Bound states in PT -symmetric layers Radek Nov ak Department of - - PowerPoint PPT Presentation
Bound states in PT -symmetric layers Radek Nov ak Department of Physics Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague Department of Theoretical Physics Nuclear Physics Institute Academy of
Department of Physics Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague Department of Theoretical Physics Nuclear Physics Institute Academy of Sciences of the Czech Republic
http://gemma.ujf.cas.cz/∼r.novak
Joint work with David Krejˇ ciˇ r´ ık
◮ Introduction
◮ PT -symmetric Quantum
mechanics
◮ Quantum waveguides
◮ PT -symmetric waveguides
◮ Model ◮ Symmetries ◮ Uniform waveguide ◮ Perturbed waveguide ◮ Essential spectrum ◮ Weakly-coupled bound
states
◮ Conclusions
◮ Hamiltonian −∆ + ix3 in L2(R) possess real spectrum [Bender, Boettcher 98] ◮ more generally: −∆ + x2(ix)ε in L2(R) for ε > 0
◮ Hamiltonian −∆ + ix3 in L2(R) possess real spectrum [Bender, Boettcher 98] ◮ more generally: −∆ + x2(ix)ε in L2(R) for ε > 0
[H, PT ] = 0 (in operator sense)
◮ Parity
(Pψ) (x) = ψ(−x)
◮ Time reversal
(T ψ) (x) = ψ(x)
◮ Hamiltonian −∆ + ix3 in L2(R) possess real spectrum [Bender, Boettcher 98] ◮ more generally: −∆ + x2(ix)ε in L2(R) for ε > 0
[H, PT ] = 0 (in operator sense)
◮ Parity
(Pψ) (x) = ψ(−x)
◮ Time reversal
(T ψ) (x) = ψ(x)
◮ Suggestions
◮ nuclear physics [Scholtz, Geyer, Hahne 92], optics [Klaiman, G¨ unther, Moiseyev 08], [Schomerus 10], solid state physics [Bendix, Fleischmann, Kottos, Shapiro 09], superconductivity [Rubinstein, Sternberg, Ma 07], electromagnetism [Ruschhaupt, Delgado, Muga 05], [Mostafazadeh 09], scattering [Hernandez-Coronado, Krejˇ ciˇ r´ ık, Siegl 11]
◮ Experiments
◮ optics [Guo et al. 09], [R¨ uter et al. 10]
◮ Suggestions
◮ nuclear physics [Scholtz, Geyer, Hahne 92], optics [Klaiman, G¨ unther, Moiseyev 08], [Schomerus 10], solid state physics [Bendix, Fleischmann, Kottos, Shapiro 09], superconductivity [Rubinstein, Sternberg, Ma 07], electromagnetism [Ruschhaupt, Delgado, Muga 05], [Mostafazadeh 09], scattering [Hernandez-Coronado, Krejˇ ciˇ r´ ık, Siegl 11]
◮ Experiments
◮ optics [Guo et al. 09], [R¨ uter et al. 10]
◮ Suggestions
◮ nuclear physics [Scholtz, Geyer, Hahne 92], optics [Klaiman, G¨ unther, Moiseyev 08], [Schomerus 10], solid state physics [Bendix, Fleischmann, Kottos, Shapiro 09], superconductivity [Rubinstein, Sternberg, Ma 07], electromagnetism [Ruschhaupt, Delgado, Muga 05], [Mostafazadeh 09], scattering [Hernandez-Coronado, Krejˇ ciˇ r´ ık, Siegl 11]
◮ Experiments
◮ optics [Guo et al. 09], [R¨ uter et al. 10]
When metric operator Θ > 0, Θ < +∞, Θ−1 < +∞ exists:
(H is then called quasi-Hermitian)
◮ H is Hermitian in Hilbert space
◮ Suggestions
◮ nuclear physics [Scholtz, Geyer, Hahne 92], optics [Klaiman, G¨ unther, Moiseyev 08], [Schomerus 10], solid state physics [Bendix, Fleischmann, Kottos, Shapiro 09], superconductivity [Rubinstein, Sternberg, Ma 07], electromagnetism [Ruschhaupt, Delgado, Muga 05], [Mostafazadeh 09], scattering [Hernandez-Coronado, Krejˇ ciˇ r´ ık, Siegl 11]
◮ Experiments
◮ optics [Guo et al. 09], [R¨ uter et al. 10]
When metric operator Θ > 0, Θ < +∞, Θ−1 < +∞ exists:
(H is then called quasi-Hermitian)
◮ H is Hermitian in Hilbert space
conservation, Stone’s theorem. . .
