DISSIPATIVE SCATTERING THEORY: AN OVERVIEW OF RECENT RESULTS (Mathematical aspects of quantum fields and related topics)

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(1)86. DISSIPATIVE SCATTERING THEORY: AN OVERVIEW OF RECENT RESULTS JÉRÉMY FAUPIN Abstract. ABSTRACT. We provide an introduction to the scattering theory of dissipative quantum systems ′ representing the long-time long‐time evolution of a system S interacting with another system SS' and susceptible of being absorbed by SS'′ . The effective dynamics of SS is generated by an operator of the form S , where H H = H0 + V − iC ∗ C on the Hilbert space of the pure states of S, H=H_{0}+V-iC^{*}C H_{0}0 is the self-adjoint self‐adjoint C is bounded. We define generator of the free dynamics of S, S, VV is symmetric and C define the basic objects of the scattering theory for the pair (H, H0 ),, next we review recent results on the spectral (H, H_{0}) singularities and the asymptotic completeness of the wave operators.. 1. Introduction INTRODUCTION 1.. When a physical quantum system interacts with another one, part of its energy may be irreversibly transferred to the other system. This phenomenon of irreversible loss of energy is usually called quantum quantum dissipation. In particular, fundamentally, quantum systems cannot be completely isolated from their environment and, therefore, any quantum system experiences quantum dissipation to some extent, due to interactions with the environment. This note is concerned with the mathematical study of effective effective or empirical models of quantum dissipation. We consider a quantum system SS interacting with another quantum system SS'′ . Our main concern is the understanding of the phenomenon of “capture”, i.e., we aim at studying models of“capture”, allowing for the description of both elastic scattering and absorption of SS by SS'′ . Such models apply to many physical situations. We focus first first on a typical example to fix fix the ideas. Consider a neutron targeted onto a complex nucleus. After interacting with the nucleus, the neutron may be elastically scattered off off the nucleus or be absorbed by the nucleus, leading to the formation of a compound nucleus. The concept of a compound nucleus was introduced by Bohr [1] [1] in 1936. In 1954, Feshbach, Porter and Weisskopf [10] [10] proposed a model describing the interaction of a neutron with a nucleus, allowing for the description of both elastic scattering and the formation of a compound nucleus. The force exerted by the nucleus on the neutron is modeled W are real-valued by a phenomenological potential of the form VV-iW − iW , where VV,, W ≥ 0. W\geq real‐valued and W The nucleus is supposed to be localized in space, which corresponds to the assumption that VV W are compactly supported or decay rapidly at infinity. and W infinity. On L L^{2}2(\mat(R hbb{R3}^{3), }) , in suitable units, the “pseudo-Hamiltonian” for the neutron is given by “pseudo‐Hamiltonian” H = −Δ + V − iW.. H=-\triangle+V-iW. (1.1) (1.1). H is called a pseudo-Hamiltonian Here, the operator H pseudo‐Hamiltonian as it generates a strongly continuous semigroup of contractions. The model of Feshbach, Porter and Weisskopf, or generalizations thereof, is called the nuclear optical optical model. \mathcal{H} In In [8, [8, 9], 9], the the scattering scattering theory theory for for aa class class of of abstract abstract pseudo-Hamiltonians pseudo‐Hamiltonians on on aa Hilbert Hilbert space space H is studied. In the abstract setting considered in [8, [8, 9], the pseudo-Hamiltonian pseudo‐Hamiltonian for the system SS is given by H = H0 + V − iC ∗ C, H=H_{0}+V-iC^{*}C,. 1.

(2) 87 J. FAUPIN. 2. \mathcal{H} with purely absolutely continuous spectrum, VV is symmetric where H H_{0}0 is a self-adjoint self‐adjoint operator on H C is bounded and relatively compact with respect and relatively compact with respect to H H_{0}0 , and C S . The main purpose to H H_{0}0 . The operator H H_{0}0 is the generator of the (unitary) (unitary) free dynamics of S. C is then to study the scattering theory for the pair (H, H0 ).. Suitable hypotheses on H H_{0}, 0 , VV and C (H, H_{0}) H is given by a are formulated in such a way that they can be verified verified in the particular case where H dissipative Schrödinger operator of the form (1.1). See the next section for more details. Prior to [8, [8, 9], mathematical scattering theory for dissipative operators on Hilbert spaces has been been considered considered by by many many authors authors (see, (see, e.g., e.g., [2, [2, 3, 3, 7, 7, 12, 12, 13, 13, 21] 21] and and references references therein). therein). In In these these H and H references, in particular, the existence of the wave operators associated to H H_{0}0 is established under various conditions. In this note, following [8, [8, 9], we go one step beyond by considering the asymptotic 9] asymptotic completeness completeness of the wave operators. We review recents results established in [8, [S, 9] showing that, under suitable assumptions, asymptotic completeness is equivalent to the absence of H. spectral singularities embedded into the essential spectrum of H. The paper is organized as follows. In Section 2, we introduce the main objects involved in dissipative scattering theory and we recall their basic properties. Section 3 is devoted to the notions of spectral singularities and asymptotic completeness. MATHEMATICAL setting SETTING 2. Mathematical. As mentioned in the introduction, we consider a quantum system SS interacting with another quantum system SS'′ and susceptible of being absorbed by SS'′ . Hence, in the main example we have in mind, SS is a neutron and SS'′ is a nucleus. The pure states of SS correspond to the normalized \mathcal{H} . The effective vectors in a complex Hilbert space H. effective dynamics of SS is supposed to be generated by \mathcal{H} , of the form a pseudo-Hamiltonian pseudo‐Hamiltonian acting on H, H = H0 + V − iC ∗ C = HV − iC ∗ C, H=H_{0}+V-iC^{*}C=H_{V}-iC^{*}C, where. \mathcal{H} corresponding to the generator of the free dynamics of S, • H H_{0}0 is a self-adjoint S, self‐adjoint operator on H ′ • VV-iC^{*}C − iC ∗ C is an effective effective interaction term due to the presence of SS' , with VV symmetric and, of ∗ C ≥ 0. course, C^{*}C\geq 0. course, C \bullet. \bullet. C which were In this section, we will state other abstract assumptions on the operators H H_{0}, 0 , VV and C introduced in [8, [8, 9] 9] in order to establish various results on the spectral and scattering theories for the pair (H, H0 ).. In the next section, we will recall that those abstract assumptions are fulfilled fulfilled in (H, H_{0}) our main example, namely the nuclear optical model. In this model HH_{0}=-\triangle 0 = −Δ on L L^{2}2(\mat(R hbb{R3}^{3)}) and VV 3 ∋ x → V (x) ∈ R, R3 ∋ x → C(x) ∈ C decaying and CC are multiplication operators by functions R \mathbb{R}^{3}\ni x\mapsto V(x)\in \mathbb{R}, \mathbb{R}^{3}\ni x\mapsto C(x)\in \mathbb{C} \infty. at ∞. H are denoted by To shorten notations below, the resolvents of the operators H H_{V} H_{0}, 0, H V and H −1 RR_{0}(z)=(H_{0}-z)^{-1} R,VR_{V}(z)=(H_{V}-z)^{-1}, (z) = (HV − z)−1 , R(z) = (H − z)−1 , 0 (z) = (H0 − z) , R(z)=(H-z)^{-1},. for any zz in the resolvent set of the corresponding operator. \mathcal{H} is denoted by 2.1. Basic assumptions and consequences. The set of bounded operators on H L(H). \mathcal{L}(\mathcal{H}) . The following basic assumptions are made:. Hypothesis Hypothesis 11 (Basic (Basic assumptions). (i) H_{0}\geq H_{0}0 is self-adjoint 0 ≥ 00 (or, (i) H (or, more generally, H self‐adjoint and and semi-bounded semi‐bounded from below), below), (ii) H_{0}, 0, (ii) VV is is symmetric symmetric and and relatively relatively compact compact with with respect respect to to H (iii) H_{0}. 0. (iii) C C\in \mat∈hcalL(H) {L}(\mathcal{H}) and and CC is relatively compact compact with respect to H.

(3) 88 DISSIPATIVE SCATTERING. 3. A on H \mathcal{H} is called dissipative We recall that an operator A D(A), Im(u, Au) ≤ \leq0. 0. dissipative if, for all uu\in∈\mathcal{D}(A), {\rm Im}(\{u, Au\rangle) A is called maximal dissipative A is dissipative and has no proper dissipative extension. Moreover, A dissipative if A Hypothesis 1 has the following simple consequences.. Proposition 2.1. Suppose that Hypothesis 11 holds. Then Then \mathcal{H} with domain (1) H H_{V}=H_{0}+V V = H0 + V is a V ) = D(H0 ).. a self-adjoint self‐adjoint operator operator on on H domain D(H \mathcal{D}(H_{V})=\mathcal{D}(H_{0}) \mathcal{H} with domain = D(H0 ).. (2) H = HV − iC ∗ C is a maximal dissipative H=H_{V}-iC^{*}C dissipative operator operator on on H domain D(H) \mathcal{D}(H)=\mathcal{D}(H_{0}) -iH generates (3) }t∈R (3) The The operator operator −iH generates aa strongly strongly continuous continuous group group {e \{e^{-−itH itH}\}_{t\in \mat hbb{R} such such that that −itH −itH e ≤ 1 if ≤ e C ∗ C |t| if e ≤ 0. ≥ 0, 0. if tt\geq 0, if tt\leq. \Vert e^{-itH}\Vert\leq 1. \Vert e^{-itH}\Vert\leq e^{\Vert C^{*}C\Vert|t|}. −itH } -iH generates the strongly continuous In particular, −iH continuous semigroup of of contractions contractions {e t≥00}.. \{e^{-itH}\}_{t\geq H is (4) The The adjoint adjoint of of H ∗ = H0 + V + iC ∗ C, H H^{*}=H_{0}+V+iC^{*}C, ∗ ∗ ) = D(H ). Moreover, iH iH^{*}∗ generates of with domain D(H domain \mathcal{D}(H^{*})=\mathcal{D}(H_{0}) of aa strongly continuous continuous group {e 0 . \{e^{ititH H^{*} \}_{t\in}\matt∈R hbb{R} ∗ itH such that {e }t≥00} is aa semigroup of of contractions. contractions. \{e^{itH^{*}}\}_{t\geq. Proof. (1) (1) is a simple consequence of the Kato-Rellich Kato‐Rellich Theorem together with the fact that VV is symmetric and relatively compact with respect to H H_{0}0 , and hence infinitesimally infinitesimally small with respect to H_{0}0 (see, to H (see, e.g., e.g., [17]). [17]). H is dissipative since, for all u To prove (2), we begin by observing that H D(H) = D(H0 ),, u\in∈\mathcal{D}(H)=\mathcal{D}(H_{0}) 2 {\rm Im} (\langle Cu Im(u, Hu) = −

