Perturbation of eigenvalues of Klein-Gordon operators. Ivica Nakić

Transcription

1 Perturbation of eigenvalues of Klein-Gordon operators Ivica Nakić Department of Mathematics, University of Zagreb (based on the joint work with K. Veselić) OTIND, Vienna, December 2016 Funded by the HRZZ project 9345

2 Se ng Klein-Gordon equation with a (real) scalar time invariant potential V with mass parameter m (normalized, c = 1): (i t V)2 ψ = ( + m 2 )ψ. Abstract Klein-Gordon equation (formal): (i t V)2 ψ = U 2 ψ, with U, V operators on a Hilbert space H, U selfadjoint and (uniformly) positive, V symmetric. 2

3 Aim and methods Aim: obtain relative perturbation bounds for the eigenvalues of the Klein-Gordon operator where we perturb the potential V V + δv: κ δλ λ κ Relative perturbation bounds are scale invariant. Methods: define operators via forms (B. Simon s Hamiltonians as quadratic forms), use symmetry of Klein-Gordon operator in indefinite geometry. We will work in the so called number or charge norm, which is the physically relevant one. 3

4 A li le bit of a historical background The idea to treat Klein-Gordon as a self-adjoint operator in an indefinite inner product setting goes back almost 50 years. Pioneering work of K. Veselić was published in Since then a number of people contributed to this topic, among them: K. Veselić H. Langer B. Najman B. Ćurgus P. Jonas C. Tretter Two papers by Langer, Najman, Tretter from 2006 and

5 Assump ons, main objects Assumptions: D(U) D(V), there exists µ R such that (V µ)u 1 =: η < 1. Quadratic pencil form: Linearisation form: q(λ)(ψ, φ) = (Uψ, Uφ) ( (V λ)ψ, (V λ)φ ), λ C. h (( ψ 1 ψ 2 ), ( φ 1 φ 2 ) ) = (VU 1 U 1/2 ψ 1, U 1/2 φ 1 ) + (U 1/2 ψ 2, U 1/2 φ 1 )+ D(q(λ)) = D(U), D(h) = D(U 1/2 ) D(U 1/2 ). (U 1/2 ψ 1, U 1/2 φ 2 ) + (U 1/2 ψ 2, VU 1 U 1/2 φ 2 ). 5

6 Assump ons, main objects (V µ)u 1 < 1 implies q(µ) > 0. q(λ) is closed and lower semi-bounded for λ R (need not be sectorial for λ C \ R). 6

7 Assump ons, main objects (V µ)u 1 < 1 implies q(µ) > 0. q(λ) is closed and lower semi-bounded for λ R (need not be sectorial for λ C \ R). Let then J = [ ] 0 I, I 0 h (( ψ 1 ψ 2 ), ( φ 1 φ 2 ) ) = g (( ψ 1 ψ 2 ), J ( φ 1 φ 2 ) ), where g is a symmetric form. The form h comes from the formal substitutions ψ 1 = U 1/2 ψ, ψ 2 = U 1/2 (i t V)ψ. Assumption (V µ)u 1 < 1 implies g is closed and g > µ. 6

8 From forms to operators Using Kato s representation theorems we construct corresponding operators Q(λ) for all λ C where q(λ) is sectorial, as well as operators G and H = JG. 7

9 From forms to operators Using Kato s representation theorems we construct corresponding operators Q(λ) for all λ C where q(λ) is sectorial, as well as operators G and H = JG. Q(λ) is a form sum of U 2 (V λ) 2. Seen as a quadratic pencil, the spectrum of Q is contained in the real line. 7

10 From forms to operators Using Kato s representation theorems we construct corresponding operators Q(λ) for all λ C where q(λ) is sectorial, as well as operators G and H = JG. Q(λ) is a form sum of U 2 (V λ) 2. Seen as a quadratic pencil, the spectrum of Q is contained in the real line. Operator H is selfadjoint in the corresponding Kreĭn space with the fundamental symmetry J. H is definitizable and is a regular critical point. Hence H has real spectrum and is similar to a selfadjoint operator on the Hilbert space H. 7