= microscopic structures of semiconductor material
◮ e.g. thin films, quantum wires, . . . [Exner, ˇ Seba 88], [Duclos, Exner 95]
= microscopic structures of semiconductor material
◮ e.g. thin films, quantum wires, . . . [Exner, ˇ Seba 88], [Duclos, Exner 95]
Mathematical description:
◮ unbounded tubular region ◮ free Laplacian ◮ boundary conditions
= microscopic structures of semiconductor material
◮ e.g. thin films, quantum wires, . . . [Exner, ˇ Seba 88], [Duclos, Exner 95]
Mathematical description:
◮ unbounded tubular region ◮ free Laplacian ◮ boundary conditions
Straight waveguides have empty discrete spectrum ⇒ study of various perturbations
◮ small bumps [Bulla, Gesztezy, Renger, Simon 97] ◮ mixing of boundary conditions [Dittrich, Kˇ r´ ıˇ z 02] ◮ twisting and bending [Krejˇ ciˇ r´ ık 08]
◮ straight waveguide
◮ 1 finite dimension (variable u) ◮ n infinite dimensions (variable x)
d u x
◮ straight waveguide
◮ 1 finite dimension (variable u) ◮ n infinite dimensions (variable x)
◮ complex boundary conditions
◮ imperfect containment of the electron
d
◮ straight waveguide
◮ 1 finite dimension (variable u) ◮ n infinite dimensions (variable x)
◮ complex boundary conditions
◮ imperfect containment of the electron
◮ uniform boundary conditions
◮ exactly solvable
d
◮ straight waveguide
◮ 1 finite dimension (variable u) ◮ n infinite dimensions (variable x)
◮ complex boundary conditions
◮ imperfect containment of the electron
◮ uniform boundary conditions
◮ exactly solvable
◮ influence of small perturbation in boundary conditions
◮ existence of bound states?
d
[Borisov, Krejˇ ciˇ r´ ık 08]
Setup:
◮ planar waveguide Ω = R × I ◮ Robin-type boundary conditions ∂uΨ + i(α0 + εβ)Ψ = 0 ◮ compactly supported perturbation β u x1 ∂uΨ + iαΨ = 0 ∂uΨ + iαΨ = 0 d
[Borisov, Krejˇ ciˇ r´ ık 08]
Setup:
◮ planar waveguide Ω = R × I ◮ Robin-type boundary conditions ∂uΨ + i(α0 + εβ)Ψ = 0 ◮ compactly supported perturbation β
Results:
◮ conditions on existence and uniqueness of the bound state ◮ eigenvalue expansion up to order of ε4 ◮ wavefunction asymptotics u x1 ∂uΨ + iαΨ = 0 ∂uΨ + iαΨ = 0 d
The waveguide Ω := Rn × (0, d) = Rn × I HαΨ := − ∆Ψ, Dom(Hα) :=
∂Ω
∂uΨ + iαΨ = 0 u d x1 x2
The waveguide Ω := Rn × (0, d) = Rn × I HαΨ := − ∆Ψ, Dom(Hα) :=
∂Ω
Let α ∈ W 1,∞(Rn) be real-valued. Then Hα is an m-sectorial
Re η Im η γ θ
The waveguide Ω := Rn × (0, d) = Rn × I HαΨ := − ∆Ψ, Dom(Hα) :=
∂Ω
Let α ∈ W 1,∞(Rn) be real-valued. Then Hα is an m-sectorial
Idea of the proof: hα[Ψ] :=
|∇Ψ(x, u)|2 dx du + i
◮ hα is sectorial and closed ◮ First representation theorem
The waveguide Ω := Rn × (0, d) = Rn × I HαΨ := − ∆Ψ, Dom(Hα) :=
∂Ω
Let α ∈ W 1,∞(Rn) be real-valued. Then Hα is an m-sectorial
Idea of the proof: hα[Ψ] :=
|∇Ψ(x, u)|2 dx du + i
◮ hα is sectorial and closed ◮ First representation theorem
Note that H ∗
α = H−α
The spatial reflection operator P and the time reversal operator T : (PΨ)(x, u) := Ψ(x, d − u), (T Ψ)(x, u) := Ψ(x, u).