(4) Cu

(5) ≤ 0. u, Hu\})=-\Vert \Vert^{2}\leq 0. H is maximal dissipative, by a theorem of Phillips [15], To verify that H [15], it then suffices to show that Ran (H-i\lambda)=\mathcal{H} Ran(H − iλ) = H for some λ\lambda>0 > 0.. We write H − iλ = (HV − iλ) Id − i(HV − iλ)−1 C ∗ C . H-i\lambda=(H_{V}-i\lambda)(Id-i(H_{V}-i\lambda)^{-1}C^{*}C) .. Since H > 0.. Moreover we have that H_{V} H_{V}-i\lambda self‐adjoint, H V is self-adjoint, V − iλ : D(H \mathcal{D}(H_{00)})arrow→ \mathcalH{H} is invertible for all λ\lambda>0 (HV − iλ)−1 C ∗ C ≤ λ−1

(6) C ∗ C

(7) ,. \Vert(H_{V}-i\lambda)^{-1}C^{*}C\Vert\leq\lambda^{-1}\Vert C^{*}C\Vert,. and hence Id H-i\lambda \mathcal{H}arrow hcal{H} is invertible for λ − i(HV − iλ)−1 C ∗ C :: H →\matH >

(8) CC^{*∗}C\Vert C

(9) .. This shows that H − iλ :: Id-i(H_{V}-i\lambda)^{-1}C^{*}C \lambda>\Vert ∗ C

(10) and hence concludes the proof of (2). D(H ) → H is invertible for λ >

(11) C \mathcal{D}(H_{00})arrow \mathcal{H} \lambda>\Vert C^{*}C\Vert −itHV } (3) H_{V} -iH_{V} V is V generates t∈RR}.. (3) Since Since H is self-adjoint, self‐adjoint, −iH generates aa strongly strongly continuous continuous unitary unitary group group {e \{e^{-itH_{V}}\}_{t\in ∗ C is bounded, a perturbation argument (see, e.g., [4, Theorem 11.4.1]) shows that Hence, since CC^{*}C (see, [4, ∗ -iH generates a strongly continuous group {e −iH }t∈R

(12) ≤ e CC^{*}C |t| R. \{e^{-−itH itH}\}_{t\in \mat hbb{R} such that

(13) e \Vert−itH e^{-itH}\Vert\l eq e^{\Vert C\Vert|t|} for all tt\in∈\mathbb{R}. −itH H is maximal The fact that ee^{-itH} is a contraction for tt\geq ≥ 0 is a consequence of the fact that H dissipative together with the Hille-Yosida Hille‐Yosida Theorem (see e.g. [4, [4, Theorem 8.3.2]). ∗ = H + V + iC ∗ C with domain H is given by H (4) H^{*}=H_{0}+V+iC^{*}C 0 (4) Standard arguments show that the adjoint of H ∗ -iH iH^{*}∗ generates of a strongly ) }= D(H0 ).. One then verifies, D(H verifies, in the same way as for −iH,, that iH \mathcal{D}(H^{* )=\mathcal{D}(H_{0}) ∗ ∗ \square }t≥00} is a semigroup of contractions continuous group {e \{e^{ititH H^{*} \}_{t\in}\matt∈R hbb{R} such that {e \{e^{iitH tH^{*}}\}_{t\geq. −itH } The contraction semigroup {e t≥0 \{e^{-itH}\}_{t\geq 0} has the interpretation of a dynamics in the following sense. H,

(14) u

(15) = 1,, represents the initial state of the quantum system SS at time tt=0 = 0,, then the If uu_{0}\in \mathcal{H}, 0 ∈ \Vert0u_{0}\Vert=1 state of SS at a positive time tt>0 > 0 is given by

(16) u e−itH u0 . Here it should be noted :=e^{-itH}u_{0} \Vert tu_{

(17) t}\−1 Vert^{-1u}u_{tt} , with uu_{t}t := −itH is a contraction, and that u = 0 since ee^{-itH} −itH is invertible. that

(18) u

(19) ≤ eq11 for all tt\geq ≥ 0 since ee^{-itH} u_{t}\neq 0 t \Verttu_{t}\Vert\l −itH u

(20) is Moreover, the semigroup property implies that, for all uu_{0}\in H,, the map [0, ∞)) ∋\ni t\mapsto\Vert t →

(21) ee^{-itH}u_{0}\Vert \mathcal{H} 0 ∈ 0 [0, \infty decreasing. Hence one can define define the probabilities of elastic scattering and absorption as follows.. H,

(22) u

(23) = 1.. The Definition 1. Suppose that Hypothesis 11 holds and let uu_{0}\in \mathcal{H}, The probability of of elastic elastic 0 ∈ \Vert0u_{0}\Vert=1 S , initially in the state u scattering of u_{0} of the system S, defined by by 0 , is defined 2 pscatt (u0 ) := lim e−itH u0 . pscatt. (u_{0}):=1\dot{ \imatt→∞ h} m\Vert e^{-itH}u_{0}\Vert^{2}tar ow\infty.

(24) 89 J. FAUPIN. 4. S , initially in the state u The u_{0} The probability of of absorption absorption of of the system S, 0 , is −itH 2 :=1 (u ) := 1 − lim u . pp_{abs}(u_{0}) e 0 abs 0. —tlt→∞ ım\infty| e‐itHu0 | 2 arrow. In scatt (u0 ) > 0,, and In the the case case where where pp_{scatt}(u_{0})>0 and if if uu_{0}0 is is orthogonal orthogonal to to the the bound bound states states of of SS (see (see the the next section for the definition definition of a bound state), a property which is expected to holds, sometimes ∈ H such that called weak asymptotic completeness, is that there exists a scattering state uu+\in + \mathcal{H}

(25) u

(26) = pscatt pscatt (u 0 ) and \Vert+u_{+}\Vert= (u_{0}) lim e−itH u0 − e−itH0 u+ = 0.. tarrow\infty 1\dot{ \imath} m\Vert e^{-itH}u_{0}-e^{-itH_{0} u+\Vert=0.. t→∞. This property will be discussed below in relation with the existence of wave operators.. H is maximal dissipative \mathcal{H} . Since H 2.2. Spectrum and spectral subspaces of H. – or equivalently dissipative—or -iH generates a strongly continuous semigroup of contractions −iH – another application of the Hillecontractions— Hille‐ H satisfies Yosida Theorem shows that the spectrum of H satisfies. σ(H) ⊂ {z ∈\mathbb{C}, C, Im(z) ≤ 0}. \sigma(H)\subset\{z\in {\rm Im}(z)\leq 0\}. H. In the remainder of this section, we review the definitions of important spectral subspaces of H. definitions \mathcal{D} is a subset of H, \overline{\mathcal{D} its closure. \mathcal{H} , we denote by D 2.2.1. The The space of of bound bound states. If D H is defined Definition 2. Suppose that Hypothesis 11 holds. The The space of of bound bound states of of H defined as as the H corresponding i.e. closure closure of of the vector space spanned by by all all eigenvectors eigenvectors of of H corresponding to real eigenvalues, eigenvalues, i.e.. Span{u ∈ D(H), ∃λ ∈\mathbb{R}, R, Hu = λu}. H b (H) :=:=Span\{u\in \mathcal{H}_{b}(H) \mathcal{D}(H), \exists\lambda\in. Hu=\lambda u\}. Similarly,. ∗ H ) := Span{u D(H ∈ R, H ∗Hu^{*}u=\l =ambda λu}. b (H \mathcal {H}_{b}(H^{* }) :=\overl ine{Span\{u\in ∈ \mathcal {D}(H^{*∗}),\),exists\l∃λ ambda \i n \mathbb{R}, u\}}.. H is self-adjoint, In the particular case were H = 0,, we see that the space of bound states self‐adjoint, i.e. CC=0 H usually denoted by H identifies (H).. In the general identifies with the pure point spectral subspace of H pp{H}_{pp}(H) \mathcal H are related as follows. case, H self‐adjoint part of H b (H) \mathcal {H}_{b}(H) and the pure point spectral subspace of the self-adjoint. Proposition 2.2. Suppose that Hypothesis 11 holds. Then Then ∗ H (HV ) ∩ Ker(C). b (H) = Hb (H ) ⊂ Hpp\mathcal{H}_{pp}(H_{V})nKer(C) \mathcal{H}_{b}(H)=\mathcal{H}_{b}(H^{*})\subset .. Proof. Proof. See See [8, [8, Lemma Lemma 3.1]. 3.1].. \square . H may be defined 2.2.2. Discrete and and essential essential spectra. The discrete and essential spectra of H defined as A on H \mathcal{H} with domain D(A) follows. We recall that an operator A λId) \mathcal{D}(A) is called Fredholm if Ran(A Ran ( A − —AId) \dimKer(A Ker ( A − co\dimRan(A Ran ( A − <\infty <\infty. is λId) < ∞ and λId) < ∞. is closed, closed, dim —AId) and codim —AId). Definition H, Definition 3 3 (Discrete (Discrete spectrum). Suppose that Hypothesis 11 holds. The The discrete discrete spectrum of of H, H with finite denoted by finite algebraic by σ\sigma_{disc}(H) of isolated eigenvalues eigenvalues of of H algebraic multiplicity. In other other disc (H),, is the set of \lambda is an Hu=\lambda D(H) \ {0} = λuu words, λ\lambda\in\sigma_{disc}(H) ∈ σdisc (H) if if λ an isolated point in σ(H), exists uu\in∈ \sigma(H) , there exists \mathcal{D}(H)\backslash \{0\} such that Hu \dimKer(H Ker ( H − <\infty. and dim λId) < ∞. —AId) Definition H, Definition 4 4 (Essential (Essential spectrum). Suppose that Hypothesis 11 holds. The The essential essential spectrum of of H, Hss not Fredholm. \lambda\i∈ n \matC hbb{C} such that H denoted by − λId by σ\sigma_{ess}(H) of λ AId is ess (H),, is the set of We mention that other possible definitions definitions of the essential spectrum may be found in the literature (see, e.g., [6, [6, Section IX]) but these different different definitions definitions coincide in our context [6, [6, Theorem IX.1.6]. H are related as follows. The discrete and essential spectra of H Proposition 2.3. Suppose that Hypothesis 11 holds. Then Then ∗ σ\sigma_{ess}(H_{0})=\sigma_{ess}(H_{V})=\sigma_{ess}(H)=\sigma_{ess}(H^{*})= \sigma(H)\backslash \ σdisc (H) = σ(H ∗ ) \\sigma_{disc}(H^{*} ) σdisc (H ∗ ). ess (H0 ) = σess (HV ) = σess (H) = σess (H ) = σ(H)\sigma_{disc}(H)=\sigma(H^{*})\backslash ..

