Branko Najman 1949 – 1996

. – p.1/25

Zvonko and Branko (1980). – p.2/25

Branko and Zvonko (1980). – p.3/25

Zvonko and me (1980). – p.4/25

Branko ( ∼ 1983). – p.5/25

Branko . – p.6/25

Branko and me (Vienna, 1995)

. – p.7/25

Branko Najman:

Born December 17, 1949

PhD (Veselic) 1979:"A contribution to the Spectral Theory of the Klein-Gordon Equation"

Assistant Professor 1979–1984Associate Professor 1984–1988Full Professor 1988–August 1996

. – p.8/25

Branko Najman:

Born December 17, 1949

PhD (Veselic) 1979:"A contribution to the Spectral Theory of the Klein-Gordon Equation"

Assistant Professor 1979–1984Associate Professor 1984–1988Full Professor 1988–August 1996

Berkeley 1980–1982 (Kato)Tulane 1985–1986Calgary 1989–1990 (Binding, Ye)Bellingham 1992–1993 (Curgus)

. – p.8/25

Klein-Gordon equation (BOM):

• self-adjoint operators in Pontryagin spaces, spectral function,critical points

• selfadjoint operators in Krein spaces, regular/singular criticalpoint ∞, scale of spaces, QUP operators

• perturbation theory of isolated eigenvalues• variational principles• nonrelativistic limit, singular perturbation

. – p.9/25

Spectral Theory of the Klein-Gordon Equation

Heinz Langer

Vienna University of Technology

The Branko Najman Conference

Dubrovnik, May 2009

(joint work with Branko Najman and Christiane Tretter)

. – p.10/25

1. The Klein-Gordon Equation((

∂

∂t− ieq

)2

− ∆ +m2

)ψ = 0, ψ = ψ(x; t), x ∈ R

n, t ≥ 0

Initial condition: ψ(·; 0) = u0 ∈ L2(Rn)

m, e mass and charge of relativistic particle with spin 0,q potential of electrostatic field, h = c = 1.

. – p.11/25

1. The Klein-Gordon Equation((

∂

∂t− ieq

)2

− ∆ +m2

)ψ = 0, ψ = ψ(x; t), x ∈ R

n, t ≥ 0

Initial condition: ψ(·; 0) = u0 ∈ L2(Rn)

m, e mass and charge of relativistic particle with spin 0,q potential of electrostatic field, h = c = 1.

ψ(·; t) = u(t) ∈ L2(Rn) = H, t ≥ 0 or t ∈ R:

((d

dt− iV

)2

+H0

)u = 0, u(0) = u0, in Hilbert space H,

where

H0 self-adjoint operator in H, H0 ≥ m2 > 0,

V symmetric operator in H.

. – p.11/25

Substitution : x = u, y = −idu

dt, x =

(x

y

):

dx

dt= iA0x, A0 =

(0 I

H0 − V 2 2V

)

. – p.12/25

Substitution : x = u, y = −idu

dt, x =

(x

y

):

dx

dt= iA0x, A0 =

(0 I

H0 − V 2 2V

)

Substitution : x = u, y =

(−i

d

dt− V

)u, x =

(x

y

):

dx

dt= iA1x, A1 =

(V I

H0 V

)

. – p.12/25

Substitution : x = u, y = −idu

dt, x =

(x

y

):

dx

dt= iA0x, A0 =

(0 I

H0 − V 2 2V

)

Substitution : x = u, y =

(−i

d

dt− V

)u, x =

(x

y

):

dx

dt= iA1x, A1 =

(V I

H0 V

)

Problem: To find assumptions such that with the formal

Block Operator Matrices closed operators can be defined,

and to study their spectral properties

. – p.12/25

Main assumptions:

(i) domH1/20 ⊂ domV (⇐⇒ S := V H

−1/20 is bounded in H)

(needed to define operators)

(ii) S := V H−1/20 = S0 + S1, ‖S0‖ < 1, S1 is compact.

(needed to establish spectral properties)

. – p.13/25

Main assumptions:

(i) domH1/20 ⊂ domV (⇐⇒ S := V H

−1/20 is bounded in H)

(needed to define operators)

(ii) S := V H−1/20 = S0 + S1, ‖S0‖ < 1, S1 is compact.