The spatial reflection operator P and the time reversal operator T : (PΨ)(x, u) := Ψ(x, d − u), (T Ψ)(x, u) := Ψ(x, u). Proposition Let α ∈ W 1,∞(Rn) be real-valued. Then Hα is PT -symmetric, i.e. [Hα, PT ] = 0. ⇒ λ ∈ σ(H) ⇔ λ ∈ σ(H)
The spatial reflection operator P and the time reversal operator T : (PΨ)(x, u) := Ψ(x, d − u), (T Ψ)(x, u) := Ψ(x, u). Proposition Let α ∈ W 1,∞(Rn) be real-valued. Then Hα is PT -symmetric, i.e. [Hα, PT ] = 0. ⇒ λ ∈ σ(H) ⇔ λ ∈ σ(H) Proposition Let α ∈ W 1,∞(Rn) be real-valued. Then Hα is T -self-adjoint, i.e T HαT = H ∗
α
⇒ σr(Hα) = ∅
Transversal operator: − ∆I
α0ψ := − ∂2
∂u2 ψ Dom(−∆I
α0) :=
at 0, d
Transversal operator: − ∆I
α0ψ := − ∂2
∂u2 ψ Dom(−∆I
α0) :=
at 0, d
α0) = σp(−∆I α0) =
kπ
d
2+∞
k=1 =:
j
+∞
j=0 ◮ eigenfunctions ψj(u) = cos(µju) − i α0 µj sin(µju),
j ≥ 0
Transversal operator: − ∆I
α0ψ := − ∂2
∂u2 ψ Dom(−∆I
α0) :=
at 0, d
α0) = σp(−∆I α0) =
kπ
d
2+∞
k=1 =:
j
+∞
j=0 ◮ eigenfunctions ψj(u) = cos(µju) − i α0 µj sin(µju),
j ≥ 0
◮
−∆I
α0
∗: same spectrum, eigenfunctions φj := Ajψj
◮ ψj and φj form a biorthonormal basis
Transversal operator: − ∆I
α0ψ := − ∂2
∂u2 ψ Dom(−∆I
α0) :=
at 0, d
α0) = σp(−∆I α0) =
kπ
d
2+∞
k=1 =:
j
+∞
j=0 ◮ eigenfunctions ψj(u) = cos(µju) − i α0 µj sin(µju),
j ≥ 0
◮
−∆I
α0
∗: same spectrum, eigenfunctions φj := Ajψj
◮ ψj and φj form a biorthonormal basis
Longitudal operator −∆′ϕ := −
n
∂2 ∂x2
k
ϕ Dom(−∆′) := W 2,2(Rn)
Transversal operator: − ∆I
α0ψ := − ∂2
∂u2 ψ Dom(−∆I
α0) :=
at 0, d
α0) = σp(−∆I α0) =
kπ
d
2+∞
k=1 =:
j
+∞
j=0 ◮ eigenfunctions ψj(u) = cos(µju) − i α0 µj sin(µju),
j ≥ 0
◮
−∆I
α0
∗: same spectrum, eigenfunctions φj := Ajψj
◮ ψj and φj form a biorthonormal basis
Longitudal operator −∆′ϕ := −
n
∂2 ∂x2
k
ϕ Dom(−∆′) := W 2,2(Rn)
◮ σ(−∆′) = σess(−∆′) = [0, +∞)
Term in boundary conditions: α(x) = α0 ∈ R
d
Term in boundary conditions: α(x) = α0 ∈ R Hα0 =
+
α0
because Ψ(x, u) = +∞
j=0 (φj, Ψ(x, ·))L2(I) ψj(u) holds for every
Ψ ∈ L2(Ω)
d
Term in boundary conditions: α(x) = α0 ∈ R Hα0 =
+
α0
because Ψ(x, u) = +∞
j=0 (φj, Ψ(x, ·))L2(I) ψj(u) holds for every
Ψ ∈ L2(Ω)
◮ resolvent can be decomposed into transversal basis:
(x, u, x′, u′) =
+∞
ψj(u) R−∆′
λ−µ2
j (x, x′) φj(u′)
d
Term in boundary conditions: α(x) = α0 ∈ R Hα0 =
+
α0
because Ψ(x, u) = +∞
j=0 (φj, Ψ(x, ·))L2(I) ψj(u) holds for every
Ψ ∈ L2(Ω)
◮ resolvent can be decomposed into transversal basis:
(x, u, x′, u′) =
+∞
ψj(u) R−∆′
λ−µ2
j (x, x′) φj(u′)
⇒ σ(Hα0) = σess(Hα0) = [µ2
0, +∞) d
Theorem Let α − α0 ∈ W 1,∞(R) with α0 ∈ R such that lim
|x|→+∞(α − α0)(x) = 0.