(27) 90 DISSIPATIVE SCATTERING. 5. ∗ C are relatively compact C^{*}C Proof. The first first two equalities are consequences of the facts that VV and C Theorem 11.2.6]). The last equality is proven (see e.g. [4, perturbations of H H_{0} perturbations of 0 (see e.g. [4, Theorem 11.2.6]). The last equality is proven e.g. e.g. in in [6, [6, Theorem Theorem \square IX.1.6]. IX. 1.6]. H is of the form pictured in Figure 1. Summing up, the spectrum of H. H . The spectrum of H H is contained in the lower half-plane. Spectrum of H. half‐plane. The H coincides with that of H ∞). essential spectrum of H H_{0}0 and is contained in [0, [0, \infty ). The discrete spectrum H consists of isolated eigenvalues of finite of H finite algebraic multiplicities which may accumulate at any point of the essential spectrum.. Figure FIGURE 1. 1.. 2.2.3. Riesz projections. Definition 5. Suppose that Hypothesis 11 holds and ∈ σdisc (H).. The and let λ\lambda\in\sigma_{disc}(H) The Riesz projection correcorre‐ \lambda , denoted sponding to λ, \pi_{\laλ mbda} , is defined denoted by by π defined by by 1 (zId − H)−1 dz, πλ := 2iπ γ. \pi_{\lambda}:=\frac{1}{2\dot{ \imath} \pi}\int_{\gamma}(zId-H)^{-i}dz,. \lambda , oriented \lambda is the only where γ\gamma is aa circle circle centered centered at at λ, oriented counterclockwise counterclockwise and and such that λ only point of of σ(H) \sigma(H) contained contained in the interior of of γ. \gamma.. ∈ H is called a generalized eigenvector corresponding Given λ\lambda\in\sigma_{disc}(H) ∈ σdisc (H),, we recall that a vector uu\in \mathcal{H} − λ)k u = 0.. As is well-known, to λ\lambda if there exists a positive integer kk such that uu\in∈\mathcal{D}(H^{k}) D(H k ) and (H well‐known, (H-\lambda)^{k}u=0 the range of the Riesz projection π \pi_{\laλ mbda} coincides with the vector space spanned by all generalized \lambda. eigenvectors corresponding to λ. Proposition 2.4. Suppose that Hypothesis 11 holds and ∈ σdisc (H).. Then \pi_{\laλ mbda} is a and let λ\lambda\in\sigma_{disc}(H) Then π a projection \dim Ran(π Ran (\pi_{\lambda})<\infty and such that dim λ ) < ∞ and k Ran(π = u n∈\mathcal D(H − λ)k u = 0,, for some kk\in ∈ N, 11\leq ≤ k\leq\dim k ≤ dim Ran(πλ ) . \mathbb{N}, Ran (\piλ_{\l)ambda})=\{u\i (H-\lambda)^{k}u=0 {D}(H^{k}),), (H Ran(\pi_{\lambda})\}. Proof. See, e.g., [4, Theorem 1.5.4].. \square . H , one can prove that the only \lambda is a real isolated eigenvalue of H, In the particular case where λ \lambda are eigenvectors in the usual sense. possible generalized eigenvectors corresponding to λ. ∩ R.. Then ∈ nσ\sidisc Proposition 2.5. Suppose that Hypothesis 11 holds and and let λ\lambda\i Then gma_{dis(H) c}(H)\cap \mathbb{R} Ran (\pi_{\lambda})=\{u\in (H −(H-\lambda)u=0\}. λ)u = 0}. Ran(π λ ) = {u ∈ D(H), \mathcal{D}(H),. Proof. Proof. See See [8, [8, Lemma Lemma 3.3]. 3.3].. \square . H^{*}∗ and verify that statements Of course, one can define define Riesz projections in the same way for H ∗. H^{*}. analogous to Propositions 2.4–2.5 2.4−2.5 hold for H.

(28) g1 91 J. FAUPIN. 6. 2.2.4. The The dissipative dissipative space. Definition 6. Suppose that Hypothesis 11 holds. The The dissipative dissipative space, or or space of of decaying decaying states H , is defined of of H, defined by by Hd (H) := u ∈ H, lim e−itH u = 0 .. \mathcal{H}_{d}(H) :=\{u\in \mathcal{H}, tar ow\it→∞ nfty 1\dot{ \imath} m\Vert e^{ -itH}u\Vert=0\}.. Likewise,. *. ∗ ∗ := ∈ H, lim eitH uu\Vert=0 H d (H \mathcal {H}_{d}(H^{*)}) := { uu\in \mathcal{H} , t→∞ tlım lleıtH = 0 }.. arrow \infty. Recalling that the probability of absorption of an initial state uu has been defined defined in Definition Definition H,

(29) u

(30) = 1,, belongs to H 1, we see that uu_{0}\in \mathcal{H}, 0 ∈ d (H) abs (u0 ) = 1.. The dissipative \Vert0u_{0}\Vert=1 \mathcal {H}_{d}(H) if and only if pp_{abs}(u_{0})=1 H corresponding to eigenvalues with a strictly negative space contains all generalized eigenvectors of H imaginary part. More precisely, we introduce the following definition. definition. Definition 7. Suppose that Hypothesis 11 holds. The The subspace H closure of of the vector \mathpcal(H) {H}_{p}(H) is the closure H corresponding space spanned by by all generalized eigenvectors eigenvectors of of H corresponding to an an eigenvalue eigenvalue with aa strictly negative imaginary part, H Ran(π disc \matphcal(H) {H}_{p}(H:= ) :=\overlinue{\{u∈\in Ran( \pi_{\lambda}λ)),, \laλmbda\i∈n\siσgma_{ d\dot(H), { \imath} sc}Im(λ) (H),{\rm Im}(\l< ambda)0<0\}}.. Likewise,. ∗ H Ran(π disc \matdhcal(H {H}_{d}(H)^{*}:= ) :=\overlinue{\{u∈\in Ran( \pi_{\lambda}λ)),, \laλmbda\i∈n\siσgma_{ d\dot(H { \imat∗h),} sc}(HIm(λ) ^{*}),{\rm Im}(\l> ambda)0>0\}}... We then have the following proposition.. Proposition 2.6. Suppose that Hypothesis 11 holds. Then Then ⊥ ∗ ∗ ⊥ H ⊆eqH ⊆eqH , , \matH )⊆ )⊆ \mathpcal(H) {H}_{p}(H)\subset \mathdcal(H) {H}_{d}(H)\subset \mathbcal(H) {H}_{b}(H) ^{\perp} hcal{pH}_(H {p}(H^{*})\subset eq \matH hcal{dH}_(H {d}(H^{*})\subset eq \matHhcalb{H(H) }_{b}(H)^{\perp}... ⊆eq H ∈ σdisc (H), Im(λ) < 0 and let uu\in∈Ran(\pi_{\lambda}) Ran(πλ ).. Proof. First, we prove that H {\rm Im}(\lambda)<0 \matphcal(H) {H}_{p}(H)\subset \mathdcal(H). {H}_{d}(H) . Let λ\lambda\in\sigma_{disc}(H), Let kk=\dim = dim Ran(πλ ) < ∞.. We compute Ran(\pi_{\lambda})<\infty k−1 −itH (−it)j e u = etIm(λ) e−it(H−λ) u = etIm(λ) (H − λ)j u → 0, t → ∞, j!. \Vert e^{-itH}u\Vert=e^{t \rm Im}(\lambda)}\Vert e^{-it(H-\lambda)}u\Vert=e^{t {\rm Im}(\lambda)}\Vert\sum_{j=0}^{k-1}\frac{(-\dot{ \imath} t)^{j} {\dot{j}! (H -\lambda)^{j}u\Vertar ow 0, tar ow\infty, j=0. since. Im(λ) < 0.. Hence {\rm Im}(\lambda)<0. uu\in∈\mathcal{H}_{d}(H) Hd (H)... ⊥ . Let u ∈ H (H) and let vv be an eigenvector of H H^{*}∗ Next, we prove that H \mathdcal(H) {H}_{d}(H)\subset⊆eq \matHhcalb{H}(H) _{b}(H)^{\perp} u\in \mathcal{H}_{d}(H) d corresponding to a real eigenvalue. We have that v, u = eitH ∗ v, e−itH u = v, e−itH u ≤

(31) v

(32) e−itH u → 0, t → ∞. |\langle v, u\rangle|=|\{e^{itH^{*}}v, e^{-itH}u\rangle|=|\{v, e^{- \dot{{\imath}}tH}u\rangle|\leq\Vert v\Vert\Vert e^{-itH}u\Vertarrow 0, tarrow\infty.. H^{*}∗ corresponding to real eigenvalues, and therefore Hence uu is orthogonal to all eigenvectors of H ∗ )⊥ . Since H (H) = H (H ∗ ) by Proposition 2.2, this concludes the proof. uu\in∈ H (H b b \mathcal{Hb}_{b}(H^{*})^{\perp} \mathcal{H}_{b}(H)=\mathcal{H}_{b}(H^{* }) ∗ ⊆ H (H ∗ ) and H (H ∗ ) ⊆ H (H)⊥ are analogous. \square The proof of H \mathdcal{H}_{d}(H^{*})\subseteq \mathcalb{H}_{b}(H)^{\perp} \mathpcal(H {H}_{p}(H^{*)})\subset eq \mathdcal{H}_{d}(H^{*}). 2.2.5. The The absolutely continuous continuous spectral subspace. Now, we turn to a possible definition definition of an “ab“ab‐ H , following solutely solutely continuous continuous spectral spectral subspace" subspace”’ for for the the non-self-adjoint non‐self‐adjoint operator operator H, following Davies Davies [3]. [3]. H is Definition 8. Suppose that Hypothesis 11 holds. The The absolutely absolutely continuous continuous spectral subspace of of H defined defined by by H ac (H) := M (H), \mathcal{H}_{ac}(H):=\overline{M(H)}, where where ∞ −itH 2 e u, v dt ≤ cu

(33) v

(34) 2 . M (H) := u ∈ H, ∃cu > 0, ∀v ∈ H,. M(H) := \{u\in \mathcal{H}, \exists c_{u}>0, \foral v\in \mathcal{H}, \int_{0} ^{\infty}|\langle e^{-itH}u, v\rangle|^{2}dt\leq c_{u}\Vert v\Vert^{2}\}. 0. ∗. −itH itH H^{*}∗ is defined The The absolutely absolutely continuous continuous spectral subspace of of H defined similarly, replacing ee^{-itH} by by ee^{itH^{*}} in the definition definition above. above..

