Resonances under Rank One Perturbations
Olivier Bourget, Victor Cortes, Rafael del Rio, Claudio Fernandez
aa r X i v : . [ m a t h . SP ] J u l RESONANCES UNDER RANK ONE PERTURBATIONS
OLIVIER BOURGET*, V´ICTOR H. CORT´ES*, RAFAEL DEL R´IO**,AND CLAUDIO FERN ´ANDEZ*
Abstract.
We study resonances generated by rank one perturbations of selfadjointoperators with eigenvalues embedded in the continuous spectrum. Instability of theseeigenvalues is analyzed and almost exponential decay for the associated resonant statesis exhibited. We show how these results can be applied to Sturm-Liouville operators.Main tools are the Aronszajn-Donoghue theory for rank one perturbations, a reductionprocess of the resolvent based on Feshbach-Livsic formula, the Fermi golden rule and acareful analysis of the Fourier transform of quasi-Lorentzian functions. We relate theseresults to sojourn time estimates and spectral concentration phenomena Introduction
The resonance phenomenon appears in several areas of physics and mathematics suchas Classical, Quantum and Wave Mechanics. Several attempts have been done to give ita precise mathematical description. We refer to [29] for a discussion about the difficultiesarising in characterizing rigorously the concept of resonance for autonomous systems inQuantum Mechanics.One of the most fruitful approaches consists in defining quantum resonances as polesof a suitable meromorphic continuation of the Hamiltonian resolvent, from the upperhalf complex plane to the lower half plane. Each pole appears as an ”eigenvalue” withnegative imaginary part, corresponding to generalized eigenfunctions outside the Hilbertspace. There is a huge literature on this subject and we refer the reader to [17] andreferences therein.Resonances can also be characterized in terms of time exponential decay of the timeevolution of the system governed by the Hamiltonian (defined as a self-adjoint operator onsome Hilbert space H ). This behaviour can be traced by means of the survival probability P ϕ for some suitable states ϕ . This quantity defined by P ϕ ( t ) = |h ϕ, e − iHt ϕ i| , Date : 22feb2017.Fondecyt Grants 1141120 and 1161732, ECOS/CONICYT Grant C10E01.Research partially supported by UNAM-DGAPA-PAPIIT IN105414. measures the probability of finding at time t the system governed by the Hamiltonian H in its initial state ϕ . On one hand, we know that exact exponential decay is impossiblefor many models of physical interest, see e.g. [29]. On the other hand, if z = λ − i Γ(Γ >
0) is a pole of the resolvent of the Hamiltonian H with ”resonant eigenfunction” ϕ (i.e. Hϕ = zϕ ), we would formally expect that, P ϕ ( t ) = e − t k ϕ k , which is incorrect if the resonant eigenfunction ϕ does not belong to the Hilbert space.Thus, in presence of a resonance z , the best one can hope is the existence of a state ψ ∈ H such that the quantity h ψ, e − iHt ψ i behaves approximately as e − izt . Both these quantitiesequal 1 at t = 0 and in most cases of interest, both approach to zero as t tends to ∞ .The main objective is then to estimate the difference, h ψ, e − iHt ψ i − e − izt , for t not to close to 0 nor to ∞ .For differential operators on the real half line, this difference can be estimated uniformlyin time [21] or in L norm [6] by means of ODE techniques. In these cases, the function ψ is a truncated resonant eigenfunction. Pointwise estimates have been exhibited when theresonance appears with the perturbation of an instable simple eigenvalue embedded insome continuous spectrum, see e.g. [10] and [16] for a review. The main ingredients are inthat case the Feshbach-Livsic reduction and the Fermi Golden rule. In [10], this approachis actually combined with some positive commutator techniques (Mourre theory) and theestimates are obtained once the eigenfunction is localized in energy.The consistent use of the Feshbach-Livsic reduction to study resonances can be tracedback at least to [18] and has been a source of several results in the last decades in differentareas. In particular, it has been used consistently in spectral theory for Non-RelativisticQuantum Electrodynamics since [1] and [2]. For the relationship between time evolution(the perspective we address in this paper) and poles of the resolvent in the context ofanalytic perturbation theory, see also [3], [4] and references therein.In this paper, we adapt the Feshbach-Livsic reduction to the context of differentialoperators on the real half line and pointwise estimates are exhibited when the resonanceappears with the perturbation of an instable simple eigenvalue embedded in the absolutelycontinuous spectrum of such operators. Although various tools developed here can beeasily adapted to a fairly wide class of perturbations, we have decided to narrow ourdiscussion to the rank-one case and to relate these results with classical results in thisfield [13], [28]. We intend to propose several extensions in a forthcoming paper.We start in Section 2 by establishing conditions which ensure that the Fourier transformof a Lorentz-like function exhibits approximate exponential time decay. The proof ofTheorem 2.1 is based on techniques of classical analysis, we have mainly singled out from[10]. The development of this section is independent from the rest of the paper. In Section ESONANCES RANK ONE 3
3, we turn our attention on rank one perturbations of the form H κ = H + κ | ψ ih ψ | , where H has a simple eigenvalue embedded in some absolutely continuous spectrum. InTheorems 3.1 and 3.2, we show how the instability of the embedded eigenvalue and thespectral properties of the operators H κ are related to the boundary values of the reducedresolvent of H and the Fermi Golden Rule. Next, we prove Theorem 3.3, which formalizesthe existence of a resonance in term of almost exponential decay in the case of rankone perturbations under suitable hypotheses on the reduced resolvent of H . The proofcombines the Feshbach-Livsic reduction process and Krein’s formula with Theorem 2.1.Corollary 3.1 discusses the relationships with Kato’s spectral concentration. In Corollary3.2, we deduce the asymptotics for the sojourn time of the corresponding eigenstate underthe evolution governed by H κ for small values of κ . Finally, we show in Section 4, how theboundary properties of the reduced resolvent of H can be deduced from the propertiesof the spectral measure of H when it has finite multiplicity. This reformulation of theproblem is summed up in Theorem 4.2 and its proof makes essential use of the propertiesof the Borel transform, see Section 6. All these results are illustrated in Section 5 by aSturm-Liouville model. In contrast with [10], the point of view developed in this paperdoes not require any positive commutator techniques.We shall use the notation C ,β ( I ) for the set of functions with first derivative β - H¨oldercontinuous in I . Given a function of complex variable F ( z ), we write F ( λ + i
0) forlim ǫ ↓ F ( λ + iǫ ). For the spectral family of orthogonal proyections of an operator T we shallwrite E T and ρ ( T ), σ ( T ), σ p ( T ), σ sc ( T ) , σ ac ( T ) denote the resolvent, the spectrum, theeigenvalues, the singular continuous and absolutely continuous spectra of T respectively[27]. The characteristic function of a Borel set ∆ will be denoted as usual by χ ∆ ( x ) where χ ∆ ( x ) = 1 if x ∈ ∆ and 0 if x / ∈ ∆. R stands for the real numbers, R + the non negativereals and C for the complex numbers. ℑ z, ℜ z stand for the imaginary and real parts of z .2. Almost exponential decay
Theorem 2.1 is the main result of this section and it is the first ingredient in the analysisdeveloped in this paper. It provides some estimates on the Fourier transform of familiesof Lorentz-like functions defined on R by: λ g ( λ ) ℑ (cid:18) λ κ − λ − κ F ( λ, κ ) (cid:19) , with κ ∈ [ − κ , κ ] for some κ > g , F and the family of real numbers ( λ κ ) κ ∈ [ − κ ,κ ] . In the following, g ∈ C ∞ ( R ) is real-valued, compactly supported on ( a, b ) for some −∞ < a < b < ∞ , and we also assumethat 0 ≤ g ≤ g ≡ a , b ] for some a < a < b < b . In addition,( H0 ) (a) lim κ → λ κ = λ , λ ∈ ( a , b ), BOURGET, CORT´ES, DEL R´IO, AND FERN ´ANDEZ (b) the complex-valued function F is bounded on [ a, b ] × [ − κ , κ ] and continuousat the point ( λ , H1 ) for any κ ∈ [ − κ , κ ], the function F ( · , κ ) is C on [ a, b ] and(a) the function F ′ := ∂ λ F is bounded on [ a, b ] × [ − κ , κ ],(b) for any κ ∈ [ − κ , κ ], the function F ( · , κ ) is C ,α on [ a , b ], uniformly in κ ∈ [ − κ , κ ] for some α ∈ (0 , H2 ) for any κ ∈ [ − κ , κ ], inf λ ∈ [ a ,b ] ℑ F ( λ, κ ) > Remark : In Assumption ( H1 ), F ′ κ is defined at a and b by taking the correspondinglateral derivatives. If the function F is continuous in both variables on [ a, b ] × [ − κ , κ ],it is necessarily bounded. Note finally that if ℑ F ( λ , > F is continuous at thepoint ( λ , a , b ], on which ( H2 ) holds forsmall values of κ . The condition ℑ F ( λ , > Theorem 2.1.
