Dissipative quantum systems: scattering theory and spectral singularities Dissipative quantum systems: scattering theory and J´ er´ emy Faupin spectral singularities Introduction: The nuclear optical model J´ er´ emy Faupin The abstract framework of Universit´ e de Lorraine, Metz dissipative scattering theory Conference: The Analysis of Complex Quantum Systems Spectral singularities CIRM, October 2019 Asymptotic complete- ness Applications J.F. and J. Fr¨ ohlich, Asymptotic completeness in dissipative scattering theory , Adv. Math., 340, 300-362, (2018). J..F. and F. Nicoleau, Scattering matrices for dissipative quantum systems , J. Funct. Anal., 9, 3062-3097, (2019).
Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy 1 Introduction: The nuclear optical model Faupin Introduction: The nuclear optical 2 The abstract framework of dissipative scattering theory model The abstract framework of 3 Spectral singularities dissipative scattering theory Spectral singularities 4 Asymptotic completeness Asymptotic complete- ness Applications 5 Applications
Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear optical model Introduction The abstract framework The nuclear optical model of dissipative scattering theory Spectral singularities Asymptotic complete- ness Applications
Dissipative quantum systems: The nuclear optical model (I) scattering theory and spectral singularities Quantum system J´ er´ emy Faupin • Neutron targeted onto a complex nucleus Introduction: • Either the neutron is elastically scattered off the nucleus The nuclear optical • Or it is absorbed by the nucleus = ⇒ Formation of a compound nucleus model The • Concept of a compound nucleus was introduced by Bohr (’36) abstract framework of dissipative Model scattering theory • Feshbach, Porter and Weisskopf (’54): nuclear optical model describing both Spectral singularities elastic scattering and absorption Asymptotic • “Pseudo-Hamiltonian” on L 2 ( R 3 ) complete- ness H = − ∆ + V ( x ) − iW ( x ) Applications with V and W real-valued, compactly supported, W ≥ 0 • Widely used in Nuclear Physics, refined versions include, e.g., spin-orbit interactions • Empirical model
Dissipative quantum systems: The nuclear optical model (II) scattering theory and spectral singularities Interpretation J´ er´ emy Faupin e − itH � • − iH generates a strongly continuous semigroup of contractions � t ≥ 0 Introduction: • Dynamics described by the Schr¨ odinger equation The nuclear optical � i ∂ t u t = Hu t model u 0 ∈ D ( H ) The abstract If the neutron is initially in the normalized state u 0 , after a time t ≥ 0, it is in framework of the unnormalized state e − itH u 0 dissipative scattering • Probability that the neutron, initially in the normalized state u 0 (supposed to be theory orthogonal to bound states), eventually escapes from the nucleus: Spectral singularities � 2 � e − itH u 0 � � p scatt ( u 0 ) = lim Asymptotic t →∞ complete- ness • Probability of absorption: Applications � 2 � e − itH u 0 � � p abs ( u 0 ) = 1 − lim t →∞ • If p scatt ( u 0 ) > 0 (and u 0 is orthogonal to bound states), one expects that there exists an (unnormalized) scattering state u + such that � u + � 2 = p scatt ( u 0 ) and � = 0 � � e − itH u 0 − e it ∆ u + � lim t →∞
Dissipative quantum systems: The nuclear optical model (III) scattering theory and spectral singularities J´ er´ emy Faupin Aim Introduction: The nuclear • Explicit expression of H rests on experimental scattering data optical model • Nuclear optical model generalizes to any quantum system S interacting with another system S ′ and susceptible of being absorbed by S ′ The abstract framework • Need to develop the full scattering theory of a class of models of dissipative scattering theory References: mathematical scattering theory for dissipative operators in Spectral Hilbert spaces singularities Asymptotic complete- • Abstract framework: Lax-Phillips [’73], Martin [’75], Davies [’79,’80], Neidhardt ness [’85], Exner [’85], Petkov [’89], Kadowaki [’02,’03], Stepin [’04], . . . Applications • Small perturbations: Kato [’66], Falconi-F-Fr¨ ohlich-Schubnel [’17], . . . • Schr¨ odinger operators: Mochizuki [’68], Simon [’79], Wang-Zhu [’14], . . .
Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear optical model The abstract framework of The abstract framework dissipative scattering theory of dissipative scattering theory Spectral singularities Asymptotic complete- ness Applications
Dissipative quantum systems: Abstract model scattering theory and spectral singularities J´ er´ emy The model Faupin • H complex Hilbert space Introduction: The nuclear • Pseudo-Hamiltonian optical model H = H 0 + V − iC ∗ C = H V − iC ∗ C , The abstract with H 0 ≥ 0, V symmetric, C ∈ L ( H ) and V , C relatively compact with framework of respect to H 0 dissipative scattering • H V is self-adjoint, H is closed and maximal dissipative, with domains theory D ( H ) = D ( H V ) = D ( H 0 ) Spectral singularities • − iH generates a strongly continuous semigroup of contractions { e − itH } t ≥ 0 . Asymptotic complete- More precisely, − iH generates a group { e − itH } t ∈ R s.t. ness � e − itH � � ≤ 1 , t ≥ 0 , � e − itH � � ≤ e � C ∗ C �| t | , t ≤ 0 Applications � � • σ ess ( H ) = σ ess ( H 0 ) and σ ( H ) \ σ ess ( H ) consists of an at most countable number of eigenvalues of finite algebraic multiplicities that can only accumulate at points of σ ess ( H )
Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear optical model The abstract framework of dissipative scattering theory Figure : Form of the spectrum of H . Spectral singularities Asymptotic complete- ness Example Applications Example to keep in mind: 1 H = L 2 ( R 3 ) , H V = − ∆ + V ( x ) = H ∗ H 0 = − ∆ , V , C = W ( x ) 2
Dissipative quantum systems: Spectral subspaces scattering theory and spectral singularities Space of bound states J´ er´ emy Faupin H b ( H ) = Span � u ∈ D ( H ) , ∃ λ ∈ R , Hu = λ u � Introduction: The nuclear optical Generalized eigenstates corresponding to non-real eigenvalues model The • For λ ∈ σ ( H ) \ σ ess ( H ), Riesz projection defined by abstract framework 1 � of ( z Id − H ) − 1 dz , Π λ = dissipative 2 i π scattering γ theory where γ is a circle centered at λ , of sufficiently small radius Spectral singularities • Ran (Π λ ) spanned by generalized eigenvectors of H associated to λ , u ∈ D ( H k ) s.t. ( H − λ ) k u = 0 Asymptotic complete- ness • Space of generalized eigenstates corresponding to non-real eigenvalues: Applications � � H p ( H ) = Span u ∈ Ran (Π λ ) , λ ∈ σ ( H ) , Im λ < 0 “Dissipative space” t →∞ � e − itH u � = 0 � � H d ( H ) = u ∈ H , lim ⊃ H p ( H )
Dissipative quantum The adjoint operator H ∗ systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: Properties of H ∗ The nuclear optical model • H ∗ = H 0 + V + iC ∗ C The abstract • λ ∈ σ ( H ∗ ) if and only if ¯ λ ∈ σ ( H ) framework of • iH ∗ generates the strongly continuous contraction semigroup { e itH ∗ } t ≥ 0 dissipative scattering • Spectral subspaces theory Spectral H b ( H ∗ ) = Span u ∈ D ( H ) , ∃ λ ∈ R , H ∗ u = λ u � � , singularities H p ( H ∗ ) = Span u ∈ Ran (Π ∗ λ ) , λ ∈ σ ( H ∗ ) , Im λ > 0 � � Asymptotic , complete- ness t →∞ � e itH ∗ u � = 0 H d ( H ∗ ) = � u ∈ H , lim � Applications
Dissipative quantum systems: The wave operators scattering theory and spectral singularities The wave operator W − ( H , H 0 ) J´ er´ emy Faupin • Defined by t →∞ e − itH e itH 0 W − ( H , H 0 ) = s-lim Introduction: The nuclear optical • If it exists, W − ( H , H 0 ) is a contraction model • W − ( H , H 0 ) H 0 = HW − ( H , H 0 ) The abstract framework of dissipative The wave operator W + ( H 0 , H ) scattering theory • Defined by Spectral t →∞ e itH 0 e − itH Π b ( H ) ⊥ singularities W + ( H 0 , H ) = s-lim Asymptotic where Π b ( H ) ⊥ denotes the orthogonal projection onto H b ( H ) ⊥ complete- ness • If it exists, W + ( H 0 , H ) is a contraction Applications • H 0 W + ( H 0 , H ) = W + ( H 0 , H ) H • Under some conditions, W + ( H 0 , H ) = W + ( H ∗ , H 0 ) ∗ • For u ∈ H b ( H ) ⊥ , � e − itH u − e − itH 0 u + � = 0 � � u + = W + ( H 0 , H ) u ⇐ ⇒ lim t →∞
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