(35) 92 DISSIPATIVE SCATTERING. 7. H is self-adjoint, In the particular case where H self‐adjoint, the definition definition of H ac{(H) \mathcal H}_{ac}(H) coincides with the usual H . Moreover, if H H is self-adjoint, one based on the nature of the spectral measures of H. (H) is self‐adjoint, M M(H) closed and hence H definition of an absolutely continuous spectral ac (H) = M (H).. Another possible definition \mathcal{H}_{ac}(H)=M(H) H follows from the theory of unitary dilations of non-self-adjoint subspace of H non‐self‐adjoint operators, see e.g., [14]. [14]. The The relevance relevance of of Definition Definition 88 may may be be supported supported by by the the following following result. result.. Proposition 2.7. Suppose that Hypothesis 11 holds. Then Then ⊥ = hH H b (H) . \matac hcal(H) {H}_{ac}(H)=\mat cal{H}_{b}(H)^{\perp}.. In particular, ∗ H ac (H) = Hac (H ). d (H) ⊆ H\mathcal{H}_{ac}(H)=\mathcal{H}_{ac}(H^{* \mathcal{H}_{d}(H)\subseteq }) .. ⊥ Proof. The fact that H = H{H}_{b}(H)^{\perp} ac (H) b (H) is proven in [3]. \mathcal {H}_{ac}(H)=\mathcal [3]. The second equation is a direct consequence \square of Propositions 2.2 and 2.6. . We mention that another natural – and relevant – definition natural‐ relevant‐ definition for the absolutely continuous spectral H would be the orthogonal complement of all generalized eigenstates of H, H , namely subspace of H ⊥ H \t˜ilde{ac\mat(H) hcal{H} _{ac}:= (H):=(\matHhcal{bH}_(H) {b}(H)\oplu⊕ s \matH hcal{Hp}_(H) {p}(H) ^{\perp} .. According to the previous proposition and Definition Definition 8, we then have that ∞ −itH 2 ˜ (H) := u ∈ Hp (H)⊥ , ∃cu > 0, ∀v ∈ H, e H u, v dt ≤ cu

(36) v

(37) 2 . \t˜ilde{\ac mathcal(H) {H} _{ac}(H) := :=\overlM in˜e{\t(H), ilde{M}(H)}, M \tilde{M}(H). := \{u\in \mathcal{H}_{p}(H)^{\perp}, \exists c_{u}>0, \foral v\in \mathcal{H} , \int_{0}^{\infty}|\langle e^{-itH}u, v\}|^{2}dt\leq c_{u}\Vert v\Vert^{2}\}. 0. 2.3. The wave and scattering operators. In this section we define define the central objects in the scattering theory for the pair (H, H0 ),, namely, the wave operators, the scattering operator and the (H, H_{0}) scattering matrices. We begin by introducing hypotheses insuring that these objects are well-defined. well‐defined. A on H, (A), H (A) and H (A) 2.3.1. Hypotheses. Recall that, given a self-adjoint \mathcal{H}, H self‐adjoint operator A pp{H}_{pp}(A), ac{H}_{ac}(A) \mathcal \mathsccal{H}_{sc}(A) \mathcal A , redenote the pure point, absolutely continuous and singular continuous spectral subspaces of A, re‐ spectively. Likewise, σ\sigma_{pp}(A), pp (A), σ ac (A) and σ sc (A) denote the pure point, absolutely continuous and \sigma_{ac}(A) \sigma_{sc}(A) A. singular continuous spectra of A. Our first hypothesis concerns the spectra of the self-adjoint H_{0}0 and H H_{V} H_{0}0 V (recall that H first self‐adjoint operators H and H H_{V} V have the same essential spectrum, assuming Hypothesis 1).. Hypothesis H_{0}0 and H H_{V} V ). Hypothesis 2 2 (Spectra (Spectra of H (i) H_{0}0 is purely absolutely ac (H0 ) = σ(H0 ), σ pp (H0 ) = ∅, (i) The The spectrum of of H absolutely continuous, continuous, i.e., i. e., σ\sigma_{ac}(H_{0})=\sigma(H_{0}), \sigma_{pp}(H_{0})=\emptyset, σ\sigma_{sc}(H_{0})=\emptyset. sc (H0 ) = ∅. (ii) H finitely many eigenvalues H_{V} V has no singular spectrum, no embedded embedded eigenvalues, eigenvalues, and and only only finitely eigenvalues ) \mat ⊂hbb{RR}\backsl \ aσ(H Hpp (HV ) < ∞. counting counting multiplicity, i.e., i. e., σ\sigma_{sc}(H_{V})=\emptyset, and dim sc (HV ) = ∅, σ \sigpp ma_{p(H p}(H_{V}V)\subset sh \sigma(H_{00)}) and \dim \mathcal{H}_{pp}(H_{V})<\infty. (HV ).. The symbol s-lim We denote by Π s‐lim stands for ac (HV ) the orthogonal projection onto H pp{H}_{pp}(H_{V}) \Pi_{ac}(H_{V}) \mathcal strong limit. Our second hypothesis concerns the unitary wave operators associated to the selfself‐ H0 ) (in the statement of Hypothesis 3 below, it is tacitly assumed that Hypothesis adjoint pair (H V , H_{0}) (H_{V}, 2 holds). Hypothesis 0, H V )). Hypothesis 3 3 (Wave (Wave operators for (H (H_{0}, H_{V}) ). The The wave operators operators W± (HV , H0 ) := s-lim eitHV e−itH0 ,. itH0 −itHV. ) := s-lim e W_{\pm}(H_{V}, H_{0}) :=tarrow\pm\int→±∞ fty s-1\dot{ \imath} me^{itH_{V} e^{- itH_{0} , W_{W\pm}±((H H_{0},0H_{,VH}) :=Vtarrow\pm\i nft→±∞ ty s-1\dot{ \imath} me^{ietH_{0} e^{-itHΠ_{V}ac\Pi_(H {ac}(HV_{V),}) ,. exist exist and and are are asymptotically asymptotically complete, complete, i.e., i. e., Ran W , H0 ) = Hac (HV ) = Hpp (HV )⊥ , ± (HV H_{0})=\mathcal{H}_{ac}(H_{V})=\mathcal{H}_{pp}(H_{V})^{\perp} W_{\pm}(H_{V},. Ran W HV ) = H. ± (H0 , H_{V})=\mathcal{H}. W_{\pm}(H_{0},.

(38) 93 J. FAUPIN. 8. In our third assumption, we require that the operator CC be relatively smooth with respect to H H_{V} V in in the the sense sense of of Kato Kato [12]. [12]. Hypothesis H_{V} V ). There V > 0 such that, for Hypothesis 4 4 (Relative smoothness of CC with respect to H There exists exists cc_{V}>0 all uu\in∈\mathcal{H}_{ac}(H) Hac (H),, −itH 2 V Ce u dt ≤ cV

(39) u

(40) 2 .. \int_{\mathb {R} \Vert Ce^{-itH_{V} u\Vert^{2}dt\leq c_{V}\Vert u\Vert^{2}. R. In the remainder of this section, we study the wave and scattering operators for the pair (H, H0 ),, (H, H_{0}) assuming that Hypotheses 1–4 1‐4 hold.. ∗ H ). Assuming that H has purely absolutely 2.3.2. The H0 ) and H_{0}0 The wave operators operators W and W − (H,H_{0}) + (H ,H_{0}) 0 . W_{-}(H, W_{+}(H^{*}, ∗ H ) in dissipative scattering thecontinuous spectrum, the wave operators W (H, H the‐ − 0 ) and W + (H ,H_{0}) 0 W_{+}(H^{*}, W_{-}(H, H_{0}) ory are defined defined in the same way as in unitary scattering theory, namely. W− (H, H0 ) := s-lim e−itH eitH0 ,. itH ∗ −itH0. ∗. W_{-}(H, H_{0}) :=s-1\dot{ \imatht→∞ } me^{-itH}e^{itH_{0} tarrow\infty, W_{+W } (H+^{*(H }, H_{0}) ,:=s-H1\dot0){ \:= imath}s-lim me^{itH^{*}ee^{-itH_{0e} tarrow\infty,, t→∞. where, recall, s-lim s‐lim stands for strong limit. ∗ H ) are stated in the following H0 ) and W The existence and basic properties of W − (H,H_{0}) + (H ,H_{0}) 0 W_{-}(H, W_{+}(H^{*}, proposition. ∗ H ) exist and Proposition 2.8. Suppose that Hypotheses 1–4 H0 ) and 1-4 hold. Then Then W and W − (H,H_{0}) + (H ,H_{0}) 0 exist and W_{-}(H, W_{+}(H^{*}, are injective contractions. Moreover, contractions. ∗. −itH ∗ ∗ ee^{-itH}W_{-}(H, W− (H,H_{0})=W_{-}(H, H0 ) = W− (H, H0 )e−itH0e^{-itH^{*}}W_{+}(H^{*}, , e−itH W+ (H H_{0})=W_{+}(H^{*}, , H0 ) = W+ (HH_{0})e^{-itH_{0}}, , H0 )e−itH0 , H_{0})e^{-itH_{0}}, \mathbb{R} for all R,, and all tt\in∈ and. ∗ ⊥ Ran ) \subset⊆ \overline{RW anW_{−-}((H, H,H_{0})}H =(\mat0 )hcal={H}_{b}(H H)\oplbu(H) s \mathcal⊕{H}_{H d}(Hd^{*}(H ) ^ {\perp} eq \matHhcalac{H(H), }_{ac}(H) ⊥ ∗ , H ) = H (H) ⊕ H (H) Ran W (H ⊆ H \overline{RanW_{++}(H^{*},H_{0})}=(\mat0 hcal{H}_{b}(H)\oplb us \mathcal{H}_{d}(Hd) ^ {\perp}\subseteq \mathcalac{H(H). }_{ac}(H) ,. .. Proof. See [8]. [8].. \square . , H ∗ ).. (H ∗ ). 2.3.3. The (H) and H The wave operators operators W and W de‐ + (H0 , H) − (H0 H^{*}) . Recall that H ac{H}_{ac}(H) ac{H}_{ac}(H^{*}) are deW_{+}(H_{0}, H) and W_{-}(H_{0}, \mathcal \mathcal ∗ fined fined in Definition Definition 8. We denote by Π ac (H),, respectively Π ac (H ),, the orthogonal projection onto \Pi_{ac}(H) \Pi_{ac}(H^{*}) H , respectively H H^{*}∗ . The wave operators W the absolutely continuous spectral subspace of H, + (H0 , H) W_{+}(H_{0}, H) ∗ and W defined by − (H0 , H W_{-}(H_{0}, H^{*})) are defined W+ (H0 , H) := s-lim eitH0 e−itH Πac (H),. ∗. −itH0 itH ∗. ∗. s-lim e W_{+}(H_{0}, H) := s-1\dot{ \imt→∞ ath} me^{itH_{0} e^{-itH}\Pi_{ac}(H) tarrow\inftyW , W_{−-}((H H_{0}0, H^{, H*}) :=s-)\li:= m_{tarrow\i nfty}e^{-itH_{0} ee^{itH^{*} Π\Piac_{a(H c}(H^{*}).) t→∞ .. itH0 , we see that the existence of W (H , H) is equivalent to the property of weak Using unitarity of ee^{itH_{0}} + 0 H) W_{+}(H_{0}, asymptotic completeness mentioned above, in the following sense: for all uu_{0}\in Hb{H}_{b}(H)^{\perp}=\mathcal (H)⊥ = Hac{H}_{ac}(H) (H),, 0 ∈\mathcal −itH u − e−itH0 u

