Spectra of definite type in waveguide models V. Lotoreichik in - PowerPoint PPT Presentation

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

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

Hamiltonian V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 3 / 17

Hamiltonian Ω = R × ( − π/ 2 , π/ 2) with opposite sides Σ ± = R × {± π/ 2 } y Σ + π Ω x O Σ − V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 3 / 17

Hamiltonian Ω = R × ( − π/ 2 , π/ 2) with opposite sides Σ ± = R × {± π/ 2 } y Σ + π Ω x O Σ − α = α ℜ + i α ℑ : R → C where α ℜ , α ℑ ∈ L ∞ ( R ; R ) V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 3 / 17

Hamiltonian Ω = R × ( − π/ 2 , π/ 2) with opposite sides Σ ± = R × {± π/ 2 } y Σ + π Ω x O Σ − α = α ℜ + i α ℑ : R → C where α ℜ , α ℑ ∈ L ∞ ( R ; R ) Hamiltonian of PT -symmetric waveguide ( Borisov-Křejčiřík-08 ) H α u = − ∆ u , dom H α = { u : u , ∆ u ∈ L 2 (Ω) , ( α ℜ ± i α ℑ ) u | Σ ± = ∂ ν u | Σ ± } ⋆ m-sectorial in L 2 (Ω) and non-selfadjoint if α ℑ � = 0 ⋆ J-selfadjoint with (J u )( x , y ) = u ( x , − y ); H ∗ α = JH α J. V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 3 / 17

Is the spectrum of H α real? V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 4 / 17

Is the spectrum of H α real? The question makes sense only for α ℑ � = 0 as otherwise H α = H ∗ α . V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 4 / 17

Is the spectrum of H α real? The question makes sense only for α ℑ � = 0 as otherwise H α = H ∗ α . σ (H α ) ⊂ R for α ℜ = 0 (Borisov-Křejčiřík-08, Novak-16) (i) α = i α ℑ for α ℑ ( x ) = − α ℑ ( − x ) ∈ C ∞ 0 ( R ). (ii) α = i( α 0 + εβ ) for α 0 ∈ (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

Is the spectrum of H α real? The question makes sense only for α ℑ � = 0 as otherwise H α = H ∗ α . σ (H α ) ⊂ R for α ℜ = 0 (Borisov-Křejčiřík-08, Novak-16) (i) α = i α ℑ for α ℑ ( x ) = − α ℑ ( − x ) ∈ C ∞ 0 ( R ). (ii) α = i( α 0 + εβ ) for α 0 ∈ (0 , 1), β ∈ C ∞ 0 ( R ; R ) & ε > 0 small. 0 ( R ) can be replaced by more general C 0 ( R ) ∩ W 1 C ∞ ∞ ( R ). V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 4 / 17

Is the spectrum of H α real? The question makes sense only for α ℑ � = 0 as otherwise H α = H ∗ α . σ (H α ) ⊂ R for α ℜ = 0 (Borisov-Křejčiřík-08, Novak-16) (i) α = i α ℑ for α ℑ ( x ) = − α ℑ ( − x ) ∈ C ∞ 0 ( R ). (ii) α = i( α 0 + εβ ) for α 0 ∈ (0 , 1), β ∈ C ∞ 0 ( R ; R ) & ε > 0 small. 0 ( R ) can be replaced by more general C 0 ( R ) ∩ W 1 C ∞ ∞ ( R ). σ (H α ) �⊂ R (Křejčiřík-Tater-08) Numerical experiment for α ℜ = 0 and α ℑ = 1 − ε exp( − x 2 10 ). V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 4 / 17

Is the spectrum of H α real? The question makes sense only for α ℑ � = 0 as otherwise H α = H ∗ α . σ (H α ) ⊂ R for α ℜ = 0 (Borisov-Křejčiřík-08, Novak-16) (i) α = i α ℑ for α ℑ ( x ) = − α ℑ ( − x ) ∈ C ∞ 0 ( R ). (ii) α = i( α 0 + εβ ) for α 0 ∈ (0 , 1), β ∈ C ∞ 0 ( R ; R ) & ε > 0 small. 0 ( R ) can be replaced by more general C 0 ( R ) ∩ W 1 C ∞ ∞ ( R ). σ (H α ) �⊂ R (Křejčiřík-Tater-08) Numerical experiment for α ℜ = 0 and α ℑ = 1 − ε exp( − x 2 10 ). A necessary & sufficient condition for σ (H α ) ⊂ R can hardly be found! V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 4 / 17