Assume ( H0 ), ( H1 ) and ( H2 ) hold. Then, given any < δ < min( | λ − a | , | b − λ | , , and κ = 0 small enough, (a) there exists a unique solution in [ a , b ] to the equation: λ = λ κ − κ ℜ F κ ( λ ) ,denoted by λ ∞ κ , which satisfies: | λ ∞ κ − λ κ | ≤ Cκ for some C > and a + δ ≤ λ ∞ κ ≤ b − δ , (b) for all t ∈ R , (1) 1 π Z ∞−∞ dλ e − iλt g ( λ ) ℑ (cid:18) λ κ − λ − κ F ( λ, κ ) (cid:19) = c κ e − iζ κ | t | + R ( t, κ ) where c − κ = 1 + κ F ′ ( λ ∞ κ , κ ) , (2) ζ κ = λ ∞ κ − iκ c κ ℑ F ( λ ∞ κ , κ ) , and the error term R ( t, κ ) satisfies: | R ( t, κ ) | ≤ Cκ if α ∈ (0 , , | t || R ( t, κ ) | ≤ Cκ α if α ∈ (0 , and | t || R ( t, κ ) | ≤ Cκ | ln | κ || if α = 1 . Remark 2.1.
By combining (3) and Hypothesis ( H0 ) in Theorem 2.1, we also deducethat: lim κ → ℜ ζ κ = λ and lim κ → κ ℑ ζ κ = −ℑ F ( λ , < . In particular we obtain using (2) that lim κ → λ ∞ κ − ℜ ζ κ κ ℑ F ( λ ∞ κ , κ ) = 1 and lim κ → ℑ ζ κ κ ℑ F ( λ ∞ κ , κ ) = − . (3) Remark 2.2.
The model for the integral described in Theorem 2.1 is the Fourier transformof Lorentzian functions. Let µ ∈ R and Γ > . Then for any λ ∈ R , ℑ ( 1 µ − λ − i Γ ) = Γ( λ − µ ) + Γ ESONANCES RANK ONE 5 and for any t ∈ R , π Z ∞−∞ dλ e − iλt ℑ (cid:18) µ − λ − i Γ (cid:19) = e − i ( µ − i Γ) | t | , which decays exponentially at infinity. This observation is one of the key to the proof ofTheorem 2.1. The strategy for the proof of Theorem 2.1 follows essentially [10]. The fixed pointargument has been borrowed to [20].2.1.
Proof of Theorem 2.1.
For simplicity, let us write for any ( λ, κ ) ∈ [ a, b ] × [ − κ , κ ], D ( λ, κ ) = λ κ − λ − κ F ( λ, κ ). Due to Hypothesis ( H2 ), we have that for any λ ∈ [ a , b ],0 < | κ | ≤ κ , | D ( λ, κ ) | ≥ κ inf λ ∈ [ a ,b ] ℑ F κ ( λ ) >
0. Now, fix δ ∈ (0 , min( | λ − a | , | b − λ | , H0 ) and ( H1 )(a), we can pick 0 < κ ≤ κ such that: • Ran ( λ κ − κ ℜ F ( · , κ )) ⊂ [ a + δ , b − δ ], for any | κ | ≤ κ , • κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ≤ − δ .In particular, for any λ ∈ [ a, a ] ∪ [ b , b ], | κ | ≤ κ , | D ( λ, κ ) | ≥ | λ κ − λ − κ ℜ F κ ( λ ) | ≥ δ .This allows us to define the function G for λ ∈ [ a, b ] and 0 < | κ | ≤ κ by: G ( λ, κ ) = D ( λ, κ ) − and then to give a sense to the integral(4) I ( t, κ ) = 1 π Z ∞−∞ dλ e − iλt g ( λ ) ℑ (cid:18) λ κ − λ − κ F ( λ, κ ) (cid:19) , for t ∈ R and 0 < | κ | ≤ κ . Since I ( − t, κ ) = I ( t, κ ), it is enough to prove Theorem 2.1for t ≥ < | κ | ≤ κ . The proof consists in adding and subtracting the Lorentz-likefunction ℑ G (which is explicited later on) on the interval [ a , b ] and reduces the problemto study I ( t, κ ) := 1 π Z b a dλ e − iλt ℑ G ( λ, κ ) . This integral contributes to the quasi-exponential behaviour term in (1) while the remain-der terms contribute to the error term R .Given δ as before and | κ | ≤ κ , we define first by a fixed point argument the realnumber λ ∞ κ for κ ∈ [ − κ , κ ]: Lemma 2.1.
Given any κ ∈ [ − κ , κ ] , there is a unique solution to the equation: λ = λ κ − κ ℜ F ( λ, κ ) in [ a , b ] . Actually, if λ ∞ κ denotes this solution, we have that: λ ∞ κ ∈ [ a + δ , b − δ ] and | λ ∞ κ − λ κ | ≤ κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ( λ, κ ) | .Proof. By hypothesis, for any | κ | ≤ κ , κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | < λ κ − κ ℜ F ( · , κ )) ⊂ [ a + δ , b − δ ]. Therefore, given such κ ∈ [ − κ , κ ], the function λ λ κ − κ ℜ F ( λ, κ ) maps [ a , b ] (resp. [ a + δ , b − δ ]) on itself and is strictlycontractive. So, we apply the Banach fixed point theorem, and the conclusions follow. (cid:3) BOURGET, CORT´ES, DEL R´IO, AND FERN ´ANDEZ
This proves statement (a) of Theorem 2.1. Now, let us define for any λ ∈ [ a , b ], | κ | ≤ κ , b D ( λ, κ ) = λ κ − λ − κ F ( λ ∞ κ , κ ) = λ ∞ κ − λ − iκ ℑ F ( λ ∞ κ , κ ) D ( λ, κ ) = λ κ − λ − κ F ( λ ∞ κ , κ ) − κ F ′ ( λ ∞ κ , κ )( λ − λ ∞ κ )= λ ∞ κ − λ − iκ F ( λ ∞ κ , κ ) − κ F ′ ( λ ∞ κ , κ )( λ − λ ∞ κ )Note that by Hypothesis ( H2 ), ℑ b D ( λ, κ ) = − κ ℑ F ( λ ∞ κ , κ ) < κ = 0. Thisallows us to define the function b G for λ ∈ [ a , b ], 0 < | κ | ≤ κ by b G ( λ, κ ) = b D ( λ, κ ) − .For λ ∈ [ a , b ] and | κ | ≤ κ , we also have that: | D ( λ, κ ) | ≥ | λ ∞ κ − λ − iκ F ( λ ∞ κ , κ ) | − κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ! | λ ∞ κ − λ |≥ − κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ! | b D ( λ, κ ) | ≥ δ | b D ( λ, κ ) | . (5)In particular, for λ ∈ [ a , b ] and 0 < | κ | ≤ κ , we define the function G by G ( λ, κ ) = D ( λ, κ ) − and it holds: | G ( λ, κ ) | ≤ δ − | b G ( λ, κ ) | . Finally, for ( λ, κ ) ∈ [ a , b ] × [ − κ , κ ], D ( λ, κ ) = λ ∞ κ − λ − iκ ℑ F ( λ ∞ κ , κ ) − κ ( F ( λ, κ ) − F ( λ ∞ κ , κ )). Hence, for λ ∈ [ a , b ] and | κ | ≤ κ , we obtain via the Mean Value Theoremthat: | D ( λ, κ ) | ≥ | λ ∞ κ − λ − iκ F ( λ ∞ κ , κ ) | − κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ! | λ ∞ κ − λ |≥ − κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ! | b D ( λ, κ ) | ≥ δ | b D ( λ, κ ) | . (6)In particular, for λ ∈ [ a , b ] and 0 < | κ | ≤ κ , it holds: | G ( λ, κ ) | ≤ δ − | b G ( λ, κ ) | . Recall that the function g vanishes outside [ a, b ] and g ≡ a , b ]. So, we can writefor t ∈ [0 , ∞ ), 0 < | κ | ≤ κ , I ( t, κ ) = I ( t, κ ) + I ( t, κ ) + I ∂ ( t, κ ) where I ( t, κ ) := 1 π Z b a dλ e − iλt ℑ ( G ( λ, κ ) − G ( λ, κ )) , I ∂ ( t, κ ) := 1 π Z a a dλ e − iλt g ( λ ) ℑ G ( λ, κ ) + 1 π Z bb dλ e − iλt g ( λ ) ℑ G ( λ, κ ) . It remains to perform the analysis of each term.
