Optimisation of the lowest eigenvalue induced by surface singular - PowerPoint PPT Presentation

Optimisation of the lowest eigenvalue induced by surface singular interactions Vladimir Lotoreichik in collaboration with Pavel Exner and David Krejčiřík Nuclear Physics Institute, Czech Academy of Sciences Chaos17, Hradec Králové, Czech Republic, 10.05.2017 V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 1 / 17

From classical to spectral isoperimetric inequality V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 2 / 17

From classical to spectral isoperimetric inequality A bounded domain Ω ⊂ R d ( d ≥ 2) with smooth boundary ∂ Ω; ball B⊂ R d . V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 2 / 17

From classical to spectral isoperimetric inequality A bounded domain Ω ⊂ R d ( d ≥ 2) with smooth boundary ∂ Ω; ball B⊂ R d . Self-adjoint Dirichlet Laplacian − ∆ Ω D in L 2 (Ω) Spectrum of − ∆ Ω 1 (Ω) > 0 – the lowest eigenvalue of − ∆ Ω D is discrete. λ D D . V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 2 / 17

From classical to spectral isoperimetric inequality A bounded domain Ω ⊂ R d ( d ≥ 2) with smooth boundary ∂ Ω; ball B⊂ R d . Self-adjoint Dirichlet Laplacian − ∆ Ω D in L 2 (Ω) Spectrum of − ∆ Ω 1 (Ω) > 0 – the lowest eigenvalue of − ∆ Ω D is discrete. λ D D . Isoperimetric inequalities   | ∂ Ω | = | ∂ B| | Ω | < |B| (geometric)   = ⇒ λ D 1 (Ω) > λ D Ω ≇ B 1 ( B ) (spectral)   V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 2 / 17

From classical to spectral isoperimetric inequality A bounded domain Ω ⊂ R d ( d ≥ 2) with smooth boundary ∂ Ω; ball B⊂ R d . Self-adjoint Dirichlet Laplacian − ∆ Ω D in L 2 (Ω) Spectrum of − ∆ Ω 1 (Ω) > 0 – the lowest eigenvalue of − ∆ Ω D is discrete. λ D D . Isoperimetric inequalities   | ∂ Ω | = | ∂ B| | Ω | < |B| (geometric)   = ⇒ λ D 1 (Ω) > λ D Ω ≇ B 1 ( B ) (spectral)   Geometric: Steiner-1842, Hurwitz-1902 ( d = 2), corollary of Brunn-Minkowski inequality ( d ≥ 3). Spectral: Faber-23 , Krahn-26 . V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 2 / 17

From classical to spectral isoperimetric inequality A bounded domain Ω ⊂ R d ( d ≥ 2) with smooth boundary ∂ Ω; ball B⊂ R d . Self-adjoint Dirichlet Laplacian − ∆ Ω D in L 2 (Ω) Spectrum of − ∆ Ω 1 (Ω) > 0 – the lowest eigenvalue of − ∆ Ω D is discrete. λ D D . Isoperimetric inequalities   | ∂ Ω | = | ∂ B| | Ω | < |B| (geometric)   = ⇒ λ D 1 (Ω) > λ D Ω ≇ B 1 ( B ) (spectral)   Geometric: Steiner-1842, Hurwitz-1902 ( d = 2), corollary of Brunn-Minkowski inequality ( d ≥ 3). Spectral: Faber-23 , Krahn-26 . Other boundary conditions The Neumann Laplacian: similar spectral inequality is trivial: λ N 1 (Ω) = 0. Non-trivial for δ -interactions on manifolds and for the Robin Laplacian. V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 2 / 17

I. Schrödinger operators with δ -interactions on hypersurfaces V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 3 / 17

Definition of Hamiltonians with surface δ -interactions V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 4 / 17

Definition of Hamiltonians with surface δ -interactions A smooth hypersurface Σ ⊂ R d , not necessarily bounded or closed. V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 4 / 17

Definition of Hamiltonians with surface δ -interactions A smooth hypersurface Σ ⊂ R d , not necessarily bounded or closed. Symmetric quadratic form in L 2 ( R d ) H 1 ( R d ) ∋ u �→ h Σ α [ u ] := �∇ u � 2 L 2 ( R d ; C d ) − α � u | Σ � 2 L 2 (Σ) for α > 0. V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 4 / 17

Definition of Hamiltonians with surface δ -interactions A smooth hypersurface Σ ⊂ R d , not necessarily bounded or closed. Symmetric quadratic form in L 2 ( R d ) H 1 ( R d ) ∋ u �→ h Σ α [ u ] := �∇ u � 2 L 2 ( R d ; C d ) − α � u | Σ � 2 L 2 (Σ) for α > 0. The quadratic from h Σ α is closed, densely defined, and semi-bounded. V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 4 / 17