(41) → 0, as ttarrow\infty. → ∞. setting uu+ := W (H , H)u , we have that

(42) e + + :=W_{+}(H_{0}, + 0 H)u_{0}0 0 \Vert e^{-itH}u_{0}-e^{-itH_{0}}u+\Vertarrow 0, ∗ The existence and basic properties of W + (H0 , H) − (H0 , H W_{+}(H_{0}, H) and W W_{-}(H_{0}, H^{*})) are stated in the following proposition. ∗ Proposition 2.9. Suppose that Hypotheses 1–4 1-4 hold. Then Then W exist and and + (H0 , H) − (H0 , H W_{-}(H_{0}, H^{*})) exist W_{+}(H_{0}, H) and W are contractions. contractions. Moreover, ∗ W = W+ (H ∗ , H W− (H0 ,H^{*})^{*}=W_{-}(H, H ∗ )∗ = W− (H, H0 ). + (H0 , H) 0 ), , W_{-}(H_{0}, W_{+}(H_{0}, H)^{*}=W_{+}(H^{*}, H_{0}). H_{0}) .. In particular, ⊥ ∗ ∗ ⊥ Ker W+ (H Hb (H) ⊕H , KerW_{ Ker}(H_{0}, W−H^{* (H = Hbus(H) Hd (H}))^{\perp} ) , 0 , H) {= 0 , H {)H}_{b}(H)\opl d (H) KerW_{+}(H_{0}, H)=(\mathcal H}_{b}(H)\opl us \mathcal {H}_{d}(H))^{\perp} })=(\mathcal \mathcal⊕{H}_{d}(H^{*. ∗ and W and W dense ranges. + (H0 , H) − (H0 , H W_{+}(H_{0}, H) and W_{-}(H_{0}, H^{*})) have dense. Proof. See [8]. [8].. \square .

(43) 94 DISSIPATIVE SCATTERING. 9. We mention that similar results can be obtained using the Kato-Birman Kato‐Birman theory of trace-class trace‐class perturbations perturbations instead instead of of relatively relatively smooth smooth perturbations, perturbations, see see [2]. [2]. 2.3.4. The The scattering operators. operators. In dissipative scattering theory, the scattering operators are defined defined by ∗ ∗ ∗ , H0 )]∗ W− (H,H_{0}) H0 ),, S(H^{*}, S(H , H0 ) = [W− (H , H0 )]∗ W+ (H ∗ H_{0}) , H0 ). S(H, H0 ) = [W+ (HH_{0})]^{*}W_{-}(H, S(H, H_{0})=[W_{+}(H^{*}, H_{0})=[W_{-} (H^{*}, H_{0})]^{*}W_{+}(H^{*}, .. Combining Propositions 2.8 and 2.9, we arrive at the following result. ∗ H ) exist and Proposition 2.10. Suppose that Hypotheses 1–4 , H0 ) and 1-4 hold. Then Then S(H, and S(H 0 exist and S(H, , H_{0}) S(H^{*},,H_{0}) are contractions. Moreover, contractions. −itH0 ∗ ee^{-itH_{0}}S(H, S(H, H H0 )e−itH0 ,e^{-itH_{0}}S(H^{*}, e−itH0 S(H ∗H_{0})= S(H^{*}, , H0 ) = S(HH_{0})e^{-itH_{0}}, , H0 )e−itH0 , 0 ) = S(H, H_{0})=S(H, H_{0})e^{-itH_{0}}, \mathbb{R} R and we have that for all all tt\in∈. , H0 ). S(H, H0 )∗ = S(H ∗H_{0}) S(H, H_{0})^{*}=S(H^{*}, .. An important question, both mathematically and physically, concerns the invertibility of the scattering operators. Regarding this question, we can state the following proposition (see (see the next section for more precise results). Proposition 2.11. Suppose that Hypotheses 1–4 1-4 hold. Then Then the following conditions conditions are are equivalent: equivalent: ∗ , H ) are invertible in L(H). (1) S(H, H ) and S(H 0 and S(H^{*}, H_{0}) 0 are invertible in \mathcal{L}(\mathcal{H}) . (1) S(H, H_{0}) ∗ H ) are given by H0 ) and (2) The − (H,H_{0}) + (H ,H_{0}) 0 The range of of the wave operators operators W W_{-}(H, and W W_{+}(H^{*}, by ⊥ ∗ ⊥ ∗ Ran W H0 )hcal={H}_{b}(H)\opl Hb (H) ) , Ran W , H0 )hcal ={H}_{b}(H)\opl Hb (H) . + (H d (H}))^{\perp} d (H) W_{-−}(H,(H, H_{0})=(\mat us \mat⊕hcal{H H}_{d}(H^{* W_{+}(H^{* }, H_{0})=(\mat us \mat⊕hcalH{H}_{d}(H))^{\perp} Proof. See [8]. [8].. \square . 2.3.5. The The scattering matrices. We recall that the multiplicity of the spectrum of a self-adjoint self‐adjoint operator is defined defined via the spectral theorem (see, e.g., [16, [16, Section VII]). To study the scattering matrices, matrices, it it is is convenient convenient to to add add the the following following condition condition to to Hypothesis Hypothesis 2(i). 2(i). Hypothesis H_{0}0 has aa constant 0 )). Hypothesis 5 5 (Multiplicity of σ(H \sigma(H_{0}) ). The The spectrum of of H constant multiplicity (which (which may be be infinite). infinite). To simplify the notations below, we set Λ := σ(H0 ). \Lambda:=\sigma(H_{0}) .. Assuming Assuming Hypotheses Hypotheses 2(i) 2(i) and and 5, 5, the the spectral spectral theorem theorem ensures ensures that that there there exists exists aa unitary unitary mapping mapping \mathcal{H} to a direct integral of Hilbert spaces, from H ⊕ H(λ)dλ, F0 : H →. \mathcal{F}_{0}:\mathcal{H}ar ow\int_{\Lambda}^{\oplus}\mathcal{H}(\lambda) d\lambda, Λ. ∗ such that F 0) \mathcal{H}(\lambda) . Moreover, since σ(H \sigma(H_{0}) \mathc0al{F}_H {0}H_{0}\0mathF cal{F}_{00}^{*} acts as multiplication by λ\lambda on each Hilbert space H(λ). has a constant multiplicity, say kk\in∈\mathbb{N}\cup\{+\infty\} N ∪ {+∞},, all spaces H(λ) identified with a fixed fixed Hilbert \mathcal{H}(\lambda) can be identified \mathcal{M} . Hence F space M. \mathcal{F0}_{0} becomes an operator ⊕ M dλ = L2 (Λ; M), F0 : H →. \mathcal{F}_{0}:\mathcal{H}ar ow\int_{\Lambda}^{\oplus}\mathcal{M}d\lambda= L^{2}(\Lambda;\mathcal{M}). ,. Λ. \Lambda to M, \dim \mathcal{M}=k \mathcal{M} , (see e.g. (Λ;\matM) where dim M = k and L L^{22}(\Lambda; hcal{M}) is the space of square integrable functions from Λ 0 [23, Chapter 0, Section 0 = −Δ,, the [23, Chapter , Section 1.3]). 1.3]). Note Note that that in in the the case case where where H \mathcal={H}=L^{L2}(\2mat(R hbb{R}3^{3)}) and and HH_{0}=-\triangle the Hilbert Hilbert 2 ), where S \mathbb{3R}^{3.}. \mathcal{M} is \mathcal{M}=L^{2} space = L2 (S space M is given given by by M (S2), where S^{2}2 stands stands for for the the sphere sphere in in R.

