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Spectra of definite type in waveguide models V. Lotoreichik in collaboration with P. Siegl Nuclear Physics Institute, Czech Academy of Sciences, e Prague, 08.06.2016 V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models

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Spectra of definite type in waveguide models

  • V. Lotoreichik

in collaboration with P. Siegl

Nuclear Physics Institute, Czech Academy of Sciences, Řež

Prague, 08.06.2016

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 1 / 17

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Outline

1

PT -symmetric waveguides

2

Main results

3

Tools and methods of the proofs

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 2 / 17

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Hamiltonian

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 3 / 17

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Hamiltonian

Ω = R × (−π/2, π/2) with opposite sides Σ± = R × {±π/2}

Ω y π x O Σ+ Σ−

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 3 / 17

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Hamiltonian

Ω = R × (−π/2, π/2) with opposite sides Σ± = R × {±π/2}

Ω y π x O Σ+ Σ−

α = αℜ + iαℑ : R → C where αℜ, αℑ ∈ L∞(R; R)

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 3 / 17

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Hamiltonian

Ω = R × (−π/2, π/2) with opposite sides Σ± = R × {±π/2}

Ω y π x O Σ+ Σ−

α = αℜ + iαℑ : R → C where αℜ, αℑ ∈ L∞(R; R)

Hamiltonian of PT -symmetric waveguide (Borisov-Křejčiřík-08)

Hαu = −∆u, dom Hα ={u : u, ∆u ∈ L2(Ω), (αℜ ± iαℑ)u|Σ± = ∂νu|Σ±} ⋆ m-sectorial in L2(Ω) and non-selfadjoint if αℑ = 0 ⋆ J-selfadjoint with (Ju)(x, y) = u(x, −y); H∗

α = JHαJ.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 3 / 17

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Is the spectrum of Hα real?

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 4 / 17

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Is the spectrum of Hα real?

The question makes sense only for αℑ = 0 as otherwise Hα = H∗

α.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 4 / 17

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Is the spectrum of Hα real?

The question makes sense only for αℑ = 0 as otherwise Hα = H∗

α.

σ(Hα) ⊂ R for αℜ = 0 (Borisov-Křejčiřík-08, Novak-16)

(i) α = iαℑ for αℑ(x) = −αℑ(−x) ∈ C∞

0 (R).

(ii) α= i(α0 + εβ) for α0 ∈ (0, 1), β ∈ C∞

0 (R; R) & ε > 0 small.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 4 / 17

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Is the spectrum of Hα real?

The question makes sense only for αℑ = 0 as otherwise Hα = H∗

α.

σ(Hα) ⊂ R for αℜ = 0 (Borisov-Křejčiřík-08, Novak-16)

(i) α = iαℑ for αℑ(x) = −αℑ(−x) ∈ C∞

0 (R).

(ii) α= i(α0 + εβ) for α0 ∈ (0, 1), β ∈ C∞

0 (R; R) & ε > 0 small.

C∞

0 (R) can be replaced by more general C0(R) ∩ W 1 ∞(R).

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 4 / 17

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Is the spectrum of Hα real?

The question makes sense only for αℑ = 0 as otherwise Hα = H∗

α.

σ(Hα) ⊂ R for αℜ = 0 (Borisov-Křejčiřík-08, Novak-16)

(i) α = iαℑ for αℑ(x) = −αℑ(−x) ∈ C∞

0 (R).

(ii) α= i(α0 + εβ) for α0 ∈ (0, 1), β ∈ C∞

0 (R; R) & ε > 0 small.

C∞

0 (R) can be replaced by more general C0(R) ∩ W 1 ∞(R).

σ(Hα) ⊂ R (Křejčiřík-Tater-08)

Numerical experiment for αℜ = 0 and αℑ = 1 − ε exp(− x2

10).

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 4 / 17

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Is the spectrum of Hα real?

The question makes sense only for αℑ = 0 as otherwise Hα = H∗

α.

σ(Hα) ⊂ R for αℜ = 0 (Borisov-Křejčiřík-08, Novak-16)

(i) α = iαℑ for αℑ(x) = −αℑ(−x) ∈ C∞

0 (R).

(ii) α= i(α0 + εβ) for α0 ∈ (0, 1), β ∈ C∞

0 (R; R) & ε > 0 small.