Theorem Let α − α0 ∈ W 1,∞(R) with α0 ∈ R such that lim
|x|→+∞(α − α0)(x) = 0.
Then (Hα − λ)−1 − (Hα0 − λ)−1 is compact in L2(Ω) for any λ ∈ ρ(Hα) ∩ ρ(Hα0).
Theorem Let α − α0 ∈ W 1,∞(R) with α0 ∈ R such that lim
|x|→+∞(α − α0)(x) = 0.
Then (Hα − λ)−1 − (Hα0 − λ)−1 is compact in L2(Ω) for any λ ∈ ρ(Hα) ∩ ρ(Hα0). Corollary σess(Hα) = σess(Hα0) = [µ2
0, +∞). essential spectrum Re λ Im λ µ2
α(x) = α0 + εβ(x), where α0 ∈ R, β ∈ W 2,∞(Rn) and ε > 0
α(x) = α0 + εβ(x), where α0 ∈ R, β ∈ W 2,∞(Rn) and ε > 0
◮ Our goal: conditions on existence and uniqueness of the bound
state
essential spectrum bound state Re λ Im λ µ2
α(x) = α0 + εβ(x), where α0 ∈ R, β ∈ W 2,∞(Rn) and ε > 0
◮ Our goal: conditions on existence and uniqueness of the bound
state
◮ bound state can be expected for small ε only for n = 1, 2
◮ due to singularity in the resolvent for λ → µ2
essential spectrum bound state Re λ Im λ µ2
α(x) = α0 + εβ(x), where α0 ∈ R, β ∈ W 2,∞(Rn) and ε > 0
◮ Our goal: conditions on existence and uniqueness of the bound
state
◮ bound state can be expected for small ε only for n = 1, 2
◮ due to singularity in the resolvent for λ → µ2
◮ singular perturbation theory - perturbation of the threshold of
the essential spectrum
essential spectrum bound state Re λ Im λ µ2
Theorem Let us assume β ∈ W 2,∞(Rn) and that β and its first and second derivations go in the infinity to 0 faster than x−4−δ for some δ > 0
Theorem Let us assume β ∈ W 2,∞(Rn) and that β and its first and second derivations go in the infinity to 0 faster than x−4−δ for some δ > 0 Let us denote β :=
If ε > 0 is sufficiently small, |α0| < π/d and α0β < 0, then Hα possesses the real and unique eigenvalue λ = λ(ε) ∈ (−∞, µ2
0).
Theorem Let us assume β ∈ W 2,∞(Rn) and that β and its first and second derivations go in the infinity to 0 faster than x−4−δ for some δ > 0 Let us denote β :=
If ε > 0 is sufficiently small, |α0| < π/d and α0β < 0, then Hα possesses the real and unique eigenvalue λ = λ(ε) ∈ (−∞, µ2
0).
The asymptotic expansion λ(ε) =
0 − ε2α2 0β2 + O(ε3)
(if n = 1), µ2
0 − e2/w(ε),
(if n = 2), where w(ε) = ε
πβα0 + O(ε2), holds as ε → 0.