(needed to establish spectral properties)

2. Operators in Krein spaces K1 and K2

Krein space K1 := H⊕H with inner product

[x,x′]1 := (x, y′) + (y, x′) = (Gx,x′), x = (x y)T, x′ = (x′ y′)T ∈ K1,

where G =

(0 I

I 0

).

A1 =

(V I

H0 V

)on

(domH0

domV

); formally: GA =

(I V

V H0

).

. – p.13/25

Theorem. If domH1/20 ⊂ domV then A1 is essentially self-adjoint in the

Krein space K1 with closure A1 given by

domA1 =

{(x

y

): x ∈ domH

1/20 , H

1/20 x+ S∗y ∈ domH

1/20

},

A1

(x

y

)=

(V x+ y

H1/20

(H

1/20 x+ S∗y

)).

. – p.14/25

Theorem. If domH1/20 ⊂ domV then A1 is essentially self-adjoint in the

Krein space K1 with closure A1 given by

domA1 =

{(x

y

): x ∈ domH

1/20 , H

1/20 x+ S∗y ∈ domH

1/20

},

A1

(x

y

)=

(V x+ y

H1/20

(H

1/20 x+ S∗y

)).

Scale of spaces: For α = 1/2 and α = 1/4 we use the Hilbert spaces

0 ≤ α ≤ 1 : Hα := dom Hα0 , ‖x‖α := ‖Hα

0 x‖, x ∈ Hα,

−1 ≤ α < 0 : Hα completion of H with respect to ‖Hα0 · ‖.

Remark. Duality between Hα and H−α, α ≥ 0:

(x, y) :=(Hα

0 x,H−α0 y

), x ∈ Hα, y ∈ H−α.

∈ H . – p.14/25

Krein space K2 := H1/4 ⊕H−1/4 with inner product

[x,x′]2 := (H1/40 x,H

−1/40 y′) + (H

−1/40 y,H

1/40 x′), x = (x y)T ∈ K2,

. – p.15/25

Krein space K2 := H1/4 ⊕H−1/4 with inner product

[x,x′]2 := (H1/40 x,H

−1/40 y′) + (H

−1/40 y,H

1/40 x′), x = (x y)T ∈ K2,

domA2 :=

{(x

y

)∈ H1/4 ⊕H−1/4 : x ∈ domH

1/20 , y ∈ H,

V x+ y ∈ H1/4, H0x+ V y ∈ H−1/4

},

A2

(x

y

):=

(V x+ y

H0x+ V y

).

. – p.15/25

Krein space K2 := H1/4 ⊕H−1/4 with inner product

[x,x′]2 := (H1/40 x,H

−1/40 y′) + (H

−1/40 y,H

1/40 x′), x = (x y)T ∈ K2,

domA2 :=

{(x

y

)∈ H1/4 ⊕H−1/4 : x ∈ domH

1/20 , y ∈ H,

V x+ y ∈ H1/4, H0x+ V y ∈ H−1/4

},

A2

(x

y

):=

(V x+ y

H0x+ V y

).

L(λ) := I −(S − λH

−1/20

)(S∗ − λH

−1/20

)

. – p.15/25

Krein space K2 := H1/4 ⊕H−1/4 with inner product

[x,x′]2 := (H1/40 x,H

−1/40 y′) + (H

−1/40 y,H

1/40 x′), x = (x y)T ∈ K2,

domA2 :=

{(x

y

)∈ H1/4 ⊕H−1/4 : x ∈ domH

1/20 , y ∈ H,

V x+ y ∈ H1/4, H0x+ V y ∈ H−1/4

},

A2

(x

y

):=

(V x+ y

H0x+ V y

).

L(λ) := I −(S − λH

−1/20

)(S∗ − λH

−1/20

)

Theorem. If domH1/20 ⊂ domV and ρ(L) 6= ∅ then A2 is

self-adjoint in the Krein space K2.