Definition of Hamiltonians with surface δ -interactions A smooth hypersurface Σ ⊂ R d , not necessarily bounded or closed. Symmetric quadratic form in L 2 ( R d ) H 1 ( R d ) ∋ u �→ h Σ α [ u ] := �∇ u � 2 L 2 ( R d ; C d ) − α � u | Σ � 2 L 2 (Σ) for α > 0. The quadratic from h Σ α is closed, densely defined, and semi-bounded. Schrödinger operator with δ -interaction on Σ of strength α H Σ α – self-adjoint operator in L 2 ( R d ) associated to the form h Σ α . V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 4 / 17

Definition of Hamiltonians with surface δ -interactions A smooth hypersurface Σ ⊂ R d , not necessarily bounded or closed. Symmetric quadratic form in L 2 ( R d ) H 1 ( R d ) ∋ u �→ h Σ α [ u ] := �∇ u � 2 L 2 ( R d ; C d ) − α � u | Σ � 2 L 2 (Σ) for α > 0. The quadratic from h Σ α is closed, densely defined, and semi-bounded. Schrödinger operator with δ -interaction on Σ of strength α H Σ α – self-adjoint operator in L 2 ( R d ) associated to the form h Σ α . The lowest spectral point for H Σ α 1 (Σ) := inf σ (H Σ µ α α ). V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 4 / 17

Motivations to study H Σ α V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 5 / 17

Motivations to study H Σ α Physics (i) ‘Leaky’ quantum systems: a particle is confined to Σ but the tunneling between different parts of Σ is not neglected. (ii) Inverse scattering problem for H Σ α is linked to the Calderon problem with non-smooth conductivity. (iii) Existence of spectral gaps for high-contrast photonic crystals. V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 5 / 17

Motivations to study H Σ α Physics (i) ‘Leaky’ quantum systems: a particle is confined to Σ but the tunneling between different parts of Σ is not neglected. (ii) Inverse scattering problem for H Σ α is linked to the Calderon problem with non-smooth conductivity. (iii) Existence of spectral gaps for high-contrast photonic crystals. Spectral geometry Characterise the spectrum of H Σ α in terms of Σ! V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 5 / 17

Motivations to study H Σ α Physics (i) ‘Leaky’ quantum systems: a particle is confined to Σ but the tunneling between different parts of Σ is not neglected. (ii) Inverse scattering problem for H Σ α is linked to the Calderon problem with non-smooth conductivity. (iii) Existence of spectral gaps for high-contrast photonic crystals. Spectral geometry Characterise the spectrum of H Σ α in terms of Σ! • An explicit mapping Σ �→ σ (H Σ α ) can not be constructed. • Particular spectral results might be very difficult to obtain. V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 5 / 17

Motivations to study H Σ α Physics (i) ‘Leaky’ quantum systems: a particle is confined to Σ but the tunneling between different parts of Σ is not neglected. (ii) Inverse scattering problem for H Σ α is linked to the Calderon problem with non-smooth conductivity. (iii) Existence of spectral gaps for high-contrast photonic crystals. Spectral geometry Characterise the spectrum of H Σ α in terms of Σ! • An explicit mapping Σ �→ σ (H Σ α ) can not be constructed. • Particular spectral results might be very difficult to obtain. Brasche-Exner-Kuperin-Šeba-94, Exner-Ichinose-01,... V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 5 / 17

δ -interactions on loops V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 6 / 17

δ -interactions on loops C ∞ -smooth loop Σ ⊂ R 2 , a circle C ⊂ R 2 . Regularity – not the main issue. Σ C V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 6 / 17

δ -interactions on loops C ∞ -smooth loop Σ ⊂ R 2 , a circle C ⊂ R 2 . Regularity – not the main issue. Σ C σ ess (H Σ α ) = R + and σ d (H Σ α ) � = ∅ for all α > 0. V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 6 / 17

δ -interactions on loops C ∞ -smooth loop Σ ⊂ R 2 , a circle C ⊂ R 2 . Regularity – not the main issue. Σ C σ ess (H Σ α ) = R + and σ d (H Σ α ) � = ∅ for all α > 0. Theorem ( Exner-05, Exner-Harrell-Loss-06 ) � | Σ | = |C| µ α 1 ( C ) > µ α = ⇒ 1 (Σ) , ∀ α > 0 . Σ ≇ C V. Lotoreichik (NPI CAS) Optimisation of the lowest eigenvalue... 10.05.2017 6 / 17