11 From forms to operators Using Kato s representation theorems we construct corresponding operators Q(λ) for all λ C where q(λ) is sectorial, as well as operators G and H = JG. Q(λ) is a form sum of U 2 (V λ) 2. Seen as a quadratic pencil, the spectrum of Q is contained in the real line. Operator H is selfadjoint in the corresponding Kreĭn space with the fundamental symmetry J. H is definitizable and is a regular critical point. Hence H has real spectrum and is similar to a selfadjoint operator on the Hilbert space H. Spectra of Q and H: Point spectra coincide and the spectrum of Q is contained in the spectrum of H. 7

12 Representa ons of Q and H Set We have for λ R: [ ] A λ = (V λ)u 1 Aλ I, A λ =, U = I A λ Q(λ) = U(A λa λ I)U, H λ = U 1/2 A λ U 1/2. [ ] U 0. 0 U 8

13 Perturba on Perturbation V = V + δv. We assume D(δV) D(U), δvu 1 < 1 η. This means that we perturb the form h by the form δh(ψ, φ) = ( δau 1/2 ψ, U 1/2 φ ). Here δa = [ ] δvu (δvu 1 ). The new operator H has the same properties as H, they have the same form domain. 9

14 Similarity to a selfadjoint operator To simplify notation from now on we assume µ = 0. Using the fact that H and H are similar to selfadjoint operators on H, we can show even more: H is similar to G 1/2 JG 1/2. H is similar to G 1/2 JG 1/2. So instead of working with selfadjoint operators in Kreĭn space, we can work with selfadjoint operators in Hilbert space which have a nice factorization representation. 10

15 Spectral gaps Let the perturbation δv be such that δvu 1 < 1 η and set κ = 1 1 η δvu 1. 11

16 Spectral gaps Let the perturbation δv be such that δvu 1 < 1 η and set κ = 1 1 η δvu 1. If (a, b) ρ(h), 0 < a < b, then ((1 + κ)a, (1 κ)b) ρ(h ). If (a, b) ρ(h), a < b < 0, then ((1 κ)a, (1 + κ)b) ρ(h ). If (a, b) ρ(h), a < 0, b > 0, then ((1 κ)a, (1 κ)b) ρ(h ). This gap cannot vanish. 11

17 Spectral gaps Let the perturbation δv be such that δvu 1 < 1 η and set κ = 1 1 η δvu 1. If (a, b) ρ(h), 0 < a < b, then ((1 + κ)a, (1 κ)b) ρ(h ). If (a, b) ρ(h), a < b < 0, then ((1 κ)a, (1 + κ)b) ρ(h ). If (a, b) ρ(h), a < 0, b > 0, then ((1 κ)a, (1 κ)b) ρ(h ). This gap cannot vanish. Essential spectral gaps: the same inclusions hold if we change ρ( ) to R \ σ ess ( ). 11

18 Rela ve perturba on es mates for eigenvalues Again the perturbation δv is such that δvu 1 < 1 η and we set κ = 1 1 η δvu 1. Let σ ± be the infimum/supremum of positive/negative spectrum of the operator H, and λ ± i, i N be the positive/negative eigenvalues of H ordered in the increasing/decreasing way, if positive/negative eigenvalues are smaller/larger than σ ± and λ ± i = σ ± otherwise. Analogous notation for H. 12

19 Rela ve perturba on es mates for eigenvalues σ λ i 0 λ + i σ + σ λ i 0 λ + i σ + Then we have κ λ + i κ λ i λ + i λ + i λ i λ i κ for all i N, κ for all i N. 13

20 Example Klein-Gordon equation in R 3 with Coulomb potential V(x) = 1. 6 x We can take µ = 0 and η = 1. 3 Let δ < 2 and δv = δ V, i.e. V (x) = 1+δ. 6 x Then we have 1 2 δ λ + i 1 2 δ λ i λ + i λ + i λ i λ i 1 δ for all i N, 2 1 δ for all i N. 2 14

21 To do get rid of the spurious spectrum in H, choice of (sub)optimal µ, perturbation estimates for eigenvalues in non-central gaps,... 15

22 5th Najman Conference in Opatija, Croatia September

23 Thanks for the attention!