. – p.15/25

3. Spectral Properties of A1 and A2.

From now on we always suppose that:

dom H1/20 ⊂ dom V ,

S := V H−1/20 = S0 + S1 with ‖S0‖ < 1 and S1 is compact.

It is also assumed that H0 ≥ m2 > 0.

ThenI − S∗S = I − (S0 + S1)

∗(S0 + S1) = I − S∗0S0 +K

with K compact

=⇒number κ of negative eigenvalues of I − S∗S is finite.

. – p.16/25

3. Spectral Properties of A1 and A2.

From now on we always suppose that:

dom H1/20 ⊂ dom V ,

S := V H−1/20 = S0 + S1 with ‖S0‖ < 1 and S1 is compact.

It is also assumed that H0 ≥ m2 > 0.

ThenI − S∗S = I − (S0 + S1)

∗(S0 + S1) = I − S∗0S0 +K

with K compact

=⇒number κ of negative eigenvalues of I − S∗S is finite.

Proposition. The number of negative squares of the Hermitian forms

[Ajx,x′]j , x,x

′ ∈ domAj , in Kj , j = 1, 2,

is finite and equals the number κ of negative eigenvalues of the opera-

tor I − S∗S.. – p.16/25

Theorem. The self-adjoint operator Aj , j = 1, 2, is definitizable in theKrein space Kj, hence the nonreal spectrum of Aj is symmetric to thereal axis and consists of at most κ pairs of eigenvalues λ, λ of finitetype; the eigenspaces at λ and λ are isomorphic.

. – p.17/25

Theorem. The self-adjoint operator Aj , j = 1, 2, is definitizable in theKrein space Kj, hence the nonreal spectrum of Aj is symmetric to thereal axis and consists of at most κ pairs of eigenvalues λ, λ of finitetype; the eigenspaces at λ and λ are isomorphic. Moreover:

(1) For the essential spectrum of Aj we have with α := (1 − ‖S0‖)m:

σess(Aj) ⊂ R \ (−α, α).

. – p.17/25

Theorem. The self-adjoint operator Aj , j = 1, 2, is definitizable in theKrein space Kj, hence the nonreal spectrum of Aj is symmetric to thereal axis and consists of at most κ pairs of eigenvalues λ, λ of finitetype; the eigenspaces at λ and λ are isomorphic. Moreover:

(1) For the essential spectrum of Aj we have with α := (1 − ‖S0‖)m:

σess(Aj) ⊂ R \ (−α, α).

(2) If V is H1/20 -compact (S0 = 0), then

σess(Aj) = {λ ∈ R : λ2 ∈ σess(H0)} ⊂ R \ (−m,m).

. – p.17/25

Theorem. The self-adjoint operator Aj , j = 1, 2, is definitizable in theKrein space Kj, hence the nonreal spectrum of Aj is symmetric to thereal axis and consists of at most κ pairs of eigenvalues λ, λ of finitetype; the eigenspaces at λ and λ are isomorphic. Moreover:

(1) For the essential spectrum of Aj we have with α := (1 − ‖S0‖)m:

σess(Aj) ⊂ R \ (−α, α).

(2) If V is H1/20 -compact (S0 = 0), then

σess(Aj) = {λ ∈ R : λ2 ∈ σess(H0)} ⊂ R \ (−m,m).

(3) If ‖V H−1/20 ‖ < 1 (S1 = 0), then Aj is uniformly positive in the

Krein space Kj and, with α := (1 − ‖V H−1/20 ‖)m,

σ(Aj) ⊂ R \ (α,−α).

. – p.17/25

Theorem. The self-adjoint operator Aj , j = 1, 2, is definitizable in theKrein space Kj, hence the nonreal spectrum of Aj is symmetric to thereal axis and consists of at most κ pairs of eigenvalues λ, λ of finitetype; the eigenspaces at λ and λ are isomorphic. Moreover:

(1) For the essential spectrum of Aj we have with α := (1 − ‖S0‖)m:

σess(Aj) ⊂ R \ (−α, α).

(2) If V is H1/20 -compact (S0 = 0), then

σess(Aj) = {λ ∈ R : λ2 ∈ σess(H0)} ⊂ R \ (−m,m).

