =2

z �B +p

(z �B)2 � 4A2,

for all z /2 [B�2A,B+2A] (a line segment in C). To verify that one can show that

s-limn!1

PnJ(A,B)Pn = J(A,B)

together with the formula

he0, (z � Jn(A,B))�1 e0i =Un

�z�B2A

�

A Un+1

�z�B2A

� , 8z /2 [B � 2A,B + 2A]

where Jn(A,B) stands for the (n+1)⇥(n+1) truncation of J(A,B), i.e., J(A,B) =Jn(A,B)� 0, and

Un(x) =

�x+px2 � 1

�n+1 ��x�px2 � 1

�n+1

2px2 � 1

are Chebyshev polynomials of the second kind. ⇤

Recall that the main result of paper [16] in Theorem 1.4. which claims thefollowing.


Theorem 12. [16, Thm. 1.4] If {an,N | n,N 2 N} ⇢ R+, {bn,N | n 2 N0, N 2N} ⇢ R and {pn,N | n 2 N0, N 2 N} associated family of orthonormal polynomials.Further let non-negative a 2 C(R+) and real b 2 C(R+) be given, such that

limn/N!t

an,N = a(t) and limn/N!t

bn,N = b(t),

for all t > 0. Then for the family of {⌫n,N | n 2 N0, N 2 N} of root-countingmeasures of polynomials {pn,N | n 2 N0, N 2 N}, one has

w-limn/N!t

⌫n,N =1

t

Z t

0![b(s)�2a(s),b(s)+2a(s)]ds

where ![↵,�] is absolutely continuous measure supported on [↵,�] with density

d![↵,�]

dt=

1

⇡p

(� � t)(t� ↵),

if ↵ < �. If ↵ = �, ![↵,�] = �{↵}.

For t > 0, let us denote

�(t) =1

t

Z t

0![b(s)�2a(s),b(s)+2a(s)]ds.

In the proof of Theorem 12, authors prove that

limn/N!t

U⌫n,N (z) = U�(t)(z) (6.3)

locally uniformly in certain neighborhood of complex 1 (i.e., for |z| > M), whereUµ denotes the logarithmic potential of the (compactly supported) Borel measureµ. Under the assumptions of Theorem 12, supports of all measures ⌫n,N , for alln,N such that n/N is close to t, are included in a real interval [�M,M ]. Thisimplies that the limit relation (6.3) holds true for all z /2 [�M,M ] and hence foralmost all z 2 C (w.r.t. the Lebesgue measure). Under these conditions one canshow (following standard methods of Potential Theory - Widom’s lemma) the weakconvergence

w-limn/N!t

⌫n,N = �(t). (6.4)

However, in the general case of complex sequences an,N and bn,N , one can getonly the relation (6.3) outside a ball, |z| > M , and not for almost all z 2 C.This however does not imply the weak convergence (6.4). To our best knowledgenor the existence of the weak limit is guaranteed (only a subsequence, by Helly’stheorem). Thus, one can not expect the validity of Theorem 12 in the complexsetting. However, one can get at least the following.

Proposition 13. Let a 2 C([0,1)) and b 2 C([0,1)) be complex-valued functionsand

limn/N!t

an,N = a(t) and limn/N!t

bn,N = b(t),

for all t > 0. Then if the weak limit of root-counting measures

⌫ = w-limn/N!t

⌫n,N

exists, then ⌫ and �(t) are equipotential measures, i.e., their logarithmic potentialscoincide outside the union of their supports.


Remark 8. Note the functions a and b are assumed to be continuous in 0 (fromthe right). This additional condition simplifies the proof considerably and we do notaim here to achieve a full generality.

Proof. Note that coe�cients an,N and bn,N are uniformly bounded if n/N is re-stricted to a compact subsets of [0,1), as it follows from the assumptions. Taket > 0 and 0 < ✏ < t, then

sup{|bn,N | | |n/N � t| ✏}+ 2 sup{|an,N | | |n/N � t| ✏} <1.

Denote by Jn(N) the (n+1)⇥ (n+1) truncation of J(N). Consequently, condition(6.2) is fulfilled and let us denote the uniform bound of operators Jn(N), for |n/N�t| ✏, by M .

The following part proceeds analogously as the proof of [16, Thm. 1.4]. Since

det (z � Jn�1(N))

det (z � Jn(N))= hen, (z � Jn(N))�1eni,

one has

det (z � Jn(N)) =nY

k=0

1

hek, (z � Jk(N))�1eki, |z| > M.

Thus,

U⌫n,N (z) =1

n+ 1log |det (z � Jn(N))| = � 1

n+ 1

nX

k=0

log��hek, (z � Jk(N))�1eki

�� ,

or equivalently

U⌫n,N (z) = �Z 1

0log��he[ns], (z � J[ns](N))�1e[ns]i

�� ds, |z| > M.

As n/N ! t, one has [sn]/N ! st. Hence, by Proposition 11,

limn/N!t

he[ns], (z � J[ns](N))�1e[ns]i =2

z � b(st) +p(z � b(st))2 � 4a(st)2

.

Further, by Lemma 9, one has

1

2|z| ��he[ns], (z � J[ns](N))�1e[ns]i

�� 1

|z|�M,

for |z| > 3M . Consequently, the Lebesgue’s dominated convergence theorem appliesand we get

limn/N!t

U⌫n,N (z) =

Z 1

0log

��z � b(st)

2+

s✓z � b(st)

2

◆2

� a(st)2

��ds

=1

t

Z t

0log

��z � b(s)

2+

s✓z � b(s)

2

◆2

� a(s)2

��ds

for |z| > 3M . The function in the last integral is known to coincide with thelogarithmic potential of ![b(s)�2a(s),b(s)+2a(s)] at z. All in all, we obtained

U⌫(z) =1

t

Z t

0U![b(s)�2a(s),b(s)+2a(s)](z)ds = U�(t)(z),


for |z| > 3M . By the harmonicity of logarithmic potentials Uµ outsides the supportµ and Identity principle for harmonic functions the last equality can be extendedto all z /2 (supp ⌫ [ supp�(t)). ⇤

References

[1] C. Bender and S. Boettcher, Quasi-exactly solvable quartic potential, J. Phys. A 31 (1998),no. 14, L273-L277.

