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 Sciences of the Czech Republic http://gemma.ujf.cas.cz/ ∼ r.novak 3rd Najman conference September 18, 2013 Joint work with David Krejˇ ciˇ r´ ık

Outline of the talk ◮ Introduction ◮ PT -symmetric Quantum mechanics ◮ Quantum waveguides ◮ PT -symmetric waveguides ◮ Model ◮ Symmetries ◮ Uniform waveguide ◮ Perturbed waveguide ◮ Essential spectrum ◮ Weakly-coupled bound states ◮ Conclusions

PT -symmetric Quantum mechanics ? Non-Hermitian observables in Quantum mechanics ?

PT -symmetric Quantum mechanics ? Non-Hermitian observables in Quantum mechanics ? ◮ Hamiltonian − ∆ + i x 3 in L 2 ( R ) possess real spectrum [Bender, Boettcher 98] ◮ more generally: − ∆ + x 2 (i x ) ε in L 2 ( R ) for ε > 0

PT -symmetric Quantum mechanics ? Non-Hermitian observables in Quantum mechanics ? ◮ Hamiltonian − ∆ + i x 3 in L 2 ( R ) possess real spectrum [Bender, Boettcher 98] ◮ more generally: − ∆ + x 2 (i x ) ε in L 2 ( R ) for ε > 0 ? Due to PT -symmetry ? [ H , PT ] = 0 (in operator sense) ◮ Parity ( P ψ ) ( x ) = ψ ( − x ) ◮ Time reversal ( T ψ ) ( x ) = ψ ( x )

PT -symmetric Quantum mechanics ? Non-Hermitian observables in Quantum mechanics ? ◮ Hamiltonian − ∆ + i x 3 in L 2 ( R ) possess real spectrum [Bender, Boettcher 98] ◮ more generally: − ∆ + x 2 (i x ) ε in L 2 ( R ) for ε > 0 ? Due to PT -symmetry ? [ H , PT ] = 0 (in operator sense) ◮ Parity ( P ψ ) ( x ) = ψ ( − x ) ◮ Time reversal ( T ψ ) ( x ) = ψ ( x ) ! Lack of techniques - no spectral theorem, no Min-max principle, . . . !

Physical relevance ◮ 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]

Physical relevance ◮ 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] ? How to make sense of PT -symmetric operators ?

Physical relevance ◮ 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] ? How to make sense of PT -symmetric operators ? When metric operator Θ > 0 , � Θ � < + ∞ , � Θ − 1 � < + ∞ exists: (H is then called quasi-Hermitian) ◮ H is Hermitian in Hilbert space � L 2 , �· , Θ ·� � ◮ h = Θ 1 / 2 H Θ − 1 / 2 is Hermitian in � L 2 , �· , ·� �

Physical relevance ◮ 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] ? How to make sense of PT -symmetric operators ? When metric operator Θ > 0 , � Θ � < + ∞ , � Θ − 1 � < + ∞ exists: (H is then called quasi-Hermitian) ◮ H is Hermitian in Hilbert space � L 2 , �· , Θ ·� � ◮ h = Θ 1 / 2 H Θ − 1 / 2 is Hermitian in � L 2 , �· , ·� � ⇒ solves problem with reality of the spectrum, probability conservation, Stone’s theorem. . .

Quantum waveguides = microscopic structures of semiconductor material ◮ e.g. thin films, quantum wires, . . . [Exner, ˇ Seba 88], [Duclos, Exner 95]

Quantum waveguides = 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

Quantum waveguides = 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]

The model ◮ straight waveguide ◮ 1 finite dimension (variable u ) ◮ n infinite dimensions (variable x ) u d x

The model ◮ straight waveguide ◮ 1 finite dimension (variable u ) ◮ n infinite dimensions (variable x ) ◮ complex boundary conditions ◮ imperfect containment of the electron d

The model ◮ 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

The model ◮ 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

Previous results [Borisov, Krejˇ ciˇ r´ ık 08] Setup: ◮ planar waveguide Ω = R × I ◮ Robin-type boundary conditions ∂ u Ψ + i( α 0 + εβ )Ψ = 0 ◮ compactly supported perturbation β ∂ u Ψ + i α Ψ = 0 u d x 1 ∂ u Ψ + i α Ψ = 0