Motivation V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 5 / 17

Motivation Previously applied methods for showing σ (H α ) ⊂ R (i) Separation of variables for α = const reduces to realness of spectra for 1-D model operators. (ii) σ ess (H α ) ⊂ R for α � = const follows by compact perturbation of H α 0 with α 0 = const . (iii) σ d (H α ) ⊂ 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

Motivation Previously applied methods for showing σ (H α ) ⊂ R (i) Separation of variables for α = const reduces to realness of spectra for 1-D model operators. (ii) σ ess (H α ) ⊂ R for α � = const follows by compact perturbation of H α 0 with α 0 = const . (iii) σ d (H α ) ⊂ 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 (H α ) ⊂ R (at least in a weak sense). V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 5 / 17

Local spectral properties of H α V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 6 / 17

Local spectral properties of H α Question I: local realness of σ (H α ) σ (H α ) ∩ U ⊂ R for U ⊂ C ? V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 6 / 17

Local spectral properties of H α Question I: local realness of σ (H α ) σ (H α ) ∩ U ⊂ R for U ⊂ C ? Reduces to realness of σ (H α ) for U = C . V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 6 / 17

Local spectral properties of H α Question I: local realness of σ (H α ) σ (H α ) ∩ U ⊂ R for U ⊂ C ? Reduces to realness of σ (H α ) for U = C . � λ ∈ ρ (T): � (T − λ ) − 1 � > ε − 1 � ∪ σ (T) ( ε -pseudospectrum) σ ε (T) := V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 6 / 17

Local spectral properties of H α Question I: local realness of σ (H α ) σ (H α ) ∩ U ⊂ R for U ⊂ C ? Reduces to realness of σ (H α ) for U = C . � λ ∈ ρ (T): � (T − λ ) − 1 � > ε − 1 � ∪ σ (T) ( ε -pseudospectrum) σ ε (T) := Normal local behaviour of σ ε (T) with σ (T) ∩ U ⊂ R σ ε (T) ∩ U ⊂ { λ ∈ U : |ℑ λ | ≤ C ε 1 / 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

Local spectral properties of H α Question I: local realness of σ (H α ) σ (H α ) ∩ U ⊂ R for U ⊂ C ? Reduces to realness of σ (H α ) for U = C . � λ ∈ ρ (T): � (T − λ ) − 1 � > ε − 1 � ∪ σ (T) ( ε -pseudospectrum) σ ε (T) := Normal local behaviour of σ ε (T) with σ (T) ∩ U ⊂ R σ ε (T) ∩ U ⊂ { λ ∈ U : |ℑ λ | ≤ C ε 1 / m } for U ⊂ C with C > 0 and m ∈ N . Question II: local behaviour of σ ε (H α ) Does σ ε (H α ) ∩ U for U ⊂ C have normal behaviour? σ ε (H α ) is important in the analysis of t �→ e − t H α and of t �→ e i t H α . V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 6 / 17

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

Non-compact perturbations V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 8 / 17

Non-compact perturbations Assumption (a) α 0 ∈ R , α 0 � = 1 , and M := { n 2 } n ∈ N ∪ { α 2 0 } (b) µ 0 := min M and µ 1 := min( M \ { µ 0 } ) , ( µ 0 < µ 1 )! σ (H i α 0 ) = [ µ 0 , ∞ ). V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 8 / 17

Non-compact perturbations Assumption (a) α 0 ∈ R , α 0 � = 1 , and M := { n 2 } n ∈ N ∪ { α 2 0 } (b) µ 0 := min M and µ 1 := min( M \ { µ 0 } ) , ( µ 0 < µ 1 )! σ (H i α 0 ) = [ µ 0 , ∞ ). Fix a compact set F ⊂ C with F ∩ R ⊂ ( −∞ , µ 1 ) F µ 0 µ 1 V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 8 / 17

Non-compact perturbations Assumption (a) α 0 ∈ R , α 0 � = 1 , and M := { n 2 } n ∈ N ∪ { α 2 0 } (b) µ 0 := min M and µ 1 := min( M \ { µ 0 } ) , ( µ 0 < µ 1 )! σ (H i α 0 ) = [ µ 0 , ∞ ). Fix a compact set F ⊂ C with F ∩ R ⊂ ( −∞ , µ 1 ) F µ 0 µ 1 Theorem ( L-Siegl-16 ) There exists γ > 0 such that for all α : R → C with � α − i α 0 � ∞ < γ σ (H α ) ∩ F ⊂ R and σ ε (H α ) ∩ F behaves normally . V. Lotoreichik (NPI CAS) Spectra of definite type in waveguide models 08.06.2016 8 / 17