Introduction Real resonances Time-decay of solutions Time-decay of solutions to dissipative Schrödinger equations Xue Ping WANG Laboratoire de Mathématiques Jean Leray Université de Nantes, France
Introduction Real resonances Time-decay of solutions Outline Introduction 1 Dissipative Schrödinger operators An overview
Introduction Real resonances Time-decay of solutions Outline Introduction 1 Dissipative Schrödinger operators An overview Real resonances 2 Absence of low energy resonances Eigenvalues near zero
Introduction Real resonances Time-decay of solutions Outline Introduction 1 Dissipative Schrödinger operators An overview Real resonances 2 Absence of low energy resonances Eigenvalues near zero Time-decay of solutions 3 A representation formula Time-decay estimates Remark on positive resonances
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Dissipative Schrödinger operators This talk is concerned with the Schrödinger operator with a complex-valued potential x ∈ R n P = − ∆ + V ( x ) V ( x ) = V 1 ( x ) − iV 2 ( x ) , (1) with V j real, V 2 ( x ) ≥ 0, V 2 � = 0 (i.e., dissipative) and | V ( x ) | ≤ C � x � − ρ , x ∈ R n , ρ > 1 . (2)
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Dissipative Schrödinger operators This talk is concerned with the Schrödinger operator with a complex-valued potential x ∈ R n P = − ∆ + V ( x ) V ( x ) = V 1 ( x ) − iV 2 ( x ) , (1) with V j real, V 2 ( x ) ≥ 0, V 2 � = 0 (i.e., dissipative) and | V ( x ) | ≤ C � x � − ρ , x ∈ R n , ρ > 1 . (2) Physical backgrounds : Optical model of nuclear scattering, propagation of waves in media with variable absorption index, ...
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Dissipative Schrödinger operators We are interested in time-decay of solutions to the time-dependent Schrödinger equation i ∂ � Pu ( t ) , on R n , ∂ t u ( t ) = (3) u ( 0 ) = u 0 , Dissipation: � u ( t ) � L 2 is decreasing.
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Selfadjoint case When V ( x ) is real, P is selfadjoint and there are many results on time-decay estimates (local energy, Strichartz, dispersive). For example, when n = 3, ρ > 3 and s > 5 / 2 appropriately large, one has e − i λ t π λ + C t µ + O ( 1 e − itP = � t µ + ǫ ) , λ ∈ σ p ( P ) as operators L 2 , s → L 2 , − s . Here π λ is the spectral projection associated with the eigenvalue λ , L 2 , s = L 2 ( � x � 2 s dx ) .
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Selfadjoint case µ > 0 depends on threshold spectral properties of P : � 3 2 , if 0 is neither eigenvalue nor resonance of P , µ = 1 2 , if 0 is eigenvalue or resonance of P . 0 is called a resonance if Pu = 0 admits a solution u ∈ L 2 , − s \ L 2 for any s > 1 2 .
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Non-selfadjoint case There are several similarities between the dissipative Schrödinger equation and the damped wave equation. Time-decay for non-selfadjoint problems are studied by several authors. Here we only mention:
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Non-selfadjoint case There are several similarities between the dissipative Schrödinger equation and the damped wave equation. Time-decay for non-selfadjoint problems are studied by several authors. Here we only mention: G. Lebau (1994), damped waves in compact manifolds.
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Non-selfadjoint case There are several similarities between the dissipative Schrödinger equation and the damped wave equation. Time-decay for non-selfadjoint problems are studied by several authors. Here we only mention: G. Lebau (1994), damped waves in compact manifolds. L. Aloui, M. Khenissi (2002-2010), wave and Schrödinger equations in exterior domains.
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Non-selfadjoint case There are several similarities between the dissipative Schrödinger equation and the damped wave equation. Time-decay for non-selfadjoint problems are studied by several authors. Here we only mention: G. Lebau (1994), damped waves in compact manifolds. L. Aloui, M. Khenissi (2002-2010), wave and Schrödinger equations in exterior domains. M. Goldberg(2010), M. Beceanu and M. Goldberg(2012), Schrödinger equation with complex-valued potentials, under the assumption of the absence of real resonances in R + .
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Real resonances Definition. Let L 2 , s = L 2 ( R n ; � x � 2 s dx ) . We call λ ∈ R + a (real) resonance of P = − ∆ + V ( x ) ( V ( x ) complex-valued) if ( P − λ ) u = 0 admits a solution u ∈ L 2 , − s \ L 2 ,
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Real resonances Definition. Let L 2 , s = L 2 ( R n ; � x � 2 s dx ) . We call λ ∈ R + a (real) resonance of P = − ∆ + V ( x ) ( V ( x ) complex-valued) if ( P − λ ) u = 0 admits a solution u ∈ L 2 , − s \ L 2 , provided that ρ > 2 and s > 1 if λ = 0; ρ > 1 and s > 1 / 2 if λ > 0 and u satisfies one of the Sommerfeld’s radiation conditions: √ u ( x ) = e ± i λ | x | | x | n / 2 − 1 ( a ( θ ) + o ( 1 )) , θ = x / | x | for some a ∈ L 2 ( S n − 1 ) and a � = 0.
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Real resonances One knows very few about real resonances. Real resonances form a compact set of measure zero.
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Real resonances One knows very few about real resonances. Real resonances form a compact set of measure zero. It is unknown if the complex potentials without any real resonances are “generic".
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Real resonances One knows very few about real resonances. Real resonances form a compact set of measure zero. It is unknown if the complex potentials without any real resonances are “generic".
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Real resonances One knows very few about real resonances. Real resonances form a compact set of measure zero. It is unknown if the complex potentials without any real resonances are “generic". The goal of this talk is to show some time-decay estimates for the dissipative Schrödinger equation without assumptions on real resonances.
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Dissipative quantum scattering Since this is a Scattering Session, we end this Introduction by a word on dissipative quantum scattering. Physical models: H. Feshbach, Unified theory of nuclear reactions, Ann. Phys., (1958).
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Dissipative quantum scattering Since this is a Scattering Session, we end this Introduction by a word on dissipative quantum scattering. Physical models: H. Feshbach, Unified theory of nuclear reactions, Ann. Phys., (1958). Mathematical analysis: Ph. Martin (1976), E.B. Davies (1979-1982), B. Simon (1980).
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Dissipative quantum scattering Since this is a Scattering Session, we end this Introduction by a word on dissipative quantum scattering. Physical models: H. Feshbach, Unified theory of nuclear reactions, Ann. Phys., (1958). Mathematical analysis: Ph. Martin (1976), E.B. Davies (1979-1982), B. Simon (1980).
Introduction Dissipative Schrödinger operators Real resonances An overview Time-decay of solutions Dissipative quantum scattering Since this is a Scattering Session, we end this Introduction by a word on dissipative quantum scattering. Physical models: H. Feshbach, Unified theory of nuclear reactions, Ann. Phys., (1958). Mathematical analysis: Ph. Martin (1976), E.B. Davies (1979-1982), B. Simon (1980). Open question. Asymptotic completeness: Is the dissipative scattering operator bijective?
Introduction Absence of low energy resonances Real resonances Eigenvalues near zero Time-decay of solutions An example of positive resonance For any λ > 0, one can show that there exits C ∞ 0 potentials V ( x ) such that λ is a resonance of the dissipative Schrödinger operator P = − ∆ + V ( x ) .
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