ESONANCES RANK ONE 7
Step 1.
We start with the term I . We write for λ ∈ [ a , b ], 0 < | κ | ≤ κ , ℑ G ( λ, κ ) = 12 i (cid:18) b κ − a κ λ − b κ − a κ λ (cid:19) , where a κ := 1 + κ F ′ ( λ ∞ κ , κ ) and b κ := λ ∞ κ a κ − iκ ℑ F κ ( λ ∞ κ ) = λ κ + κ ( λ ∞ κ F ′ ( λ ∞ κ , κ )) − F ( λ ∞ κ , κ )). Note that δ ≤ − κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ≤ | a κ | for any | κ | ≤ κ .Now, for 0 < | κ | ≤ κ , we consider the function g ( · , κ ) defined by(7) g ( z, κ ) = 12 i (cid:18) b κ − a κ z − b κ − a κ z (cid:19) , which is meromorphic in the complex plane, with poles at ζ κ and ζ κ : ζ κ = λ ∞ κ − iκ ℑ F ( λ ∞ κ , κ ) a κ . In particular, for λ ∈ [ a , b ], 0 < | κ | ≤ κ , g ( λ, κ ) = ℑ G ( λ, κ ). Note that for any0 < | κ | ≤ κ , | λ ∞ κ − ℜ ζ κ ||ℑ ζ κ | (cid:27) ≤ κ δ − ℑ F ( λ ∞ κ , κ ) ≤ κ δ − sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | . (8)Explicit calculations also yield: ℑ ζ κ = − κ (1 + κ ℜ F ′ ( λ ∞ κ , κ )) ℑ F ( λ ∞ κ , κ ) | a κ | . Once observed that for 0 < | κ | ≤ κ , δ ≤ − κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ≤ κ ℜ F ′ ( λ ∞ κ , κ ), we also deduce that: ℑ ζ κ ≤ − κ δ ℑ F ( λ ∞ κ , κ )(1 + κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ) < . In other words, for any 0 < | κ | ≤ κ , the pole ζ κ lies in the lower half-plane and: − κ δ − ℑ F ( λ ∞ κ , κ ) ≤ ℑ ζ κ ≤ − κ δ ℑ F ( λ ∞ κ , κ )(1 + κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ) < . Now, fix 0 < δ ′ < δ and 0 < κ ′ ≤ κ such that: κ ′ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | ≤ δ ′ δ . In view of the bound (8) and the fact that λ ∞ κ ∈ [ a + δ , b − δ ] for any 0 < | κ | ≤ κ ,this implies that for any 0 < | κ | ≤ κ ′ , ℜ ζ κ ∈ [ a + ( δ − δ ′ ) , b − ( δ − δ ′ )]. Let γ be afixed smooth curve in the lower half plane, joining the endpoints of the interval [ a , b ]and staying at positive distance from the closure of the bounded sets { ζ κ ; 0 < | κ | ≤ κ ′ } and { ζ κ ; 0 < | κ | ≤ κ ′ } . Then, for 0 < | κ | ≤ κ ′ , the closed curve J ∪ γ enclose only thepole ζ κ and so,(9) 1 π I J ∪ γ − e − izt g ( z, κ ) dz = c κ e − iζ κ t , BOURGET, CORT´ES, DEL R´IO, AND FERN ´ANDEZ with c κ = a − κ . Therefore,(10) I ( t, κ ) = 1 π Z b a e − iλt ℑ G ( λ, κ ) dλ = c κ e − iζ κ t + 1 π Z γ e − izt g ( z, κ ) dz Now, for all z ∈ γ , g ( z, κ ) = κ h ( z, κ ) where h ( z, κ ) = p κ z + q κ | a κ | ( z − ζ κ )( z − ζ κ ) , with p κ = ℑ F ′ ( λ ∞ κ , κ ) and q κ = ℑ F ( λ ∞ κ , κ ) − λ ∞ κ ℑ F ′ ( λ ∞ κ , κ ).By construction, inf z ∈ γ, < | κ |≤ κ ′ | z − ζ κ | > z ∈ γ, < | κ |≤ κ ′ | z − ζ κ | >
0, so the func-tions h ( · , κ ) are analytic in some fixed open region containing γ for any 0 < | κ | ≤ κ ′ .Once combined with Hypotheses ( H0 )(b) and ( H1 )(a), this implies that sup z ∈ γ, < | κ |≤ κ ′ | h ( z, κ ) | < ∞ and sup z ∈ γ, < | κ |≤ κ ′ | h ′ ( z, κ ) | < ∞ .Now, note that for any t ≥ z ∈ γ , | e − izt | ≤
1, since the curve γ is containedin the lower half plane. We have that for any t ≥ < | κ | ≤ κ ′ , (cid:12)(cid:12)(cid:12)(cid:12)Z γ e − izt g ( z, κ ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cκ , (11) (cid:12)(cid:12)(cid:12)(cid:12)Z γ e − izt g ′ ( z, κ ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cκ . (12)for some C > Step 2.
In order to conclude, first define for t ≥ < | κ | ≤ κ ′ ,R ( t, κ ) := I ( t, κ ) − c κ e − iζ κ | t | = ( I ( t, κ ) − c κ e − iζ κ | t | ) + I ( t, κ ) + I ∂ ( t, κ )(13) = 1 π Z γ e − izt g ( z, κ ) dz + I ( t, κ ) + I ∂ ( t, κ ) , due to (10). According to (11), Corollary 2.1 and Proposition 2.1, all the terms on theRHS are of order κ (for κ small enough), which yields our first estimate on R . ESONANCES RANK ONE 9
Now, note that integration by parts yields for t ≥ < | κ | ≤ κ ′ , Z γ e − izt g ′ ( z, κ ) dz = it Z γ e − izt g ( z, κ ) dz + e − ib t ℑ G ( b , κ ) − e − ia t ℑ G ( a , κ ) , Z b a dλ e − iλt ℑ ( G ′ ( λ, κ ) − G ′ ( λ, κ )) = it Z b a dλ e − iλt ℑ ( G ( λ, κ ) − G ( λ, κ ))+ e − ib t ℑ ( G ( b , κ ) − G ( b , κ )) − e − ia t ℑ ( G ( a , κ ) − G ( a , κ )) Z a a dλ e − iλt ( g ( λ ) ℑ G ( λ, κ )) ′ = it Z a a dλ e − iλt g ( λ ) ℑ G ( λ, κ ) + e − ia t ℑ G ( a , κ ) , Z bb dλ e − iλt ( g ( λ ) ℑ G ( λ, κ )) ′ = it Z bb dλ e − iλt g ( λ ) ℑ G ( λ, κ ) − e − ib t ℑ G ( b , κ ) , where we have used g ( a ) = 0 = g ( b ) and g ( a ) = 1 = g ( b ). It follows from (13) that for t ≥ < | κ | ≤ κ ′ , (14) itR ( t, κ ) = J ( t, κ ) + J ( t, κ ) + J ∂ ( t, κ ) , where J ( t, κ ) = 1 π Z γ e − izt g ′ ( z, κ ) dz J ( t, κ ) = 1 π Z b a dλ e − iλt ℑ ( G ′ ( λ, κ ) − G ′ ( λ, κ )) J ∂ ( t, κ ) = 1 π (cid:18)Z a a dλ e − iλt ( g ( λ ) ℑ G ( λ, κ )) ′ + Z bb dλ e − iλt ( g ( λ ) ℑ G ( λ, κ )) ′ (cid:19) . According to (12) and Proposition 2.1, the first and third terms on the RHS of (14) areof order κ . In view of Corollary 2.2, the second one is of order κ α (resp. κ | log | κ || ) if α ∈ (0 ,
1) (resp. if α = 1). This completes the proof of statement (b).The last part of Theorem 2.1 is a direct consequence of formula (2).2.2. Technicalities.