Previous results [Borisov, Krejˇ ciˇ r´ ık 08] Setup: ◮ planar waveguide Ω = R × I ◮ Robin-type boundary conditions ∂ u Ψ + i( α 0 + εβ )Ψ = 0 ◮ compactly supported perturbation β Results: ◮ conditions on existence and uniqueness of the bound state ◮ eigenvalue expansion up to order of ε 4 ◮ wavefunction asymptotics ∂ u Ψ + i α Ψ = 0 u d x 1 ∂ u Ψ + i α Ψ = 0

Definition of the Hamiltonian The waveguide Ω := R n × (0 , d ) = R n × I H α Ψ := − ∆Ψ , � ∂ u Ψ + i α Ψ = 0 Ψ ∈ W 2 , 2 (Ω) � � � Dom( H α ) := on ∂ Ω u x 1 x 2 ∂ u Ψ + i α Ψ = 0 d ∂ u Ψ + i α Ψ = 0

Definition of the Hamiltonian The waveguide Ω := R n × (0 , d ) = R n × I H α Ψ := − ∆Ψ , Ψ ∈ W 2 , 2 (Ω) � ∂ u Ψ + i α Ψ = 0 � � � Dom( H α ) := on ∂ Ω Theorem Let α ∈ W 1 , ∞ ( R n ) be real-valued. Then H α is an m-sectorial operator on L 2 (Ω). Im η θ γ Re η

Definition of the Hamiltonian The waveguide Ω := R n × (0 , d ) = R n × I H α Ψ := − ∆Ψ , � ∂ u Ψ + i α Ψ = 0 � Ψ ∈ W 2 , 2 (Ω) � � Dom( H α ) := on ∂ Ω Theorem Let α ∈ W 1 , ∞ ( R n ) be real-valued. Then H α is an m-sectorial operator on L 2 (Ω). Idea of the proof: � |∇ Ψ( x , u ) | 2 d x d u h α [Ψ] := Ω � � R n α ( x ) | Ψ( x , d ) | 2 d x − i R n α ( x ) | Ψ( x , 0) | 2 d x + i ◮ h α is sectorial and closed ◮ First representation theorem

Definition of the Hamiltonian The waveguide Ω := R n × (0 , d ) = R n × I H α Ψ := − ∆Ψ , � ∂ u Ψ + i α Ψ = 0 � Ψ ∈ W 2 , 2 (Ω) � � Dom( H α ) := on ∂ Ω Theorem Let α ∈ W 1 , ∞ ( R n ) be real-valued. Then H α is an m-sectorial operator on L 2 (Ω). Idea of the proof: � |∇ Ψ( x , u ) | 2 d x d u h α [Ψ] := Ω � � R n α ( x ) | Ψ( x , d ) | 2 d x − i R n α ( x ) | Ψ( x , 0) | 2 d x + i ◮ h α is sectorial and closed ◮ First representation theorem Note that H ∗ α = H − α

Symmetries of H α The spatial reflection operator P and the time reversal operator T : ( P Ψ)( x , u ) := Ψ( x , d − u ) , ( T Ψ)( x , u ) := Ψ( x , u ) .

Symmetries of H α The spatial reflection operator P and the time reversal operator T : ( P Ψ)( x , u ) := Ψ( x , d − u ) , ( T Ψ)( x , u ) := Ψ( x , u ) . Proposition Let α ∈ W 1 , ∞ ( R n ) be real-valued. Then H α is PT -symmetric, i.e. [ H α , PT ] = 0. ⇒ λ ∈ σ ( H ) ⇔ λ ∈ σ ( H )

Symmetries of H α The spatial reflection operator P and the time reversal operator T : ( P Ψ)( x , u ) := Ψ( x , d − u ) , ( T Ψ)( x , u ) := Ψ( x , u ) . Proposition Let α ∈ W 1 , ∞ ( R n ) be real-valued. Then H α is PT -symmetric, i.e. [ H α , PT ] = 0. ⇒ λ ∈ σ ( H ) ⇔ λ ∈ σ ( H ) Proposition Let α ∈ W 1 , ∞ ( R n ) be real-valued. Then H α is T -self-adjoint, i.e T H α T = H ∗ α ⇒ σ r ( H α ) = ∅