Unitary transformation: U −1
ε
HαUε = Hα0 + εZε, Dom(U −1
ε
HαUε) = Dom(Hα0), where . . . (UεΨ) (x, u) := e−iεβ(x)uΨ(x, u) for any Ψ ∈ L2(Ω) . . . Zε := 2iu∇′β(x) · ∇′ + 2iβ(x) ∂
∂u +
Unitary transformation: U −1
ε
HαUε = Hα0 + εZε = Hα0 + εC ∗
ε D,
Dom(U −1
ε
HαUε) = Dom(Hα0), where . . . (UεΨ) (x, u) := e−iεβ(x)uΨ(x, u) for any Ψ ∈ L2(Ω) . . . Zε := 2iu∇′β(x) · ∇′ + 2iβ(x) ∂
∂u +
. . . Cε is a multiplication operator . . . D is a first order differential operator
Unitary transformation: U −1
ε
HαUε = Hα0 + εZε = Hα0 + εC ∗
ε D,
Dom(U −1
ε
HαUε) = Dom(Hα0), where . . . (UεΨ) (x, u) := e−iεβ(x)uΨ(x, u) for any Ψ ∈ L2(Ω) . . . Zε := 2iu∇′β(x) · ∇′ + 2iβ(x) ∂
∂u +
. . . Cε is a multiplication operator . . . D is a first order differential operator Birman-Schwinger principle: λ ∈ σp(Hα) ⇔ −1 ∈ σp
ε
Separation of the resolvent singularity ⇒ implicit equation for the eigenvalues
1 2 3 4 5 x x 0.05 0.10 0.15 0.20 0.25
R z x , x
n = 1
1 2 3 4 5 x x 0.02 0.04 0.06 0.08
R z x , x
n = 2
Separation of the resolvent singularity ⇒ implicit equation for the eigenvalues −1 = ε Fn
0 − λ Ω
ψ0(u)
ε (I + M λ ε )−1Dφ0
where . . . Fn is remainder of the singular part . . . M λ
ε is remainder of the regular part
1 2 3 4 5 x x 0.05 0.10 0.15 0.20 0.25
R z x , x
n = 1
1 2 3 4 5 x x 0.02 0.04 0.06 0.08
R z x , x
n = 2
Separation of the resolvent singularity ⇒ implicit equation for the eigenvalues −1 = ε Fn
0 − λ Ω
ψ0(u)
ε (I + M λ ε )−1Dφ0
where . . . Fn is remainder of the singular part . . . M λ
ε is remainder of the regular part
Banach fixed point theorem
◮ existence and uniqueness of the solution ◮ alternatively Rouch´
e theorem
Separation of the resolvent singularity ⇒ implicit equation for the eigenvalues −1 = ε Fn
0 − λ Ω
ψ0(u)
ε (I + M λ ε )−1Dφ0
where . . . Fn is remainder of the singular part . . . M λ
ε is remainder of the regular part
Banach fixed point theorem
◮ existence and uniqueness of the solution ◮ alternatively Rouch´
e theorem PT -symmetry
◮ reality of the bound state
◮ α0β < 0
◮ perturbation acts against uniform boundary conditions
d
◮ α0β < 0
◮ perturbation acts against uniform boundary conditions
◮ the ground state energy is real
◮ other eigenvalues can be complex!
d
◮ the case n = 1 [Krejˇ
ciˇ r´ ık, Tater 08]
◮ the case n = 1 [Krejˇ
ciˇ r´ ık, Tater 08]
◮ Free Laplace operator in a tubular neighborhood of a hyperplane
with imposed non-Hermitian boundary conditions
◮ space-time symmetry
◮ Free Laplace operator in a tubular neighborhood of a hyperplane
with imposed non-Hermitian boundary conditions
◮ space-time symmetry
◮ Uniform boundary conditions
◮ spectrum of free Laplacian shifted by the lowest eigenvalue of the
transversal operator
◮ Free Laplace operator in a tubular neighborhood of a hyperplane
with imposed non-Hermitian boundary conditions
◮ space-time symmetry
◮ Uniform boundary conditions
◮ spectrum of free Laplacian shifted by the lowest eigenvalue of the
transversal operator
◮ Stability of the essential spectrum
◮ under perturbation vanishing in infinity
◮ Free Laplace operator in a tubular neighborhood of a hyperplane
with imposed non-Hermitian boundary conditions
◮ space-time symmetry
◮ Uniform boundary conditions
◮ spectrum of free Laplacian shifted by the lowest eigenvalue of the
transversal operator
◮ Stability of the essential spectrum
◮ under perturbation vanishing in infinity
◮ Existence and uniqueness of a bound state for weak coupling
◮ perturbation acting in the mean against uniform boundary
conditions
◮ strip: λ(ε) ∼ ε2 ◮ layer: λ(ε) ∼ e2/ε
◮ Free Laplace operator in a tubular neighborhood of a hyperplane
with imposed non-Hermitian boundary conditions
◮ space-time symmetry
◮ Uniform boundary conditions
◮ spectrum of free Laplacian shifted by the lowest eigenvalue of the
transversal operator
◮ Stability of the essential spectrum
◮ under perturbation vanishing in infinity
◮ Existence and uniqueness of a bound state for weak coupling
◮ perturbation acting in the mean against uniform boundary
conditions
◮ strip: λ(ε) ∼ ε2 ◮ layer: λ(ε) ∼ e2/ε
? Existence of the metric operator