(44) 95 J. FAUPIN. 10. Using that the scattering operator S(H, H0 ) commutes with H H_{0}0 , by Proposition 2.10, one can S(H, H_{0}) verify that ⊕ S(λ)dλ. F0 S(H, H0 )F0∗ =. \mathcal{F}_{0}S(H, H_{0})\mathcal{F}_{0}^{*}=\int_{\Lambda}^{\oplus} S(\lambda)d\lambda. Λ. The bounded operators. S(λ) ∈hcalL(M), S(\lambda)\in \mat {L}(\mathcal{M}) \l a mbda\i n \Lambda defined for a.e. λ ∈ Λ, are called the scattering matrices (for the pair (H, H0 )). defined , (H, H_{0}) ). ∗ ∗ H ) by the relation One can define define in the same way the scattering matrices S 0 S^{*}(\l(λ) ambda) for the pair (H (H^{*},, H_{0}) ⊕ S ∗ (λ)dλ. F0 S(H ∗ , H0 )F0∗ = ,. \mathcal{F}_{0}S(H^{*}, H_{0})\mathcal{F}_{0}^{*}=\int_{\Lambda}^{\oplus}S^{*} (\lambda)d\lambda. Λ. Under the conditions of Proposition 2.11, we then have that ∗ [S(λ)] = S ∗ (λ), [S(\lambda)]^{*}=S^{*}(\lambda) ,. a.e. λ \lambda\i for a.e. ∈n\Lambda Λ.. We set. ∗ F →L^ {L2}(\2Lambda; (Λ;\matM). \mat±hcal{F:= }_{\pm}:F =\mat0hW cal{F}_±{0}W(H _{\pm}V^{*}(H,_{VH }, H_{00})):\mat: hcalH{H}arrow hcal{M}) .. s (X; H) the set of Hölder \mathcal{H} , we denote by C Given ss\geq ≥ 0, and a Hilbert space H, 0 , an interval C^{S}(X;\mathcal{H}) X of order s. s. \mathcal{H} ‐valued functions on X continuous H-valued In order to insure that the map λ\lambda\mapsto → S(λ) S(\lambda) is continuous, it is convenient to require that the operators VV and CC are strongly smooth with respect to H H_{0}0 and H H_{V} V , respectively, in the following sense. sense. X X. Hypothesis H_{0}0 ). Hypothesis 6 6 (Strong (Strong smoothness smoothness of of VV with with respect respect to to H ). K :: G G :: H (i) There exist an auxiliary Hilbert space G and operators \ m a t h c a l { G } \mathcal{H→ }arrow \mathG cal{G} and \mathcal{→ G}ar ow \mathG cal{G} such (i) There exist an auxiliary Hilbert space and operators G and K such that that 1/2 −1 ∈ L(H; G) and K ∈ L(G). + 1) VV=G^{*}KG = G∗ KG,, with G(H G(H_{0}^{1/2}0 +1)^{-1}\in \mathcal{L}(\math∗cal{H};\mathcal{G}) and K\in \mathcal{L}(\mathcal{G}) . (ii) For all C, Im(z) = 0, \mathbb{C}, 0 (z)G is compact. all zz\in ∈ {\rm Im}(z)\neq 0, GR GR_{0}(z)G^{*} compact. 1 i.e. G is ⋐ Λ, i.e. (iii) H_{0}0 -smooth X\Subset\Lambda, 0 ∈ (( \f2rac{,1}1) (iii) The The operator operator G is strongly strongly H ‐smooth with with exponent exponent ss_{0}\in on any any compact compact set set X {2},1) on ∗ 0 continuous. →{G}arrow CsC^{s_{0}}(X; (X;\mathcal M) F 0 [G1X (H0 )] } : G \mathcal{F}_{0}[GI_{X}(H_{0})]^{* \mathcal {M}) is continuous.. Hypothesis H_{V} V ). Hypothesis 7 7 (Strong (Strong smoothness of CC with respect to H i.e. (i) ∈ (0, 1) on ⋐ Λ, i.e. H_{V} X\subset\Lambda, V -smooth (i) The The operator operator CC is strongly H ‐smooth with exponent exponent ss\in(0,1) on any any compact compact set X ∗ s continuous. F (Λ;\matH) ± [C1X (HV )] } : H \mathcal{F}_{\pm}[CI_{X}(H_{V})]^{* \mathcal→ {H}arrowC C^{s}(\Lambda; hcal{H}) is continuous. ∗ (ii) C, Im(z) = 0, \mathbb{C}, V (z)C is (ii) For For all all zz\in∈ {\rm Im}(z)\neq 0, CR CR_{V}(z)C^{*} is compact. compact. (iii) The map (iii) The ∗ ˚ Λ → \C RV (λo C(R_{ +V}i0) −0)-R L(H), \mat∈hring{λ\Lambda} in\lambda\mapst (\lambda+i R_{VV}(\l(λ ambda-−i0))C^{i0) *} \inC\mathcal∈{L}(\mat hcal{H}) ,. bounded. is bounded.. We refer to [22, [22, 23] for details on the theory of strongly smooth operators. In the statement below, Ss\#♯ stands for SS or SS^{*}∗ . Based on a generalization of Kuroda’s represenrepresen‐ tation tation formula, formula, the the following following result result was was established established in in [9]. [9]. Λ, Proposition 2.12. Suppose that Hypotheses 1–7 1‐7 hold. Then, Then, for all all λ \lambda\∈ in\mathring˚ {\Lambda}, SS\#(\l♯ (λ) contraction ambda) is aa contraction ♯ (λ) − Id is compact. If, in addition, dim M = +∞, then for all λ ∈ ˚ ♯ (λ)

(45) = 1 and, in and SS\#(\lambda)-Id Λ,

(46) S \dim \mathcal{M}=+\infty compact. addition, , all \lambda\in\mathring{\Lambda}, \Vert S\#(\l ambda)\Vert=1 and, ∗ , H )

(47) . particular,

(48) S(H, 0 )

(49) = 1 =

(50) S(H 0 \Vert S(H,H H_{0})\Vert=1=\Vert S(H^{*}, H_{0})\Vert. Proof. See [9]. [9].. \square .

(51) 96 DISSIPATIVE SCATTERING. 11. SPECTRAL singularities SINGULARITIES and AND asymptotic ASYMPTOTIC completeness COMPLETENESS 3. Spectral. Our next concern is to study more precisely the invertibility of the scattering matrices and operaopera‐ tor. Invertibility of S(λ) S(\lambda) is a strongly relevant physical property since it shows that to any incoming state at energy λ\lambda corresponds a unique outgoing state and vice versa. In Section 3.1, we explain that \lambda . Section non-invertibility non‐invertibility of S(λ) S(\lambda) is equivalent to the presence of a spectral singularity at energy λ. 3.2 is devoted to the property of asymptotic asymptotic completeness completeness of the wave operators. 3.1. Spectral singularities. Recall that, under our assumptions and notations, the essential specspec‐ H is given by σ trum of H ess (H) = σ(H0 ) = Λ.. We recall the notion of a spectral singularity introduced \sigma_{ess}(H)=\sigma(H_{0})=\Lambda \Lambda and points in the boundary Λ in [8, 9], distinguishing points in the interior of Λ Λ. \Lambda\back\slash\math˚ ring{\Lambda}. Definition 9. H if circ} . We (i) Λ. (i) Let λ\lambda\i∈n A^{\˚ We say that λ\lambda is aa regular spectral point of of H if there exists exists aa compact compact interval \lambda , such that K K ⊂\matR K_{\lambda} hbb{R} whose interior contains K_{\lambda} contains λ, does not contain contain any any accumulation point of of λ \subset λ does H , and eigenvalues eigenvalues of of H, and such that the limits CR(µ − i0)C ∗ := lim CR(µ − iε)C ∗ ε↓0 0}CR(\mu-i\varepsilon)C^{*} CR( \mu-i0)C^{*} :=\lim_{\varepsilon\downarrow. \lambda is not a regular spectral point of exist exist uniformly in µ of L(H). If λ of \mu\in∈K_{\lK ambda}λ in the norm topology of \mathcal{L}(\mathcal{H}) . If H , we say that λ H. \lambda is a spectral singularity of H, of H. H if (ii) Λ \ l˚ Λ. (ii) Let Let λ\lambda\i∈n\Lambda\backs ash A . We We say say that that λ\lambda is is aa regular regular spectral spectral point point of of H if there there exists exists aa compact compact interval interval \lambda , such K ⊂\matR Λ K_{\lambda} hbb{R} whose \mu\i∈ n K_{\laK mbda}\λcap\mat∩hri˚ ng{A} are λ \subset whose interior interior contains contains λ, such that that all all µ are regular regular in in the the sense sense of of (i) (i) and and such that the map ˚ K Λ ∋hring{µ\Lambda} →\niCR(µ K_{\lλambda}∩\cap\mat \mu\mapsto CR(− \mu-i0i0)C )C^{*}\in \mat∗hcal∈{L} L(H) (\mathcal{H}) bounded. is bounded. \Lambda is right-unbounded, (iii) > 0 such that all ∈ +\infty is regular if (iii) If If Λ right‐unbounccled, we say that +∞ if there exists exists mm>0 all µ\mu\in [m, ∩ \˚ Λ [m, \infty)∞) \cap\Lambda^{ circ} are are regular regular in in the the sense sense of of (i) (i) and and such such that that the the map map. L(H) [m, Λ ∋ µ → CR(µ [m, \inft∞) y)\cap\mat∩hri˚ ng{\Lambda}\ni \mu\mapst o CR(\mu-− i0)C^{*i0)C }\in \math∗cal{∈ L} (\mat hcal{H}) bounded. is bounded.. Note that our definition definition of a regular spectral point is local. One can rephrase this definition definition H if the limiting absorption principle for H H holds in a \lambda is a regular spectral point of H saying that λ ∗ \lambda , for the weighted resolvent CR(z)C neighborhood of λ, CR(z)C^{*} , for values of the spectral parameter zz in the lower half-plane. half‐plane. It should be noted that we do not need to require that the limiting absorption principle hold for values of the spectral parameter in the upper half-plane: half‐plane: This is due to the fact H is supposed to be dissipative. We also mention that there is a natural definition that H definition of a spectral H^{*}∗ , such that λ H if and only if λ \lambda is a spectral singularity of H \lambda is singularity for the adjoint operator H ∗. H^{*}. a spectral singularity of H W bounded and compactly supported, In the case where H = −Δ+V −iW on L H=-\triangle+V-iW L^{2}2(\mat(R hbb{R3}^{3), }) , with VV and W H corresponds to a resonance embedded in the essential spectrum [0, a spectral singularity of H ∞)) [0, \infty (see, e.g., [5] [5] for the theory of resonances for Schrödinger operators, and [8] [8] for a comparison between the the notions notions of of resonances resonances and and spectral spectral singularities). singularities). rΛ. ing{A} . It is based, The next theorem provides several characterizations of a spectral singularity λ\lambda\i∈n\math˚ in particular, on a generalization of Kuroda’s representation formula to the context of dissipative scattering theory.. \lambda\i∈n\mathr˚ ing{A} . Then Theorem 3.1. Suppose that Hypotheses 1–7 Λ. 1‐7 hold. Let λ Then the following conditions conditions are equivalent: equivalent:.