In this section, the results are stated under Hypotheses ( H0 ), ( H1 )and ( H2 ). The quantities δ ∈ (0 , min( | λ − a | , | b − λ | , < κ ≤ κ are fixedaccording to conditions explicited in the proof of Theorem 2.1.First, we provide upper bounds on the terms I ∂ and J ∂ : Proposition 2.1.
There exists
C > such that for all t ∈ R , < | κ | ≤ κ , |I ∂ ( t, κ ) | ≤ Cκ and |J ∂ ( t, κ ) | ≤ Cκ . Proof:
We deduce from Lemma 2.1 that for all λ ∈ [ a, a ], | κ | ≤ κ , | D ( λ, κ ) | ≥ a + δ − λ ≥ δ >
0. On the other hand, for λ ∈ [ a, a ], 0 < | κ | ≤ κ , (15) ℑ G ( λ, κ ) = κ ℑ F ( λ, κ ) | D ( λ, κ ) | In view of ( H0 )(b), we deduce that for t ∈ R , 0 < | κ | ≤ κ , (cid:12)(cid:12)(cid:12)(cid:12)Z a a dλ e − iλt g ( λ ) ℑ G ( λ, κ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ κ sup λ ∈ [ a,a ] |ℑ F ( λ, κ ) | Z a a dλ ( a + δ − λ ) ≤ Cκ . For all λ ∈ [ a, a ], 0 < | κ | ≤ κ , ( g ℑ G ) ′ = g ′ ℑ G + g ℑ G ′ and we deduce from (15) that ℑ G ′ ( λ, κ ) = κ ℑ F ′ ( λ, κ ) | D ( λ, κ ) | − κ ℜ ( D ( λ, κ ) D ′ ( λ, κ )) ℑ F ( λ, κ ) | D ( λ, κ ) | with D ′ ( λ, κ ) = − − κ F ′ ( λ, κ ). It follows that for t ∈ R , 0 < | κ | ≤ κ , (cid:12)(cid:12)(cid:12)(cid:12)Z a a dλ e − iλt ( g ( λ ) ℑ G ( λ, κ )) ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ κ Z a a dλ g ( λ ) |ℑ F ′ ( λ, κ ) | + | g ′ ( λ ) ||ℑ F ( λ, κ ) | ( a + δ − λ ) + 2 κ Z a a dλ g ( λ ) (1 + κ | F ′ ( λ, κ ) | ) |ℑ F ( λ, κ ) | ( a + δ − λ ) ≤ Cκ , in view of Hypotheses ( H0 )(b) and ( H1 )(a). A similar procedure applies to the term Z bb dλ e − iλt g ( λ ) ℑ G ( λ, κ )and the conclusion of the proposition follows. (cid:3) Now, we provide some upper bounds on the terms I and J , which rely on the followinglemma: Lemma 2.2.
Let ( α, β ) ∈ [0 , ∞ ) , < | κ | < κ and z κ = λ ∞ κ − iκ ℑ F ( λ ∞ κ , κ ) = λ κ − κ F ( λ ∞ κ , κ ) . There exist C > and < κ ≤ κ , such that for any < | κ | ≤ κ , Z b a | λ − ℜ z κ | α | λ − z κ | β dλ ≤ Cκ α − β +1) if α − β + 1 < C | log | κ || if α − β + 1 = 0 C if α − β + 1 > Proof.
We start with some preliminary remarks. By Lemma 2.1, ℜ z κ = λ ∞ κ ∈ [ a + δ , b − δ ] for any 0 < | κ | ≤ κ . Lemma 2.1 and Hypothesis ( H0 )(a) also imply: lim κ → λ ∞ κ = λ .Finally, ℑ z κ = − iκ ℑ F ( λ ∞ κ , κ ) < H2 ). Since F is continuous at ( λ , κ → ℑ z κ κ = −ℑ F ( λ , < . ESONANCES RANK ONE 11
So, given 0 < δ < ℑ F ( λ , < κ ≤ κ , such that for any 0 < | κ | < κ , (16) − κ sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ( λ, κ ) | ≤ ℑ z κ ≤ − κ δ . Now, with the change of variables λ − ℜ z κ = µ |ℑ z κ | , we obtain that: Z b a | λ − ℜ z κ | α | λ − z κ | β dλ = |ℑ z κ | α − β +1 Z b κ a κ | µ | α ( µ + 1) β dµ where a κ = ℜ z κ − a |ℑ z κ | and b κ = ℜ z κ − b |ℑ z κ | We denote γ = α − β . Case γ = − . We split the integral, integrating on the intervals [ a κ , max( − , a κ )],[max( − , a κ ) , min(1 , b κ )] and [min(1 , b κ ) , b κ ]. On the interval [max( − , a κ ) , min(1 , b κ )],the integral is bounded by the same integral on [ − ,
1] for which we observe that theintegrand is bounded by | µ | α ≤
1. This term is finally bounded by 2 |ℑ z κ | γ +1 . Theintegral on [min(1 , b κ ) , b κ ] is bounded by: |ℑ z κ | γ +1 Z b κ | µ | α ( µ + 1) β dµ ≤ |ℑ z κ | γ +1 Z b κ µ γ dµ = ( b − ℜ z κ ) γ +1 ( γ + 1) − |ℑ z κ | γ +1 γ + 1 . We manage the integral on the interval [ a κ , max( − , a κ )] analogously. Estimates for thecase γ = − Case γ = − . We split again the integral, integrating on the intervals [ a κ , max( − , a κ )],[max( − , a κ ) , min(1 , b κ )] and [min(1 , b κ ) , b κ ]. On the interval [max( − , a κ ) , min(1 , b κ )],the integral is bounded by the same integral on [ − , , b κ ) , b κ ] is bounded by: Z b κ | µ | α ( µ + 1) β dµ ≤ Z b κ µ dµ = ln( b − ℜ z κ ) − ln |ℑ z κ | . We manage the integral on the interval [ a κ , max( − , a κ )] analogously. Estimates for thecase γ = − (cid:3) Lemma 2.3.
There exists
C > such that for any λ ∈ [ a , b ] , < | κ | ≤ κ , | G ( λ, κ ) − G ( λ, κ ) | ≤ Cκ | b G ( λ, κ ) | sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | , | G ( λ, κ ) − G ( λ, κ ) | ≤ Cκ | b G ( λ, κ ) | | λ − λ ∞ κ | α +1 . Proof.
For λ ∈ [ a , b ], 0 < | κ | ≤ κ , we note that G − G = G ( D − D ) G and so | G − G | ≤ δ − | b G | | D − D | (see e.g. (5) and (6)). On the other hand, for λ ∈ [ a , b ], | κ | ≤ κ , D ( λ, κ ) − D ( λ, κ ) = κ ( F ( λ, κ ) − F ( λ ∞ κ , κ ) − F ′ ( λ ∞ κ , κ )( λ − λ ∞ κ ))(17) = κ Z λλ ∞ κ ( F ′ ( µ, κ ) − F ′ ( λ ∞ κ , κ )) dµ . ( H1 )(a) entails: | D ( λ, κ ) − D ( λ, κ ) | ≤ Cκ | λ − λ ∞ κ | sup ( λ,κ ) ∈ [ a ,b ] × [ − κ ,κ ] | F ′ ( λ, κ ) | , whichleads to the first estimate once noted that | b G ( λ, κ ) || λ − λ ∞ κ | ≤ λ ∈ [ a , b ] and0 < | κ | ≤ κ . ( H1 )(b) entails: | D ( λ, κ ) − D ( λ, κ ) | ≤ Cκ | λ − λ ∞ κ | α +1 , which leads to thesecond estimate. (cid:3) Corollary 2.1.