(3) If ‖V H−1/20 ‖ < 1 (S1 = 0), then Aj is uniformly positive in the

Krein space Kj and, with α := (1 − ‖V H−1/20 ‖)m,

σ(Aj) ⊂ R \ (α,−α).

⊃ σess(Aj)

⊃±√σ(H0)

•

••λ1

•λ2

• •· · ·

0 ‖S0‖m‖S0‖m

−m m. – p.17/25

⊃ σess(Aj)

⊃±√σ(H0)

•

••λ1

•λ2

• •· · ·

0 ‖S0‖m‖S0‖m

−m m

. – p.18/25

⊃ σess(Aj)

⊃±√σ(H0)

•

••λ1

•λ2

• •· · ·

0 ‖S0‖m‖S0‖m

−m m

(4) The number of positive eigenvalues of Aj that have a nonpositiveeigenvector plus the number of negative eigenvalues of Aj thathave a nonnegative eigenvector plus the number of all eigenvaluesin C

+ is at most κ.

. – p.18/25

⊃ σess(Aj)

⊃±√σ(H0)

•

••λ1

•λ2

• •· · ·

0 ‖S0‖m‖S0‖m

−m m

(4) The number of positive eigenvalues of Aj that have a nonpositiveeigenvector plus the number of negative eigenvalues of Aj thathave a nonnegative eigenvector plus the number of all eigenvaluesin C

+ is at most κ.(5) With the exception of the real eigenvalues in (4), all the spectrum

of Aj on the positive half axis is of positive type and all thespectrum on the negative half axis is of negative type.

. – p.18/25

⊃ σess(Aj)

⊃±√σ(H0)

•

••λ1

•λ2

• •· · ·

0 ‖S0‖m‖S0‖m

−m m

(4) The number of positive eigenvalues of Aj that have a nonpositiveeigenvector plus the number of negative eigenvalues of Aj thathave a nonnegative eigenvector plus the number of all eigenvaluesin C

+ is at most κ.(5) With the exception of the real eigenvalues in (4), all the spectrum

of Aj on the positive half axis is of positive type and all thespectrum on the negative half axis is of negative type.

Distinction between A1 and A2:Theorem. Infinity is a regular critical point of A2, but in general asingular critical point of A1.

. – p.18/25

Theorem. The operator A2 is the infinitesimal generator of a stronglycontinuous group

(eitA2

)t∈R

in the Krein space K2. If x0 ∈ dom A2, theCauchy problem

dx(t)

dt= iA2x(t), x(0) = x0,

has a unique classical solution given by

x(t) = eitA2x0, t ∈ R.

. – p.19/25

4. Klein-Gordon Gleichung in Rn.

H = L2(Rn), H0 = −∆ +m2, Hα = W 2α

2 (Rn), α ∈ [0, 1].V = eq maximal operator of multiplication by eq : R

n → R.

. – p.20/25

4. Klein-Gordon Gleichung in Rn.

H = L2(Rn), H0 = −∆ +m2, Hα = W 2α

2 (Rn), α ∈ [0, 1].V = eq maximal operator of multiplication by eq : R

n → R.

Assumption (i): dom H1/20 ⊂ dom V now becomes

∃a, b ≥ 0 : ‖V u‖2 ≤ a‖u‖2 + b‖(−∆ +m2)1/2u‖2, u ∈ W 12 (Rn).

. – p.20/25

4. Klein-Gordon Gleichung in Rn.

H = L2(Rn), H0 = −∆ +m2, Hα = W 2α

2 (Rn), α ∈ [0, 1].V = eq maximal operator of multiplication by eq : R

n → R.

Assumption (i): dom H1/20 ⊂ dom V now becomes

∃a, b ≥ 0 : ‖V u‖2 ≤ a‖u‖2 + b‖(−∆ +m2)1/2u‖2, u ∈ W 12 (Rn).

Assumption (ii): V = V0 + V1 such that

∃a, b ≥ 0 : ‖V0u‖2 ≤ a‖u‖2 + b‖(−∆ +m2)1/2u‖2, u ∈ W 12 (Rn),

with a/m+ b < 1 and V1 is (−∆ +m2)1/2-compact.