C∞

0 (R) can be replaced by more general C0(R) ∩ W 1 ∞(R).

σ(Hα) ⊂ R (Křejčiřík-Tater-08)

Numerical experiment for αℜ = 0 and αℑ = 1 − ε exp(− x2

10).

A necessary & sufficient condition for σ(Hα) ⊂ R can hardly be found!

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 4 / 17

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Motivation

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 5 / 17

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Motivation

Previously applied methods for showing σ(Hα) ⊂ R

(i) Separation of variables for α = const reduces to realness of spectra for 1-D model operators. (ii) σess(Hα) ⊂ R for α = const follows by compact perturbation of Hα0 with α0 = const. (iii) σd(Hα) ⊂ R for α = const can be shown via ad hoc tricks for special cases with extra symmetries or via Birman-Schwinger principle.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 5 / 17

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Motivation

Previously applied methods for showing σ(Hα) ⊂ R

(i) Separation of variables for α = const reduces to realness of spectra for 1-D model operators. (ii) σess(Hα) ⊂ R for α = const follows by compact perturbation of Hα0 with α0 = const. (iii) σd(Hα) ⊂ R for α = const can be shown via ad hoc tricks for special cases with extra symmetries or via Birman-Schwinger principle.

Main motivation

To demonstrate applicability of definite type spectra to proving σ(Hα) ⊂ R (at least in a weak sense).

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 5 / 17

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Local spectral properties of Hα

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 6 / 17

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Local spectral properties of Hα

Question I: local realness of σ(Hα)

σ(Hα) ∩ U ⊂ R for U ⊂ C?

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 6 / 17

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Local spectral properties of Hα

Question I: local realness of σ(Hα)

σ(Hα) ∩ U ⊂ R for U ⊂ C? Reduces to realness of σ(Hα) for U = C.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 6 / 17

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Local spectral properties of Hα

Question I: local realness of σ(Hα)

σ(Hα) ∩ U ⊂ R for U ⊂ C? Reduces to realness of σ(Hα) for U = C. σε(T) :=

λ ∈ ρ(T): (T − λ)−1 > ε−1 ∪ σ(T) (ε-pseudospectrum)

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 6 / 17

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Local spectral properties of Hα

Question I: local realness of σ(Hα)

σ(Hα) ∩ U ⊂ R for U ⊂ C? Reduces to realness of σ(Hα) for U = C. σε(T) :=

λ ∈ ρ(T): (T − λ)−1 > ε−1 ∪ σ(T) (ε-pseudospectrum)

Normal local behaviour of σε(T) with σ(T) ∩ U ⊂ R

σε(T) ∩ U ⊂ {λ ∈ U : |ℑλ| ≤ Cε1/m} for U ⊂ C with C > 0 and m ∈ N.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 6 / 17

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Local spectral properties of Hα

Question I: local realness of σ(Hα)

σ(Hα) ∩ U ⊂ R for U ⊂ C? Reduces to realness of σ(Hα) for U = C. σε(T) :=

λ ∈ ρ(T): (T − λ)−1 > ε−1 ∪ σ(T) (ε-pseudospectrum)

Normal local behaviour of σε(T) with σ(T) ∩ U ⊂ R

σε(T) ∩ U ⊂ {λ ∈ U : |ℑλ| ≤ Cε1/m} for U ⊂ C with C > 0 and m ∈ N.

Question II: local behaviour of σε(Hα)

Does σε(Hα) ∩ U for U ⊂ C have normal behaviour? σε(Hα) is important in the analysis of t → e−tHα and of t → eitHα.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 6 / 17

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Outline

1

PT -symmetric waveguides

2

Main results

3

Tools and methods of the proofs

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 7 / 17

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Non-compact perturbations

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 8 / 17

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Non-compact perturbations

Assumption

(a) α0 ∈ R, α0 = 1, and M := {n2}n∈N ∪ {α2

0}

(b) µ0 := min M and µ1 := min(M \ {µ0}), (µ0 < µ1)! σ(Hiα0) = [µ0, ∞).