There exists
C > such that for any t ∈ R , < | κ | ≤ κ , |I ( t, κ ) | ≤ Z b a | G ( λ, κ ) − G ( λ, κ ) | dλ ≤ Cκ . Proof.
We integrate the second statement of Lemma 2.3 for 0 < | κ | ≤ κ and obtain that, Z b a | G ( λ, κ ) − G ( λ, κ ) | dλ ≤ Cκ Z b a | λ − λ ∞ κ | α +1 | λ − λ ∞ κ + iκ ℑ F ( λ ∞ κ , κ ) | dλ . We use Lemma 2.2, with β = 2 and α + 1 instead of α . Then, the number α − β + 1 inLemma 2.2 is just α , which is positive. The first and second statements follow. (cid:3) Lemma 2.4.
There exists
C > such that for any λ ∈ [ a , b ] , < | κ | ≤ κ , | G ′ ( λ, κ ) − G ′ ( λ, κ ) | ≤ Cκ | b G ( λ, κ ) | | λ − λ ∞ κ | α . Proof.
For λ ∈ [ a , b ], 0 < | κ | ≤ κ , we can write the difference between the difference ofthe derivatives of G and G w.r.t. λ : G ′ − G ′ = L + L + L where L = G ( G − G )(1 + κ F ′ ( λ, κ )), L = κ GG ( F ′ ( λ ∞ κ , κ ) − F ′ ( λ, κ )) and L = G ( G − G )(1 + κ F ′ ( λ ∞ κ , κ )).Recall that for λ ∈ [ a , b ], 0 < | κ | ≤ κ , • G − G = G ( D − D ) G and so | G − G | ≤ δ − | b G | | D − D | (see (5) and (6)), • | D ( λ, κ ) − D ( λ, κ ) | ≤ Cκ | λ − λ ∞ κ | α +1 due to Hypothesis ( H1 )(b) and (17).Up to some positive multiplicative constant, the quantities L and L can be bounded by κ | b G ( λ, κ ) | | λ − λ ∞ κ | α +1 , while the term L is bounded by κ | b G ( λ, κ ) | | λ − λ ∞ κ | α . The prooffollows from the fact that | b G ( λ, κ ) | | λ − λ ∞ κ | ≤ λ ∈ [ a , b ] and 0 < | κ | ≤ κ . (cid:3) Corollary 2.2.
There exists
C > such that for any t ∈ R , < | κ | ≤ κ , |J ( t, κ ) | ≤ Z b a | G ′ ( λ, κ ) − G ′ ( λ, κ ) | dλ ≤ C κ α if α ∈ (0 , κ | log | κ || if α = 1 κ if α > . ESONANCES RANK ONE 13
Proof.
We integrate Lemma 2.4 for 0 < | κ | ≤ κ and obtain that: Z b a | G ′ ( λ, κ ) − G ′ ( λ, κ ) | dλ ≤ Cκ Z b a | λ − λ ∞ κ | α | λ − λ ∞ κ + iκ ℑ F ( λ ∞ κ , κ ) | dλ . Then, we apply Lemma 2.2 with β = 2. Indeed, if α ∈ (0 ,
1) (resp. α = 1, resp. α > α − β + 1 = α − < > (cid:3) Rank one perturbations
In this section, we shall prove of results concerning rank one perturbations of self–adjoint operators. In 3.1 we show how the positivity on the imaginary part of the unper-turbed reduced resolvent implies pure absolutely continuous spectrum. In 3.2 we describehow a simple embedded eigenvalue turns into a resonance. Here smoothness of the resol-vent is required. In 3.3 we relate this to dynamic behavior of the system.3.1.
Behavior of Spectra.
Let H be a self-adjoint operator on Hilbert space H , ψ ∈ H a normalized vector and define(18) H κ = H + κ | ψ ih ψ | , κ ∈ R . Let H ψ := span { ( H κ − z ) − ψ/z / ∈ R } be the cyclic subspace generated by ψ . This space is independent of κ and reduces theoperator H κ , for every κ , see [13].Let us denote by H ψκ the part of H κ on H ψ , i.e. H ψκ : H ψ → H ψ is given by H ψκ γ = H κ γ, for all γ ∈ DomH κ ∩ H ψ . Let P be an orthogonal projection such that P H ⊂ H P , that is the range of P reduces H (see [19] p. 278) and denote P ⊥ := I − P , H ⊥ κ := P ⊥ H P ⊥ . For κ ∈ R then define(19) F κ ( z ) := h ψ, P ⊥ ( H ⊥ κ − z ) − P ⊥ ψ i and F κ ( λ + i
0) := lim ǫ ↓ F κ ( λ + iǫ )Recall that σ ac ( H ), σ s ( H ) and σ p ( H ) denote the absolutely and singular and point spec-trum respectively. With the definitions introduced above, our results on the behavior ofthe spectra read as follows, Theorem 3.1.
Let J ⊂ R be an open interval. If for every λ ∈ J (20) ℑF ( λ + i > then J ⊂ σ ac ( H ψκ ) and J ∩ σ s ( H ψκ ) = ∅ , for all κ = 0 . Theorem 3.2.
Let J ⊂ R be an open interval. Suppose that (1) ℑF ( λ + i > for every λ ∈ J . (2) If H ϕ = λϕ , for some λ ∈ J , then h ϕ, ψ i 6 = 0 and λ is a simple eigenvalue.Then J ⊂ σ ac ( H κ ) and J ∩ σ p ( H κ ) = ∅ , for all κ = 0 . Remark 3.1.
It could happen that σ sc ( H κ ) = φ . Before going into the proofs, we shall need a preliminary result. Let us recall that ρ ( H ) := C \ σ ( H ), where σ ( H ) is the spectrum of H . Lemma 3.1.
Let H be a self-adjoint operator and P an orthogonal projection such that P H ⊂ HP , that is, P H reduces H . Then for all z ∈ ρ ( H ) , P ⊥ ( H − z ) − P ⊥ = P ⊥ ( P ⊥ HP ⊥ − z ) − P ⊥ Proof.
Since H is self–adjoint and the range of P ⊥ reduces H , we have that H | P ⊥ H is alsoself–adjoint. By the basic criterium for self–adjointness Ran( H | P ⊥ H − z ) = P ⊥ H (see [27]p. 256).Let ψ ∈ P ⊥ H be arbitrary. Then, there exists ϕ ∈ DomH | P ⊥ H = P ⊥ H ∩
DomH suchthat ψ = ( H − z ) P ⊥ ϕ . Now, P ⊥ ( H − z ) − P ⊥ ψ = P ⊥ ( H − z ) − P ⊥ ( H − z ) P ⊥ ϕ = P ⊥ ϕ and P ⊥ ( P ⊥ HP ⊥ − z ) − P ⊥ ψ = P ⊥ ( P ⊥ HP ⊥ − z ) − P ⊥ ( H − z ) P ⊥ ϕ = P ⊥ ( P ⊥ HP ⊥ − z ) − ( P ⊥ HP ⊥ − z ) P ⊥ ϕ = P ⊥ ϕ. (cid:3) Now, let us prove Theorems 3.1 and 3.2.
Proof of Theorem 3.1.