. – p.20/25

4. Klein-Gordon Gleichung in Rn.

H = L2(Rn), H0 = −∆ +m2, Hα = W 2α

2 (Rn), α ∈ [0, 1].V = eq maximal operator of multiplication by eq : R

n → R.

Assumption (i): dom H1/20 ⊂ dom V now becomes

∃a, b ≥ 0 : ‖V u‖2 ≤ a‖u‖2 + b‖(−∆ +m2)1/2u‖2, u ∈ W 12 (Rn).

Assumption (ii): V = V0 + V1 such that

∃a, b ≥ 0 : ‖V0u‖2 ≤ a‖u‖2 + b‖(−∆ +m2)1/2u‖2, u ∈ W 12 (Rn),

with a/m+ b < 1 and V1 is (−∆ +m2)1/2-compact.

Example (Coulomb potential): Assumption (ii) is satisfied for

V (x) =γ

|x|+ V1(x), x ∈ R

n \ {0},

with n ≥ 3, γ ∈ R, |γ| < (n− 2)/2, and V1 ∈ Lp(Rn), n ≤ p <∞.

. – p.20/25

4. Klein-Gordon Gleichung in Rn.

H = L2(Rn), H0 = −∆ +m2, Hα = W 2α

2 (Rn), α ∈ [0, 1].V = eq maximal operator of multiplication by eq : R

n → R.

Assumption (i): dom H1/20 ⊂ dom V now becomes

∃a, b ≥ 0 : ‖V u‖2 ≤ a‖u‖2 + b‖(−∆ +m2)1/2u‖2, u ∈ W 12 (Rn).

Assumption (ii): V = V0 + V1 such that

∃a, b ≥ 0 : ‖V0u‖2 ≤ a‖u‖2 + b‖(−∆ +m2)1/2u‖2, u ∈ W 12 (Rn),

with a/m+ b < 1 and V1 is (−∆ +m2)1/2-compact.

Example (Coulomb potential): Assumption (ii) is satisfied for

V (x) =γ

|x|+ V1(x), x ∈ R

n \ {0},

with n ≥ 3, γ ∈ R, |γ| < (n− 2)/2, and V1 ∈ Lp(Rn), n ≤ p <∞.

For details and more examples (e.g. Rollnik and bounded potentials):H.L, B.Najman, C.Tretter:Proc.Edinburgh Math.Soc.51(2008),711-750

. – p.20/25

5. Semi-groups associated with A1.Assume in this section that V is bounded and can be decomposed asV = V0 + V1 with

‖V0‖ < m, V1H−1/20 is compact.

κ – number of negative eigenvalues of I − S∗S = I −H−1/20 V 2H

−1/20

A1 =

(V I

H0 V

)in K1 = H⊕H, inner product gen. by G =

(0 I

I 0

).

. – p.21/25

5. Semi-groups associated with A1.Assume in this section that V is bounded and can be decomposed asV = V0 + V1 with

‖V0‖ < m, V1H−1/20 is compact.

κ – number of negative eigenvalues of I − S∗S = I −H−1/20 V 2H

−1/20

A1 =

(V I

H0 V

)in K1 = H⊕H, inner product gen. by G =

(0 I

I 0

).

Proposition. Let µ ∈ ρ(A1) ∩ R. Then there exists a maximal nonne-gative subspace L+ ⊂ K1 that is invariant under (A1 − µ)−1 and suchthat

Im σ((A1 − µ)−1

∣∣L+

)≤ 0.

dimL+ ∩ L[⊥]+ ≤ κ, L+ can be chosen such that it contains all algebraic

eigenspaces of A1 in C+.

. – p.21/25

L+ admits angular operator representation

L+ =

{(x

Kx

): x ∈ domK

}

with a maximal accretive operator K in H: Re(Kx, x) ≥ 0, x ∈ domK.

. – p.22/25

L+ admits angular operator representation

L+ =

{(x

Kx

): x ∈ domK

}

with a maximal accretive operator K in H: Re(Kx, x) ≥ 0, x ∈ domK.