[2] S. Beraha, J. Kahane, N. J. Weiss, Limits of zeros of recursively defined families of polynomi-als, in “Studies in Foundations and Combinatorics”, pp. 213–232, Advances in MathematicsSupplementary Studies Vol. 1, ed. G.-C. Rota, Academic Press, New York, 1978.

[3] M. Bertola, T. Bothner, Zeros of large degree Vorob’ev-Yablonski polynomials via a Hankeldeterminant identity. Int. Math. Res. Not. IMRN 2015, no. 19, 9330D9399

[4] J. Borcea, R. Bøgvad, and B. Shapiro, On rational approximation of algebraic functions,Adv. Math. vol 204, issue 2 (2006) 448–480.

[5] J. Borcea, R. Bøgvad and B. Shapiro, Homogenized spectral pencils for exactly solvable op-erators: asymptotics of polynomial eigenfunctions, Publ. RIMS, vol 45 (2009) 525–568.

[6] R. Bøgvad, B. Shapiro, On motherbody measures with algebraic Cauchy transform, Enseign.Math. 62 (2016), no. 1-2, 117–142.

[7] A. Bourget, T.McMillen, On the distribution and interlacing of the zeros of Stieltjes polyno-mials, Proc. AMS 138 (2010), 3267–3275.

[8] A. Bourget, T.McMillen, A. Vargas, Interlacing and nonorthogonality of spectral polynomialsfor the Lame operator, Proc. AMS 137 (2009), 1699–1710.

[9] R. J. Buckingham, P. D. Miller, Large-degree asymptotics of rational Painleve II functions:noncritical behaviour. Nonlinearity 27 (2014), no. 10, 2489–2578.

[10] R. J. Buckingham, P. D. Miller, Large-degree asymptotics of rational Painleve II functions:critical behaviour. Nonlinearity 28 (2015), no. 6, 1539–1596.

[11] P. A. Clarkson, E. L. Mansfield, The second Painleve equation, its hierarchy and associatedspecial polynomials, Nonlinearity 16 (2003), R1–R26.

[12] E. Coussement, J. Coussement and W.Van Assche, Asymptotic zero distribution for a classof multiple orthogonal polynomials,

[13] A. Eremenko, A. Gabrielov, Quasi-exactly solvable quartic: elementary integrals and asymp-totics, J. Phys. A: Math. Theor. 44 (2011) 312001.

[14] S.Fisk, Polynomials, roots, and interlacing, arXiv:math/0612833.[15] Y. Kametaka, On Poles of the Rational Solution of the Toda Equation of Painleve-II Type,

Proc. Japan Acad. Ser A, 59(A) (1983) 358–360.[16] A. B. J. Kuijlaars, W. Van Assche, The asymptotic zero distribution of orthogonal polynomials

with varying recurrence coe�cients, J. Approx. Theory 99 (1999), 167–197.[17] D. Masoero, Poles of integrale tritronguee and anharmonic oscillators. A WKB approach. J.

Phys. A: Math. Theor., 43(9), 5201, 2010.[18] D. Masoero, V. De Benedetti, Poles of integrale tritronguee and anharmonic oscillators.

Asymptotic localization from WKB analysis. Nonlinearity 23:2501, 2010.[19] E. Mukhin, V. Tarasov, On conjectures of A. Eremenko and A. Gabrielov for quasi-exactly

solvable quartic. Lett. Math. Phys. 103 (2013), no. 6, 653–663.[20] N. Nilsson, Some growth and ramification properties of certain integrals on algebraic mani-

fold,. Ark. Mat. 5 1965 463–476 (1965).[21] M. Petkovsek, H. Zakrajsek, Pascal-like determinants are recursive, Adv. Appl. Math., 33

(2004), 431–450.[22] B. Shapiro, and M. Tater, Asymptotics of spectral polynomials, Acta Polytechnica vol 47(2-3)

(2007) 32–35.[23] B. Shapiro, and M. Tater, On spectral polynomials of the Heun equation. I, JAT, 162 (2010)

766–781.[24] K. Shin, Eigenvalues of PT-symmetric oscillators with polynomial potentials, J. Phys. A 38

(2005), no. 27, 6147–6166.[25] K. Strebel, Quadratic di↵erentials, Ergebnisse der Mathematik und ihrer Grenzgebiete, 5,

Springer-Verlag, Berlin, (1984), xii+184 pp.[26] B. Shapiro, K. Takemura, and M. Tater, On spectral polynomials of the Heun equation. II,

Comm. Math. Phys. 311(2) (2012), 277–300.


[27] M. Taneda, Remarks on the Yablonskii-Vorob’ev polynomials, Nagoya Math. J., 159 (2000),87–111.

[28] A. Vorob’ev, On rational solutions of the second Painleve equations, Di↵. Eqns (1) (1965),58–59 (in Russian).

[29] A. Yablonskii, On rational solutions of the second Painleve equation, Vesti Akad. Navuk.BSSR Ser. Fiz. Tkh. Nauk. 3 (1959), 30–35 (in Russian).

Department of Mathematics, Stockholm University, S-10691, Stockholm, Sweden

Email address: [email protected]

Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sci-

ences, 250 68 Rez near Prague, Czech Republic

Email address: [email protected]

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