Since P and P ⊥ reduce H , we have that for any z ∈ C \ R , P ( H − z ) − = ( H − z ) − P , and the corresponding commutation relation with P ⊥ (see[19] Theorem.6.5, p.173, Ch.3.). Therefore, ℑh ψ, ( H − λ − iǫ ) − ψ i = ℑh P ψ, ( H − λ − iǫ ) − P ψ i + ℑh P ⊥ ψ, ( H − λ − iǫ ) − P ⊥ ψ i = ℑh P ψ, ( H − λ − iǫ ) − P ψ i + ℑF ( λ + iǫ ) , where for the last equality we use Lemma 3.1.Next we note that the term ℑh P ψ, ( H − λ − iǫ ) − P ψ i is always nonnegative. Actually,for any self-adjoint operator H , setting u = ( H − z ) − γ we have ℑh γ, ( H − z ) − γ i = ℑh ( H − z ) u, u i = ℑ z k u k . ESONANCES RANK ONE 15
Hence, ℑh γ, ( H − z ) − γ i is nonnegative when ℑ z >
0. Holomorphic functions which mapthe upper half plane into itself called Herglotz, Nevanlinna or Pick functions. The limitsof this functions when approaching the real axis from above exists a.e., see [28] Thm I.4.Using hypothesis (20), we deduce that for every λ ∈ J ,lim ǫ ↓ ℑh ψ, ( H − λ − iǫ ) − ψ i ≥ ℑF ( λ + i > . Now, by the spectral theorem, there exists a measure µ ψ such that h ψ, ( H − z ) − ψ i = Z R dµ ψ ( t ) t − z and H , restricted to the the subspace of cyclicity generated by ψ , that is H ψ definedabove, is unitarily equivalent to multiplication by the identity in L ( R , dµ ψ ).A result essentially due to Aronszajn and Donoghue (see [13], Theorem 2 and also [28]Theorem II.2 (iii)) states that the absolutely continuous part of µ ψ is given by µ ψac (∆) = µ ψ (∆ ∩ A ), where A = { λ | < ℑh ψ, ( H − λ − i − ψ i < ∞} . Also, the singular part of µ ψ is given by µ ψs (∆) = µ ψ (∆ ∩ B ), where B = { λ |ℑh ψ, ( H − λ − i − ψ i = ∞} , for any Borel set ∆.In fact due to Krein’s formula (see I.13 in [28]), we have that:a) A is the support of the a.c. part of the measure µ κψ in the expression h ψ, ( H κ − z ) − ψ i = Z R dµ κψ ( t ) t − z , for all κ (see [13]).b) If ℑh ψ, ( H − λ − i − ψ i = ∞ then ℑh ψ, ( H κ − λ − i − ψ i = 0 for κ = 0.Therefore the result follows. (cid:3) Proof of Theorem 3.2.
We have the decomposition H = H ψ ⊕ H ⊥ ψ , where H ψ is the cyclicsubspace generated by ψ . This subspace reduces H κ , for all κ ([13]). Let us denote asabove by H ψκ the part of H κ on H ψ and by H ψκ ⊥ the part of H κ on H ⊥ ψ . Then, H ψκ : H ψ → H ψ H ψκ ⊥ : H ⊥ ψ → H ⊥ ψ Since by Theorem 3.1 J ⊂ σ ac ( H ψκ ) ⊂ σ ( H κ ), we conclude J ⊂ σ ac ( H κ ) . Now assume H κ θ = λθ , with λ ∈ J . Write θ = θ + θ , where θ ∈ H ψ , θ ∈ H ⊥ ψ , H κ θ = λθ and H κ θ = λθ ( H ψ and H ⊥ ψ reduce H κ , for all κ ).Since H ψκ has pure a.c. spectrum in J , by Theorem 3.1 it follows that θ = 0. Since θ ∈ H ⊥ ψ , we deduce that θ ⊥ ψ and h θ, ψ i = 0. Hence, λθ = H κ θ = H θ. Therefore, λ is an eigenvalue of H and λ ∈ J . From the fact that λ is a simple eigenvalueof H , we conclude that θ = cϕ which contradicts the hypothesis h ϕ, ψ i 6 = 0. Therefore, λ cannot be an eigenvalue of H κ . (cid:3) Existence of resonance.
Now, we state the result concerning the existence of aresonance described by an approximate exponential decay of the survival probability.For the next theorem consider the situation where H has a simple eigenvalue λ , with H ϕ = λ ϕ . As above, given a normalized vector ψ ∈ H such that h ϕ, ψ i 6 = 0, we define H κ := H + κ | ψ ih ψ | , κ ∈ R . Let us assume there is an interval I containing λ suchthat F κ ( λ + i
0) exists, for all λ ∈ I , where(21) F κ ( z ) := h ψ, P ⊥ ( H ⊥ κ − z ) − P ⊥ ψ i and P = h ϕ, ·i ϕ , P ⊥ = I − P . By C ,β ( J ) we denote the set of functions F such that F ′ is β -H¨older continuous in J , cf. section 6. Then we have, Theorem 3.3.
Let H ϕ = λ ϕ where λ is a simple eigenvalue. Assume that F ( λ + i ∈ C ,β ( I ) , with β > and it satisfies (22) ℑ F ( λ + i > . Let J ⊂ I be a closed interval such that λ is an interior point of J and ℑ F ( λ + i > ,for all λ ∈ J .Given g a smooth characteristic function supported on J which is identically in aneighborhood of λ , we have that for all t ∈ R and all small enough κ = 0 , (23) h ϕ, e − iH κ t g ( H κ ) ϕ i = c κ e − iζ κ | t | + R ( t, κ ) with ζ κ and c κ described in Theorem 2.1.In particular, ℑ ζ κ < , c κ = 1 + O ( κ ) and | R ( t, κ ) | ≤ Cκ . Moreover, | t | | R ( t, κ ) | ≤ Cκ α , if α ∈ (0 , and | t | | R ( t, κ ) | ≤ Cκ | log | κ || , if α = 1 . Before proving Theorem 3.3, we need the following lemmas.Let H be a self-adjoint operator defined in a Hilbert space H and P be an orthogonalprojector with Ran P ⊂ Dom( H ), with Ran P of finite dimension. ESONANCES RANK ONE 17
Lemma 3.2. (Feshbach-Livsic formula) Consider z ∈ C , ℑ z = 0 . Then the operator M := P ( H − z ) P − P HP ⊥ ( H ⊥ − z ) − P ⊥ HP is invertible and as an operator M : Ran P → Ran P and P ( H − z ) − P = P (cid:0) P ( H − z ) P − P HP ⊥ ( H ⊥ − z ) − P ⊥ HP (cid:1) − P .
See [18] for a proof.
Lemma 3.3.
Let f ∈ C ,β ( I ) where I ⊂ R an open interval. Assume that f ( x ) = 0 forall x ∈ I . Then (1 /f ) ∈ C ,β ( J ) where J is a closed interval contained in I .Proof. Since f ( x ) = 0 for all x ∈ I , the function f is C . To prove the β - H¨older continuityin I , we compute,( 1 f ) ′ ( x ) − ( 1 f ) ′ ( y ) = f ′ ( y )( f ( x ) − f ( y )) + f ( y )( f ′ ( y ) − f ( x )) f ( x ) f ( y )For x, y in J , by continuity the denominator is bounded away from zero.On the other hand, by continuity of f and f ′ and the Mean Value Theorem, the firstterm in the numerator satisfies, | f ′ ( y ) | | f ( x ) + f ( y ) | | f ( x ) − f ( y ) | ≤ c | x − y | , Because Lipschitz continuity implies H¨older continuity, we conclude that the first term is β -H¨older continuous.The second term is bounded above by c | f ′ ( x ) − f ′ ( y ) | which finishes the proof. (cid:3) Proof of Theorem 3.3.
Following Krein’s formula, see [28] I.13 and [13], we obtain that F κ ( z ) = F ( z )1 + κ F ( z ) . Therefore F κ ( λ + i
0) exists and is finite for all λ ∈ ¯ J and κ ∈ R . By hypothesis andLemma 3.3 the function λ → F κ ( λ + i
0) belongs to C ,β ( J ). Actually, ( λ, κ ) → F κ ( λ + i C ,β ( J × R ) function. Note that for all ( λ, κ ) ∈ J × R ,(24) ℑF κ ( λ + i
0) = ℑF ( λ + i | κ F ( λ + i | > λ is simple, we can use the Feshbach-Livsic formula, see Lemma 3.2, to obtain h ϕ, ( H κ − z ) − ϕ i = 1 h ϕ, ( H κ − z ) ϕ i − h ϕ, P H κ P ⊥ ( H ⊥ κ − z ) − P ⊥ H κ P ϕ i = 1 λ + κ |h ϕ, ψ i| − z − κ |h ϕ, ψ i| F κ ( z ) From the above identity and (22) we obtain that for all κ = 0, for all λ ∈ ¯ J the limit h ϕ, ( H κ − λ − i − ϕ i exists and λ → h ϕ, ( H κ − λ − i − ϕ i is a C ,β ( I ) function.Also, by the Stone’s formula we know that(25) h ϕ, e − iH κ t g ( H κ ) ϕ i = lim ǫ ↓ π Z R g ( λ ) e − iλt ℑh ϕ, ( H κ − λ − iǫ ) − ϕ i dλ = 1 π Z R g ( λ ) e − iλt ℑ (cid:18) λ − λ + κ |h ϕ, ψ i| − κ |h ϕ, ψ i| F κ ( λ + i (cid:19) dλ . since the limit can be taken inside the integral by (24).To finish the proof, apply Theorem 2.1 to (25) with λ κ = λ + κ |h ϕ, ψ i| and F ( λ, κ ) = |h ϕ, ψ i| F κ ( λ + i (cid:3) Spectral Concentration and Sojourn time.