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 8 / 17

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Non-compact perturbations

Assumption

(a) α0 ∈ R, α0 = 1, and M := {n2}n∈N ∪ {α2

0}

(b) µ0 := min M and µ1 := min(M \ {µ0}), (µ0 < µ1)! σ(Hiα0) = [µ0, ∞). Fix a compact set F ⊂ C with F ∩ R ⊂ (−∞, µ1) µ0 µ1 F

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 8 / 17

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Non-compact perturbations

Assumption

(a) α0 ∈ R, α0 = 1, and M := {n2}n∈N ∪ {α2

0}

(b) µ0 := min M and µ1 := min(M \ {µ0}), (µ0 < µ1)! σ(Hiα0) = [µ0, ∞). Fix a compact set F ⊂ C with F ∩ R ⊂ (−∞, µ1) µ0 µ1 F

Theorem (L-Siegl-16)

There exists γ > 0 such that for all α: R → C with α − iα0∞ < γ σ(Hα) ∩ F ⊂ R and σε(Hα) ∩ F behaves normally.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 8 / 17

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Non-compact perturbations

Assumption

(a) α0 ∈ R, α0 = 1, and M := {n2}n∈N ∪ {α2

0}

(b) µ0 := min M and µ1 := min(M \ {µ0}), (µ0 < µ1)! σ(Hiα0) = [µ0, ∞). Fix a compact set F ⊂ C with F ∩ R ⊂ (−∞, µ1) µ0 µ1 F

Theorem (L-Siegl-16)

There exists γ > 0 such that for all α: R → C with α − iα0∞ < γ σ(Hα) ∩ F ⊂ R and σε(Hα) ∩ F behaves normally. Non-real spectrum of Hα does not appear near low-lying real spectrum.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 8 / 17

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Necessity of assumption α0 = 1

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 9 / 17

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Necessity of assumption α0 = 1

Assume α0 = 1.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 9 / 17

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Necessity of assumption α0 = 1

Assume α0 = 1.

Example

For any γ > 0 there exists α ∈ C such that |α − i| < γ and that σ(Hα) = ({λ0} + R+) ∪ ({λ1} + R+) ∪ ({λ1} + R+). λ0 ∈ R and ℑλ1 = 0. ℑλ1 → 0 holds as γ → 0+.

ℑλ ℜλ λ0 λ1 λ1

Local realness of σ(Hα) is fully violated

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 9 / 17

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Compact perturbations

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 10 / 17

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Compact perturbations

α0 = 1, M := {n2}n∈N ∪ {α2

0}, µ0 := min M, and µ1 := min(M \ {µ0}).

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 10 / 17

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Compact perturbations

α0 = 1, M := {n2}n∈N ∪ {α2

0}, µ0 := min M, and µ1 := min(M \ {µ0}).

Further, assume α − iα0 ∈ L∞

∞(R) := {f ∈ L∞(R): {x : |f (x)| > ε} is bounded ∀ε > 0}.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 10 / 17

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Compact perturbations

α0 = 1, M := {n2}n∈N ∪ {α2

0}, µ0 := min M, and µ1 := min(M \ {µ0}).

Further, assume α − iα0 ∈ L∞

∞(R) := {f ∈ L∞(R): {x : |f (x)| > ε} is bounded ∀ε > 0}.

σess(Hα) = σess(Hiα0) = [µ0, ∞).

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 10 / 17

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Compact perturbations

α0 = 1, M := {n2}n∈N ∪ {α2

0}, µ0 := min M, and µ1 := min(M \ {µ0}).

Further, assume α − iα0 ∈ L∞

∞(R) := {f ∈ L∞(R): {x : |f (x)| > ε} is bounded ∀ε > 0}.

σess(Hα) = σess(Hiα0) = [µ0, ∞).

Theorem (L-Siegl-16)

For any [a, b] ⊂ (−∞, µ1) exists open U ⊂ C, [a, b] ⊂ U σ(Hα) ∩ U ⊂ R and σε(Hα) ∩ U behaves normally.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 10 / 17

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Compact perturbations

α0 = 1, M := {n2}n∈N ∪ {α2

0}, µ0 := min M, and µ1 := min(M \ {µ0}).

Further, assume α − iα0 ∈ L∞

∞(R) := {f ∈ L∞(R): {x : |f (x)| > ε} is bounded ∀ε > 0}.

σess(Hα) = σess(Hiα0) = [µ0, ∞).

Theorem (L-Siegl-16)

For any [a, b] ⊂ (−∞, µ1) exists open U ⊂ C, [a, b] ⊂ U σ(Hα) ∩ U ⊂ R and σε(Hα) ∩ U behaves normally. We exclude accumulation of non-real eigenvalues of Hα to [µ0, µ1) ⊂ σess(Hα)!