(52) 97 12. J. FAUPIN. H. (1) λ\lambda is aa regular spectral point of of H. H located \ l a m b d a (2) λ is not an accumulation point − i(0,\infty) ∞) and (2) is not an accumulation point of of eigenvalues eigenvalues of of H located in in λ\lambda-i(0, and the the limit limit. CR(µ − i0)C ∗ = lim CR(µ − iε)C ∗. CR( \mu-i0)C^{*}=\lim_{\varepsiloε↓0 n\downarrow 0}CR(\mu-ie)C^{*}. exists exists in the norm topology of of L(H). \mathcal{L}(\mathcal{H}) . (3) − iCRV (λ − i0)C ∗ is (3) The The operator operator Id Id-iCR_{V}(\lambda-i0)C^{*} is invertible invertible in in L(H). \mathcal{L}(\mathcal{H}) . (4) The The scattering matrix S(λ) S(\lambda) is invertible in L(M). \mathcal{L}(\mathcal{M}) . Proof. See [9]. [9].. \square . In general, it is a difficult problem to identify explicitly the spectral singularities of a given dissipative operator. Nevertheless, one can show that the set of spectral singularities is not too large in the following sense. H is Proposition 3.2. Suppose that Hypotheses 1–7 1‐7 hold. Then Then the set of of spectral singularities of of H \Lambda of 0. a closed closed subset of of Λ of Lebesgue measure 0.. Proof. See [9]. [9].. \square . ∗ H ) Recall from Propositions 2.8 and 2.11 that the scattering operators S(H, H0 ) and S(H 0 S(H, H_{0}) S(H^{*},,H_{0}) H0 ) and W H0 ) have closed ranges. The are invertible if and only if the wave operators W − (H,H_{0}) + (H,H_{0}) W_{-}(H, W_{+}(H, following proposition shows that the study of spectral singularities is also relevant in order to answer the question of the invertibility of the scattering operators.. Λ finite and Proposition 3.3. Suppose that Hypotheses 1–7 1‐7 hold. Suppose in addition addition that Λ and \Lambda\backs\lash\mathrin˚ g{\Lambda} is finite ˚ \Lambda that all all λ\lambda\i∈ n\Lambda\bΛ ackslash \m\athring{\Λ Lambda} are are regular in the sense of of Definition Definition 99 (if (if Λ is right-unbounded, right‐unbounded, we also also assume assume that +∞ is following conditions that+\infty is regular). regular). Then Then the the following conditions are are equivalent: equivalent: (1) S(H, H0 ) is invertible in L(H), S(H, H_{0}) \mathcal{L}(\mathcal{H}) , ∗ H ) is invertible in L(H), (2) S(H 0 S(H^{*},,H_{0}) \mathcal{L}(\mathcal{H}) , H has no spectral singularities in ˚ \mathring{A}. (3) H Λ. Proof. See [9]. [9].. \square . To conclude this section, we propose the following definition definition of the “order” order” of a spectral singularity H . It will be relevant in the next section. of H. H of \lambda is a \lambda\in∈ \mathring{\˚ Lambda} is aa spectral singularity of \mathbb{N}^{* Definition 10. We Λ N∗} if We say that λ of H of order order ν\nu\in∈ if λ a spectral H \lambda , such that the singularity of K_{\lambda}λ , whose interior contains of H and and there exists exists aa compact compact interval K contains λ, limits. lim(µ − λ)ν CR(µ − iε)C ∗. 1ε↓0 \dot{ \imath} m(\mu\varepsilon\downar ow 0-\lambda)^{\nu}CR(\mu-i\varepsilon) C^{*}. exist exist uniformly in µ of L(H). \mu\in∈K_{\lK ambda}λ in the norm topology of \mathcal{L}(\mathcal{H}) . H=-\triangle+V-iC^{*}C As mentioned above, if one considers the nuclear optical model H = −Δ+V −iC ∗ C with bounded C , then a spectral singularity corresponds to a resonance and compactly supported potentials VV and C, in the usual sense (see, (see, e.g., [5]). [5]). One can then verify that the order of a spectral singularity in the sense sense of of Definition Definition 10 10 corresponds corresponds to to the the multiplicity multiplicity of of the the corresponding corresponding resonance, resonance, see see [8, [8, Section Section 6]..

(53) 98 DISSIPATIVE SCATTERING. 13. 3.2. Asymptotic completeness. We are interested in this section in the property of asymptotic completeness of the wave operators. In our context, this property can be defined defined as follows. ∗ H ) are said to be asymptotically H0 ) and Definition 11. The The wave operators operators W and W be asymptotically − (H,H_{0}) + (H ,H_{0}) 0 are W_{-}(H, W_{+}(H^{*}, H complete complete if if their ranges coincide coincide with the orthogonal orthogonal complements complements of of all all generalized eigenstates eigenstates of of H H^{*}∗ , respectively. In other and H other words, ⊥ ∗ ⊥ ∗ H0 )) = Hb (H) ⊕ {H}_{p}(H^{* Hp (H}))^{\perp} ) , Ran(W , H0 )) = Hb (H) ⊕H . Ran(W Ran (W_{-}− Ran (W_{+}(H^{* + (H p (H) (H,(H, H_{0}))=(\mathcal {H}_{b}(H)\opl us \mathcal }, H_{0}))=(\mathcal {H}_{b}(H)\opl us \mathcal {H}_{p}(H))^{\perp}. With the alternative definition definition H sug‐ \t˜ilde{\ac mathcal(H) {H} _{ac}(H) of the absolutely continuous spectral subspace of HH suggested at the end of Section 2.2, we see that asymptotic completeness means that Ran(W Ran (W_{+}(H, + (H, H 0 )) = H_{0}))= ∗ , H )) = H ˜ ac (H).. Ran (W_{+}(H^{* H + (H 0 \t˜ilde{\ac mathcal(H {H} _{ac}(∗H^{)*}) and Ran(W }, H_{0}))=\tilde{\mathcal{H}}_{ac}(H) In In [8], [8], asymptotic asymptotic completeness completeness is is proven proven under under the the following following further further assumption. assumption.. Hypothesis 8. H has at (i) finitely many (discrete) (i) H at most finitely (discrete) eigenvalues. eigenvalues. H has \mathring{\Lambda} and (ii) finitely many Λ finite (ii) H has at at most most finitely many spectral spectral singularities singularities in in ˚ and each each spectral spectral singularity singularity is is of of finite order. order. \Lambda is right-unbounded, +\infty is (iii) Λ finite and Λ \ a˚ Λ (iii) Λ \Lambda\b\ackslash˚ \mathring{\Lais mbda}iS finite and all all λ\lambda\i∈n\Lambda\backsl sh A are regular Moreover, ifif Λ right‐unbounded, then +∞ regular. We then have the following result. 1-8 hold. Then Theorem 3.4. Suppose that Hypotheses 1–8 Then. Hp (H ∗}))=\mathcal{H}_{d} (H^{* = Hd (H ∗ ). H p (H) = Hd (H), , \mathcal{H}_{p}(H^{* \mathcal{H}_{p}(H)=\mathcal{H}_{d}(H) }) .. Moreover, ∗ W H0 ) and H0 ) are and W are asymptotically asymptotically complete complete − (H,H_{0}) + (H ,H_{0}) W_{-}(H, W_{+}(H^{*}, ˚ H has no spectral singularities in Λ. \Lambda^{\circ}. \Leftrightarrow ⇐⇒ H. If If these equivalent equivalent conditions conditions are are satisfied, satisfied, then H ‐invariant direct (1) There There is an an H-invariant direct sum decomposition decomposition ⊥ H \mathcal={H}=\{\H mathcalb{(H) H}_{b}(H)\opl⊕us \matHhcalp{(H) H}_{p}(H)\}\opl⊕us(\mathHcal{Hb} (H) _{b}(H)\opl⊕ us \matHhcalp{H}(H _{p}(H^{∗*}))^{\perp} , ⊥ H to Hb (H) ⊕ Hp (H ∗ ) and and the restriction of of H analogous statement ( \ mat h cal { H } _ { b } ( H ) \ opl u s \mat h cal { H } _ { p } ( H ^{ * } ) ^ { \perp} is similar to HH_{0}0 .. An analogous ∗. H^{*}. holds for H ∗ (2) The + (H0 , H) − (H0 , H The wave operators operators W W_{+}(H_{0}, H) and and W W_{-}(H_{0}, H^{*})) are are surjective and and their kernels are are given by by ⊥ ∗ ∗ ⊥ Ker W+ (H Hb (H) ⊕H , KerW_{ Ker}(H_{0}, W−H^{* (H = Hbus(H) Hp (H}))^{\perp} ) , 0 , H) {= p (H) 0 , H {)H}_{b}(H)\opl KerW_{+}(H_{0}, H)=(\mathcal H}_{b}(H)\opl us \mathcal {H}_{p}(H))^{\perp} })=(\mathcal \mathcal⊕{H}_{p}(H^{* ∗ H ) are bijective. (3) H0 ) and 0 are bijective. (3) The The scattering scattering operators operators S(H, S(H, H_{0}) and S(H S(H^{*},,H_{0}). Proof. See [8, [8, 9].. \square . 3.3. Application to the nuclear optical model. Now, we describe the main consequences of the abstract results previously stated to the nuclear optical model mentioned in the introduction. We refer to [8, 9] 9] for details showing that the abstract hypotheses 1–8 1‐8 are indeed satisfied satisfied in the case of the nuclear optical model, under the conditions on the potentials imposed in the following theorems. In this section, on L L^{2}2(\mat(R hbb{R3}^{3), }) , we set H HV H_{V}=-\triangle+V(x) = −Δ + V (x), H = HV − iW (x). 0 = −Δ, H_{0}=-\triangle, , H=H_{V}-iW(x) .. Recall that the sphere in. \mathb {R3}^{3} R. is denoted by. 2. SS^{2}..