The results of Section 3.2 bear thetwo following straightforward consequences:
Corollary 3.1.
Under the hypotheses of Theorem 3.3, we have that for any t ∈ R , (26) lim κ → h ϕ, e − i κ κ ( H κ −ℜ ζ κ ) | t | g ( H κ ) ϕ i = e −| t | where Γ κ = |h ψ, ϕ i| ℑF ( λ ∞ κ , κ ) .Proof. By multiplying the equation (23) by e i ℜ ζ κ | t | , we obtain, h ϕ, e − i ( H κ −ℜ ζ κ ) | t | g ( H κ ) ϕ i = c κ e − κ Γ κ | t | + e i ℜ ζ κ | t | R ( t, κ ) . After scaling the time, i.e. replacing t by tκ Γ κ , it follows that h ϕ, e − i κ κ ( H κ −ℜ ζ κ ) | t | g ( H κ ) ϕ i = c κ e −| t | + e i ℜ ζ κ | t | Γ κ R ( tκ Γ κ , κ ) . The corollary follows now from the estimates on the error term in Theorem 3.3. (cid:3)
Remark 3.2.
Note that | Γ κ − Γ | ≤ C | κ | with Γ = ℑ F ( λ , . Formula (26) is related to Kato’s spectral concentration, see [12], [14], [19]. The con-nection between this type of formula and the spectral concentration has been establishedfor isolated eigenvalues (of the unperturbed operator) in [12] and extended to embeddedeigenvalues in [14].Given any Hamiltonian H , the quantity |h ϕ, e − iHt ϕ i| measures the probability of find-ing the system in its initial state ϕ , at time t . Hence,(27) τ H ( ϕ ) ≡ Z ∞−∞ |h ϕ, e − iHt ϕ i| dt represents the expected amount of time the system spends in its initial state. As remarkedin [22], one expects that in presence of a resonance near λ , there exists a state ϕ whose ESONANCES RANK ONE 19 sojourn time τ H ( ϕ ) is very large and such that the spectral measure of H in the state ϕ is concentrated near λ . An explicit lower bound on the sojourn time appears in [ ? ] alsoin the case of rank one perturations. Corollary 3.2.
Under the hypotheses of Theorem 3.3, we have that for any < | κ | ≤ κ ∗ , (cid:12)(cid:12)(cid:12)(cid:12) τ H κ ( ϕ κ ) − κ Γ κ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:26) | κ | α − if < α < / if α ≥ / , for some C > , where ϕ κ := √ g ( H κ ) ϕ and Γ κ = |h ψ, ϕ i| ℑF ( λ ∞ κ , κ ) . Observe that ϕ κ → ϕ when κ → Proof.
We denote by k · k the Hilbert norm on L ( R , dt ). For any 0 < | κ | ≤ κ ∗ , |kI ( t, κ ) k − | c κ | k e − iζ κ | t | k | ≤ k R ( t, κ ) k (cid:0) kI ( t, κ ) k + | c κ |k e − iζ κ | t | k (cid:1) ≤ k R ( t, κ ) k (cid:0) k R ( t, κ ) k + 2 | c κ |k e − iζ κ | t | k (cid:1) . where I is given by (4). Now, k e − iζ κ | t | k = κ κ . The result follows from the estimates onthe error term in Theorem 3.3. (cid:3) Finite spectral multiplicity
First, we recall the concept of spectral representation and spectral multiplicity for ageneral self-adjoint operator.4.1.
Reduced operator. Spectral measures.
Let H : H → H be a self-adjointoperator in a Hilbert space H . From the spectral representation, see [31] Theorem 7.18p.195, we know that there exists a family of measures { ρ α : α ∈ Λ } and a unitary operator˜ U such that the following diagram commutes, H H −−−→ H ˜ U y x ˜ U − ⊕ α ∈ Λ L ( R , ρ α ) M id −−−→ ⊕ α ∈ Λ L ( R , ρ α )That is, H = ˜ U − M id ˜ U where M id is the maximal operator of multiplication by theidentity id , i.e., M id f ( x ) = xf ( x ).Assume that the operator H has finite spectral multiplicity, that is Λ is a finite setwith N elements. We can construct a matrix measure distribution ρ and a space of vectorvalued functions L ( R , C N , ρ ), such that H = U − H M id U H where U H : H → L ( R , C N , ρ )is unitary and M id f ( x ) = xf ( x ) is defined from L ( R , C N , ρ ) → L ( R , C N , ρ ). For theseconcepts see [5] Vol. 2 section 71, [32] section 10, [24] p.101 , [5] Section 72 and [32]Theorem 8.7. Multiplicity of eigenvalues.
Since H is unitarily equivalent to the operator multi-plication for the identity on L ( R , ρ ), with ρ a matrix measure distribution, it is interestingto characterize the multiplicity of a given eigenvalue of H in terms of ρ . For fixed λ , denoteby δ λ the measure(28) δ λ (∆) := ( λ ∈ ∆0 λ ∆where ∆ ⊂ R is a Borel set. In what follows M = ( m ij ) shall denote a non-negativesymmetric matrix with m ij ∈ C where i, j = 1 , , ...N . We shall use the notation M δ λ forthe matrix measure distribution with entries ( m ij δ λ ). Lemma 4.1.
Consider λ ∈ R , µ a matrix measure distribution defined on R , with µ ( { λ } ) = 0 and M a non-negative constant matrix as above. Let us take M id defined onthe space L ( R , C N , dµ + M δ λ ) .Then λ is an eigenvalue of M id of multiplicity p ∈ { , , . . . , N } if and only if Rank M = p if and only if dim (Ker M ) = N − p .Proof. λ is an eigenvalue of M id if and only if there exists a non zero vector ϕ ∈ L ( R , C N , dµ + M δ λ ) such that ( M id − λ ) ϕ ( x ) = 0 almost everywhere. This impliesthe existence of ~u ∈ C N such that the eigenfunction has the form ϕ ( x ) = χ { λ } ( x ) ~u ,where χ { λ } ( x ) = 1 if x = λ and 0 otherwise. Thus λ is an eigenvalue of M id if andonly if k ϕ k L = h ~u, M ~u i 6 = 0 and ϕ = χ { λ } ( x ) ~u . Now, for the difference of two eigen-vectors we have k χ { λ } ( x ) ~u − χ { λ } ( x ) ~v k L = h ~w, M ~w i with ~w = ~u − ~v . Since M is non-negative, h ~w, M ~w i = 0 if and only if M w = 0. Therefore the eigenvector ϕ = χ { λ } ( x ) ~u isdifferent from the eigenvector χ { λ } ( x ) ~v in the space L ( R , C N , dµ + M δ λ ) if and only if M ~u = M~v . Now let us construct an isomorphism between the subspace of eigenvectorsand the Range of M in the following way: I : Ker ( M id − λ ) → { ~v : M ~u = ~v } I ( χ { λ } ( x ) ~u ) = M ~u
Since I is biyective and linear the two spaces have the same dimension and the lemma isproven. (cid:3) Remark 4.1.
If rank
M > , then the matrix distribution dµ + M δ λ does not correspondto a Sturm-Liouville operator with limit point case conditions, since the singular spectrumof such operators is simple see [30] . Reduction process.
Let B ⊂ R be a fixed Borel set and let P B := E H ( B ) where E H is the spectral family associated to H . Let us decompose the matrix function ρ asfollows: ρ (∆) = ρ B (∆) + ρ B c (∆) ESONANCES RANK ONE 21 with ∆ ⊂ R a Borel set where ρ B (∆) := ρ (∆ ∩ B ) , ρ B c (∆) := ρ (∆ ∩ B c ) . The following result will be useful to prove almost exponential decay of resonant states.
Theorem 4.1.