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 10 / 17

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Compact perturbations

α0 = 1, M := {n2}n∈N ∪ {α2

0}, µ0 := min M, and µ1 := min(M \ {µ0}).

Further, assume α − iα0 ∈ L∞

∞(R) := {f ∈ L∞(R): {x : |f (x)| > ε} is bounded ∀ε > 0}.

σess(Hα) = σess(Hiα0) = [µ0, ∞).

Theorem (L-Siegl-16)

For any [a, b] ⊂ (−∞, µ1) exists open U ⊂ C, [a, b] ⊂ U σ(Hα) ∩ U ⊂ R and σε(Hα) ∩ U behaves normally. We exclude accumulation of non-real eigenvalues of Hα to [µ0, µ1) ⊂ σess(Hα)!

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 10 / 17

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Outline

1

PT -symmetric waveguides

2

Main results

3

Tools and methods of the proofs

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 11 / 17

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Definite type spectra

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 12 / 17

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Definite type spectra

Let T be a J-self-adjoint operator in a Hilbert space (H, (·, ·)).

Definite type spectra (Langer-Markus-Matsaev-97)

λ ∈ σ++(T) (λ ∈ σ−−(T)) if and only if (i) there exists {un} ⊂ dom T, un = 1, (T − λ)un → 0, (ii) lim infn→∞(±Jun, un) > 0 for any such sequence {un}.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 12 / 17

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Definite type spectra

Let T be a J-self-adjoint operator in a Hilbert space (H, (·, ·)).

Definite type spectra (Langer-Markus-Matsaev-97)

λ ∈ σ++(T) (λ ∈ σ−−(T)) if and only if (i) there exists {un} ⊂ dom T, un = 1, (T − λ)un → 0, (ii) lim infn→∞(±Jun, un) > 0 for any such sequence {un}. The concept has been earlier applied to: (i) Laplacian −w∆ with an indefinite weight w. (ii) Damping equations for thin beams. (iii) PT -symmetric ordinary differential operators.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 12 / 17

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Abstract part of the proof

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 13 / 17

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Abstract part of the proof

m-sectorial operators T1, T2

T∗

1 = J1T1J1 in H1 and T∗ 2 = J2T2J2 in H2.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 13 / 17

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Abstract part of the proof

m-sectorial operators T1, T2

T∗

1 = J1T1J1 in H1 and T∗ 2 = J2T2J2 in H2.

m-sectorial operator T := T1 ⊗ I2 + I1 ⊗ T2

T∗ = JTJ in H = H1 ⊗ H2 with J := J1 ⊗ J2

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 13 / 17

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Abstract part of the proof

m-sectorial operators T1, T2

T∗

1 = J1T1J1 in H1 and T∗ 2 = J2T2J2 in H2.

m-sectorial operator T := T1 ⊗ I2 + I1 ⊗ T2

T∗ = JTJ in H = H1 ⊗ H2 with J := J1 ⊗ J2 We develop a way of finding σ±±(T) relying on the properties of T1, T2.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 13 / 17

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Abstract part of the proof

m-sectorial operators T1, T2

T∗

1 = J1T1J1 in H1 and T∗ 2 = J2T2J2 in H2.

m-sectorial operator T := T1 ⊗ I2 + I1 ⊗ T2

T∗ = JTJ in H = H1 ⊗ H2 with J := J1 ⊗ J2 We develop a way of finding σ±±(T) relying on the properties of T1, T2. This method applies to various classes of operators having the structure T1 ⊗ I2 + I1 ⊗ T2.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 13 / 17

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Applying abstract method to PT -symmetric waveguide

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 14 / 17

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Applying abstract method to PT -symmetric waveguide

Hiα0 = T1 ⊗ I2 + I1 ⊗ T2 in L2(Ω) = L2(R) ⊗ L2(I), I = (−π/2, π/2).