Theorem. Under the above assumptions on V there exists a maximalaccretive operator K in H such that, with the semi-group

(T (τ)

)τ≥0

given by T (τ) := e−τ(K+V ), τ ≥ 0, for any initial valuev0 ∈ dom

((K + V )2

), the function

v(τ) := T (τ)v0, τ ≥ 0,

is a classical solution of the Cauchy problem

v(τ) + 2V v(τ) + V 2v(τ) −H0v(τ) = 0, τ ≥ 0, v(0) = v0.

σ(K + V ) ⊂ C+ ∪ R contains all eigenvalues of A1 in C

+, the spectralpoints of positive type of A1 and the real eigenvalues of A1 that are notof negative type.

. – p.22/25

6. The Pontryagin space operator A0

Suppose again that S := V H−1/20 = S0 + S1, ‖S0‖ < 1, S1 compact.

Additional assumption: 0 /∈ σp(I − S∗S)

A0 =

(0 I

H0 − V 2 2V

)

. – p.23/25

6. The Pontryagin space operator A0

Suppose again that S := V H−1/20 = S0 + S1, ‖S0‖ < 1, S1 compact.

Additional assumption: 0 /∈ σp(I − S∗S)

A0 =

(0 I

H0 − V 2 2V

)

We introduce the space G := H1/2 ⊕H(

= dom H1/20 ⊕H

)with norm

‖x‖G =(‖H

1/20 x‖2 + ‖y‖2

)1/2

and the (in general indefinite) inner product (‘energy inner product’)

〈x,x′〉 :=((I − S∗S)H

1/20 x,H

1/20 x′

)+ (y, y′), x,x′ ∈ G.

(G, 〈·, ·〉) is a Pontryagin space with index of negativity κ

(a Hilbert space if ‖S‖ < 1).

. – p.23/25

The operator

H := H1/20 (I−S∗S)H

1/20 , dom H =

{x ∈ H1/2 : (I−S∗S)H

1/20 x ∈ H1/2

},

is self-adjoint and boundedly invertible in H.

. – p.24/25

The operator

H := H1/20 (I−S∗S)H

1/20 , dom H =

{x ∈ H1/2 : (I−S∗S)H

1/20 x ∈ H1/2

},

is self-adjoint and boundedly invertible in H.With A0 we associate the block operator matrix

A0 :=

(0 I

H 2V

), dom A0 = domH ⊕ dom H

1/20 .

Theorem. The operator A0 is self-adjoint in the Pontryagin space G. Itgenerates a strongly continuous group

(eitA0

)t∈R

of unitary operatorsin G and hence the Cauchy problem

dx

dt= iA0x, x(0) = x0,

has the unique solution x(t) = eitA0x0, t ∈ R, for all initial values x0 ∈ G.

. – p.24/25

H.L., B. Najman, C. Tretter: Spectral Theory of the Klein–GordonEquation in Pontryagin spaces. Comm. Math. Phys. 267 (2006),159–180.H.L., B. Najman, C. Tretter: Spectral Theory of the Klein–GordonEquation in Krein spaces. Proc. Edinburgh Math. Soc. 51(2008),711–750.

. – p.25/25

H.L., B. Najman, C. Tretter: Spectral Theory of the Klein–GordonEquation in Pontryagin spaces. Comm. Math. Phys. 267 (2006),159–180.H.L., B. Najman, C. Tretter: Spectral Theory of the Klein–GordonEquation in Krein spaces. Proc. Edinburgh Math. Soc. 51(2008),711–750.

P. Jonas

. – p.25/25

H.L., B. Najman, C. Tretter: Spectral Theory of the Klein–GordonEquation in Pontryagin spaces. Comm. Math. Phys. 267 (2006),159–180.H.L., B. Najman, C. Tretter: Spectral Theory of the Klein–GordonEquation in Krein spaces. Proc. Edinburgh Math. Soc. 51(2008),711–750.

P. Jonas

Ongoing work:

M.Langer, C.Tretter: Variational principles

M.Winkelmeier, M.Koppen: Simplicity of ground states

. – p.25/25