(54) 99 J. FAUPIN. 14. Theorem 3.5. Suppose that α V (x) = > 3 such for all (i) ∈ C2 (R3 ) and (i) VV is is real-valued, real‐valued, VV\in C^{2}(\mathbb{R}^{3}) and there there exists exists ρ\rho>3 such that, that, for all |α| |\alpha|≤\leq2,2, ∂\partial^{\alpha}V(x)= −ρ−|α| ), |x| → ∞, O(x \mathcal{O}(\langle x\rangle^{-\rho-|\alpha|}), |x|arrow\infty, W is non-negative, (ii) W (x) > 0 on > 2 such that non‐negative, W W(x)>0 on aa non-trivial non‐trivial open open set and and there exists exists δ\delta>2 −δ ), |x| → ∞, W (x) = O(x W(x)=\mathcal{O}(\langle x\}^{-\delta}), |x|arrow\infty, (iii) H_{V}. V. (iii) 00 is neither an an eigenvalue eigenvalue nor aa resonance of of H Then, for all λ > 0, Then, all \lambda>0, 2 (S 2 )) S(λ) S(\lambda) is invertible in L(L \mathcal{L}(L^{2}(S2)). \Leftrightarrow ⇐⇒. H. λ\lambda is not a spectral singularity of of H.. Moreover, 2 3 S(H, H0 ) is invertible in L(L S(H, H_{0}) \mathcal{L}(L^{2}(R (\mathbb{R})) ^{3}). \Leftrightarrow ⇐⇒. H has no spectral singularities in (0, H ∞),, (0, \infty). H0 ) = Hd (H ∗ )⊥ . and if if these conditions conditions hold, then Ran W − (H, W_{-}(H, H_{0})=\mathcal{H}_{d}(H^{*})^{\perp}. Proof. See [8, [8, 9].. \square . \mathb {R3}^{3} R. L L_{c}^∞ (\mathbb{3R}^{3).}) . c{\infty}(R. \mathb {C} is denoted by to C If The set of bounded and compactly supported potentials from W belong to L we suppose that VV and W L_{c}^∞ (\mathbb{R3}^{3),}) , we have in addition the following more precise results. c{\infty}(R. Theorem 3.6. Suppose that (i) (i) VV is is real-valued real‐valud’ and and VV\in L_{∈c}^{\L infty∞ hbb{R}3^{3). }) . c}(\mat(R W is non-negative, (ii) W (x) > 0 on non‐negative, W W(x)>0 on a non-trivial non‐trivial open open set and and W W\in L_{∈c}^{\iL nfty}∞ hbb{R}3^{3). }) . c(\mat(R 0 (iii) H_{V}. V. (iii) 0 is neither an an eigenvalue eigenvalue nor aa resonance of of H Then, Then, H p (H) = Hd (H).. Moreover, \mathcal{H}_{p}(H)=\mathcal{H}_{d}(H) W H0 ) is asymptotically complete complete − (H,H_{0}) W_{-}(H,. \Leftrightarrow ⇐⇒. ∗ ⊥ Ran W H0 ) = Hp (H ) − (H, W_{-}(H, H_{0})=\mathcal{H}_{p}(H^{* })^{\perp}. \Leftrightarrow ⇐⇒. H has no spectral singularities in (0, H ∞).. (0, \infty). If If these conditions conditions hold, then (1) H0 ) is (1) S(H, S(H, H_{0}) is invertible invertible in in L(L \mathcal{L}(L2^{2}(R (\mathbb{3R})), ^{3}) , (2) For all λ > 0, S(λ) is invertible in L(L (S 2 )),, \lambda>0, all S(\lambda) \mathcal{2L}(L^{2}(S^{2})) ∗ ⊥ H to H (3) The H_{0}. 0. The restriction of of H \matphcal(H {H}_{p}(H^{)*})^{\perp}isis similar to H Proof. Proof. See See [8, [8, 9]. 9].. \square . We mention that the fact that H b (H) = {0} in the context of the present section follows from \mathcal{H}_{b}(H)=\{0\} unique continuation arguments. Moreover, it is proven in [19] [19] that 00 cannot be a spectral singularity H . On the other hand, for any λ \lambda>0 of H. > 0,, one can construct smooth and compactly supported H (see [20]). \lambda is a spectral singularity of H W such that λ potentials VV and W Acknowledgements. I am grateful to S. Richard for hospitality at the University of Nagoya, and to F. Hiroshima for the invitation to give a talk at the conference Mathematical aspects aspects of of quantum quantum fields and fields and related topics, RIMS, Kyoto, July 2018. I warmly thank J. Fröhlich and F. Nicoleau for fruitful collaborations. REFERENCES References. [1] [1] N. N. Bohr. Bohr. Neutron Neutron capture capture and and nuclear nuclear constitution. constitution. Nature, Nature, 137:344–348, 137:344−348, 1936. 1936. [2] E. B. Davies. Two-channel Two‐channel Hamiltonians and the optical model of nuclear scattering. Ann. Inst. H. Poincaré Sect. A (N.S.), (N.S.), 29(4):395–413, 29(4):395-413 , 1978. [3] [3] E. E. B. B. Davies. Davies. Nonunitary Nonunitary scattering scattering and and capture. capture. I. I. Hilbert Hilbert space space theory. theory. Comm. Comm. Math. Math. Phys., Phys., 71(3):277–288, 71(3):277-288 , 1980..

(55) 100 DISSIPATIVE SCATTERING. 15. [4] E. B. Davies. Linear operators operators and and their spectra, volume 106 of Cambridge Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007. [5] S. Dyatlov and M. Zworski. Mathematical theory of of scattering resonances. In preparation, http://math.mit.edu/dyatlov/res/res_20170228.pdf. http://math.mit.edu/dyatlov/res/res‐20170228.pdf. [6] [6] D. D. E. E. Edmunds Edmunds and and W. W. D. D. Evans. Evans. Spectral Spectral theory theory and and differential differential operators. operators. Oxford Oxford MatheMathe‐ matical Monographs. The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. [7] [7] M. M. Falconi, Falconi, J. J. Faupin, Faupin, J. J. Fröhlich, Fröhlich, and and B. B. Schubnel. Schubnel. Scattering Scattering Theory Theory for for Lindblad Lindblad Master Master Equations. Comm. Comm. Math. Phys., 350(3):1185–1218, 350(3):1185-1218 , 2017. [8] J. Faupin, J. Fröhlich. Asymptotic completeness in dissipative scattering theory Scattering. Adv. Adv. Math., Math., 340:300–362, 340:300−362, (2018). (2018). [9] J. Faupin, F. Nicoleau. Scattering matrices for dissipative quantum systems. arXiv:1808.09179, arX_{l}v:1808.\theta 9179, (2018). (2018). [10] [10] H. H. Feshbach, Feshbach, C. C. Porter, Porter, and and V. V. Weisskopf. Weisskopf. Model Model for for nuclear nuclear reactions reactions with with neutrons. neutrons. Phys. Phys. Rev., 96:448–464, 96:448−464, 1954. [11] T. Kato. Wave operators and similarity for some non-selfadjoint non‐selfadjoint operators. Math. Ann., 162:258– 162:25S− 279, 279, 1965/1966. 1965/1966. [12] T. Kato. Perturbation theory for linear operators. operators. Die Grundlehren der mathematischen WisWis‐ senschaften, Band 132. Springer-Verlag Springer‐Verlag New York, Inc., New York, 1966. [13] [13] P. P. A. A. Martin. Martin. Scattering Scattering theory theory with with dissipative dissipative interactions interactions and and time time delay. delay. Nuovo Nuovo Cimento Cimento B (11), 30(2):217–238, B(11), 30(2):217-238 , 1975. [14] B. Sz.Nagy, C. Foias, H. Bercovici and L. Kérchy. Harmonic analysis analysis of of operators operators on on Hilbert spaces Springer, New York, 2010. [15] R. S. Phillips. Dissipative operators and hyperbolic systems of partial differential differential equations Trans. Trans. Amer. Math. Soc., 90:193–254, 90:193−254, 1959. [16] [16] M. M. Reed Reed and and B. B. Simon. Simon. Methods Methods of of modern modern mathematical mathematical physics. physics. I. I. Functional Functional analysis. analysis. Second Second edition. edition. Academic Press [Harcourt [Harcourt Brace Jovanovich, Publishers], New York-London, York‐London, 1980. [17] [17] M. M. Reed Reed and and B. B. Simon. Simon. Methods Methods of of modern modern mathematical mathematical physics. physics. II. II. Fourier Fourier Analysis, Analysis, selfself‐ adjointness. adjointness. Academic Press [Harcourt [Harcourt Brace Jovanovich, Publishers], New York-London, York‐London, 1975. [18] M. Reed and B. Simon. Methods of of modern mathematical physics. III. Scattering theory. AcaAca‐ demic demic Press Press [Harcourt [Harcourt Brace Brace Jovanovich, Jovanovich, Publishers], Publishers], New New York-London, York‐London, 1979. 1979. [19] X. P. Wang. Number of eigenvalues for dissipative Schrödinger operators under perturbation. J. J. Math. Pures Appl. (9), (9), 96(5):409–422, 96(5):409-422 , 2011. [20] [20] X. X. P. P. Wang. Wang. Time-decay Time‐decay of of semigroups semigroups generated generated by by dissipative dissipative Schrödinger Schrödinger operators. operators. J. J. Differential Equations, 253(12):3523–3542, 253(12):3523-3542 , 2012. [21] X. P. Wang and L. Zhu. On the wave operator for dissipative potentials with small imaginary part. Asymptot. Anal., 86(1):49–57, 86(1):49-57 , 2014. [22] D. R. Yafaev. Mathematical scattering theory, volume 105 of Translations Translations of of Mathematical Monographs. American Mathematical Society, Providence, RI, 1992. General theory, Translated from the Russian by J. R. Schulenberger. [23] D. R. Yafaev. Mathematical scattering theory, Analytic theory, Mathematical Surveys and Monographs 158. American Mathematical Society, Providence, RI, 2010. (J. Faupin) Faupin) Institut INSTITUT Elie ELIE Cartan CARTAN de DE Lorraine, LORRAINE, Université UNIVERSITÉ de DE Lorraine, LORRAINE, 57045 57045 Metz METZ Cedex CEDEX 1, 1, France FRANCE E-mail E‐mail address: address: jeremy.faupin@univ-lorraine.fr j eremy.faupin@univ‐lorraine.fr.

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