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 14 / 17

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Applying abstract method to PT -symmetric waveguide

Hiα0 = T1 ⊗ I2 + I1 ⊗ T2 in L2(Ω) = L2(R) ⊗ L2(I), I = (−π/2, π/2). (i) T1f = −f ′′, dom T1 = H2(R) (ii) T2f = −f ′′, dom T2 = {f ∈ H2(I): iα0f (±π/2) + f ′(±π/2) = 0}

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 14 / 17

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Applying abstract method to PT -symmetric waveguide

Hiα0 = T1 ⊗ I2 + I1 ⊗ T2 in L2(Ω) = L2(R) ⊗ L2(I), I = (−π/2, π/2). (i) T1f = −f ′′, dom T1 = H2(R) (ii) T2f = −f ′′, dom T2 = {f ∈ H2(I): iα0f (±π/2) + f ′(±π/2) = 0} α0 = 1, M := {n2}n∈N ∪ {α2

0}, µ0 := min M, and µ1 := min(M \ {µ0}).

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 14 / 17

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Applying abstract method to PT -symmetric waveguide

Hiα0 = T1 ⊗ I2 + I1 ⊗ T2 in L2(Ω) = L2(R) ⊗ L2(I), I = (−π/2, π/2). (i) T1f = −f ′′, dom T1 = H2(R) (ii) T2f = −f ′′, dom T2 = {f ∈ H2(I): iα0f (±π/2) + f ′(±π/2) = 0} α0 = 1, M := {n2}n∈N ∪ {α2

0}, µ0 := min M, and µ1 := min(M \ {µ0}).

Proposition (L-Siegl-16)

σ++(Hiα0) = [µ0, µ1) and σ−−(Hiα0) = ∅. µ0 µ1 ++ 00

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 14 / 17

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Applying abstract method to PT -symmetric waveguide

Hiα0 = T1 ⊗ I2 + I1 ⊗ T2 in L2(Ω) = L2(R) ⊗ L2(I), I = (−π/2, π/2). (i) T1f = −f ′′, dom T1 = H2(R) (ii) T2f = −f ′′, dom T2 = {f ∈ H2(I): iα0f (±π/2) + f ′(±π/2) = 0} α0 = 1, M := {n2}n∈N ∪ {α2

0}, µ0 := min M, and µ1 := min(M \ {µ0}).

Proposition (L-Siegl-16)

σ++(Hiα0) = [µ0, µ1) and σ−−(Hiα0) = ∅. µ0 µ1 ++ 00 Results on Hα, α = const are obtained via properties and perturbation theory for σ++ (Azizov, Behrndt, Jonas, Philipp, Trunk,...)

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 14 / 17

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Generalized PT -symmetric waveguides

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 15 / 17

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Generalized PT -symmetric waveguides

Hiα0,V0 = T1 ⊗ I2 + I1 ⊗ T2 in L2(Ω) = L2(R) ⊗ L2(I). (i) T1f = −f ′′ + V0f , dom T1 = H2(R), V0 ∈ L∞(R; R); (ii) T2 stays the same.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 15 / 17

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SLIDE 55

Generalized PT -symmetric waveguides

Hiα0,V0 = T1 ⊗ I2 + I1 ⊗ T2 in L2(Ω) = L2(R) ⊗ L2(I). (i) T1f = −f ′′ + V0f , dom T1 = H2(R), V0 ∈ L∞(R; R); (ii) T2 stays the same. We compute explicitly σ++(Hiα0,V0) and σ−−(Hiα0,V0) in terms of σ(T1)!

++ ++ ++ −− −− −− 00 00

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 15 / 17

slide-56
SLIDE 56

Generalized PT -symmetric waveguides

Hiα0,V0 = T1 ⊗ I2 + I1 ⊗ T2 in L2(Ω) = L2(R) ⊗ L2(I). (i) T1f = −f ′′ + V0f , dom T1 = H2(R), V0 ∈ L∞(R; R); (ii) T2 stays the same. We compute explicitly σ++(Hiα0,V0) and σ−−(Hiα0,V0) in terms of σ(T1)!

++ ++ ++ −− −− −− 00 00

Again applying perturbation argument we obtain similar statements for Hα,V with α, V = const.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 15 / 17

slide-57
SLIDE 57

Reference

  • P. Siegl and V. Lotoreichik,

Spectra of definite type in waveguide models, to appear in Proc. Amer. Math. Soc., arXiv:1602.08883.

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 16 / 17

slide-58
SLIDE 58

Thank you! Thank you for your attentention! Happy Birthday, Miloš!

  • V. Lotoreichik (NPI CAS)

Spectra of definite type in waveguide models 08.06.2016 17 / 17