Let ψ be a vector in H and H be a self-adjoint operator with finite spectralmultiplicity N . Then h P B ψ, ( P B HP B − z ) − P B ψ i = Z R h ( U H ψ )( x ) , dρ B ( x )( U H ψ )( x ) i C N x − z = Z R h ( U H P B ψ )( x ) , dρ ( x )( U H P B ψ )( x ) i C N x − z . Remark 4.2.
In case de matrix dρ ( x ) has only one entry h ( U H ψ )( x ) , dρ ( x )( U H ψ )( x ) i = | ( U H ψ )( x ) | dρ ( x ) . Before we prove the above theorem we need the following lemmas.
Lemma 4.2.
For any Borel set ∆ ⊂ R the following identity holds, (29) P B E P B HP B (∆) P B = E H (∆ ∩ B ) = P B E H (∆) P B . Proof.
We first note that P B HP B = HP B = f ( H ) where f ( x ) = xχ B ( x ) , with χ B thecharacteristic function on the Borel set B . Also we know that E f ( H ) (∆) = E H ( f − (∆)),see [9] Theorem 4, Chapter 6 p.158. Thus, P B E P B HP B (∆) P B = E P B HP B (∆) P B = E HP B (∆) P B = E HP B (∆) E H ( B ) = E H ( f − (∆)) E H ( B )= E H ( f − (∆) ∩ B ) . We claim that for f ( x ) = xχ B ( x ) one has that f − (∆) ∩ B = ∆ ∩ B . Clearly, if x ∈ ∆ ∩ B then f ( x ) = xχ B ( x ) = x ∈ ∆, so x ∈ f − (∆).Also, x ∈ f − (∆) ∩ B , implies that f ( x ) ∈ ∆. But for x ∈ B , g ( x ) = 1 so f ( x ) = x and x ∈ ∆, ending the proof. (cid:3) Lemma 4.3.
For any vector ψ ∈ H (30) h E H (∆) ψ, ψ i = Z ∆ h ( U H ψ )( x ) , dρ ( x )( U H ψ )( x ) i . Proof.
From the spectral representation theorem we know that χ ∆ ( H ) = U − H M χ ∆ U H where ( M χ ∆ h )( x ) = χ ∆ ( x ) h ( x ). Then h E H (∆) ψ, ψ i H = h χ ∆ ( H ) ψ, ψ i H = h U H χ ∆ ( H ) ψ, U H ψ i L ( ρ ) = h M χ ∆ U H ψ, U H ψ i L ( ρ ) = Z R χ ∆ ( x ) h ( U H ψ )( x ) , dρ ( x )( U H ψ )( x ) i = Z ∆ h ( U H ψ )( x ) , dρ ( x )( U H ψ )( x ) i . (cid:3) Proof of Theorem 4.1.
Let us define the measure µ g (∆) = h E H (∆ ∩ B ) g, g i . Accordingto Lemma 4.3 and the definition of ρ B (∆) we deduce the following identity: µ ψ (∆) = Z ∆ ∩ B h ( U H ψ )( x ) , dρ ( x )( U H ψ )( x ) i = Z ∆ h ( U H ψ )( x ) , dρ B ( U H ψ )( x ) i ( x ) . On the other hand, using Lemma 4.2 we get that(31) µ ψ (∆) = h P B E P B HP B (∆) P B ψ, ψ i = Z ∆ h ( U H ψ )( x ) , dρ B ( x )( U H ψ )( x ) i By the spectral theorem, h P B ψ, ( P B HP B − z ) − P B ψ i = Z R x − z d h E P B HP B P B ψ, P B ψ i = Z R x − z d h P B E P B HP B P B ψ, ψ i . So, by identity (31) one has that h P B ψ, ( P B HP B − z ) − P B ψ i = Z R x − z dµ ψ ( x )= Z R x − z h ( U H ψ )( x ) , dρ B ( x )( U H ψ )( x ) i . To prove the second equality in the theorem we apply again Lemma 4.2 to obtain that h P B ψ, ( P B HP B − z ) − P B ψ i = Z R x − z d h P B E P B HP B P B ψ, ψ i = Z R x − z d h E H ( x ) P B ψ, P B ψ i . We finish the proof by using Lemma 4.3. (cid:3)
ESONANCES RANK ONE 23
Let H be a self-adjoint operator defined on H with spectrum of finite multiplicity N .Define H κ = H + κ | ψ ih ψ | where ψ a normalized vector on H and κ ∈ R . Let U H be a unitary operator and µ amatrix measure distribution defined on R such that U H H U − H = M id on L ( R , C N , dµ ).Now we can present a useful result about almost exponential decay. It will be appliedin next section to Sturm-Liouville operators. Theorem 4.2. a) Assume that H has a simple eigenvalue λ embedded in somecontinuous spectrum. Precisely, a1) H ϕ = λ ϕ , k ϕ k = 1 . a2) There exists an open interval I of R with dµ = γ ( λ ) dλ + M δ λ , λ ∈ I where for each λ ∈ I , γ ( λ ) is a non-negative matrix and M is a constantnon-negative matrix of rank one. b) The map λ → h ( U H ψ )( λ ) , γ ( λ )( U H ψ )( λ ) i is C ,α ( I ) , with < α ≤ . c) For ϕ and ψ we assume that h ϕ, ψ i H = h U H ϕ, U H ψ i L = 0 and h ( U H ψ )( λ ) , γ ( λ )( U H ψ )( λ ) i C N = 0 . Then there exists an open interval J , λ ∈ J, ¯ J ⊂ I such that (1) For all κ = 0 , σ p ( H κ ) ∩ J = ∅ , J ⊂ σ ac . (2) Given g a smooth characteristic function supported on J which is identically ina neighborhood of λ , we have that for all t ∈ R and all κ = 0 but small enough h ϕ, e − iH κ t g ( H κ ) ϕ i = c κ e − iζ κ | t | + R ( t, κ ) with ζ κ and c κ described in Theorem 2.1.In particular, ℑ ζ κ < , c κ = 1+ O ( κ ) and | R ( t, κ ) | ≤ Cκ . Moreover, | t || R ( t, κ ) | ≤ Cκ α ,if α ∈ (0 , and | t || R ( t, κ ) | ≤ Cκ | log | κ || , if α = 1 . Remark 4.3.
There is a normalized vector ~u such that Ran ( M ) = C ~u and U H ϕ = 1 p h ~u, M ~u i C N χ { λ } ~u . Now we can use the tools we have developed to prove the theorem.
Proof.
Applying Theorem 4.1 with B = R \ { λ } and ρ = µ we obtain h ψ, P ⊥ ( H ⊥ − z ) − P ⊥ ψ i = Z h ( U H ψ )( λ ) , γ ( λ )( U H ψ )( λ ) i C N dλλ − z . Then (1) follows applying Theorem 3.2, taking into account that ℑ F ( λ + i
0) = ℑh ψ, P ⊥ ( H ⊥ − λ − i − P ⊥ ψ i = h ( U H ψ )( λ ) , γ ( λ )( U H ψ )( λ ) i C N and (2) follows from Theorem 3.3 together with Theorem 6.1. (cid:3) A Sturm Liouville model with an embedded eigenvalue
First let us construct an operator acting on functions defined on the half axis R + whichhas an embedded eigenvalue. The function h ( x ) := cos x + k sin x is solution of the initial value problem − u ′′ = uu (0) = 1 , u ′ (0) = k . We assume k > k = 1). Let us define q ( x ) := − k ddx h h ( x )1 + k R x h ( t ) dt i . The operator L N ( N stands for the Neumann boundary condition) acting on a densesubspace of L ( R + ) generated by( lu )( x ) = − u ′′ + q ( x ) u, u ′ (0) = 0 , ≤ x < ∞ has the following spectral function see [23] p. 46:(32) ρ N ( λ ) = ˆ ρ ( λ ) + ks ( λ − s ( t ) = (cid:26) t <
01 if t ≥ ρ ( λ ) is the spectral function of the operator L k generated by( l u )( x ) = − u ′′ , ≤ x < ∞ u ′ (0) = ku (0)that is(33) d ˆ ρ ( λ ) = ( √ λπ ( λ + k ) dλ if λ ≥
00 if λ < L N .The operator L N defined as above will take the place of H in Theorem 4.2. This operatorsatisfies the hypothesis a1) and a2) of that theorem. Fix a vector ψ ∈ L ( R + ), R + = { x ∈ R : x ≥ } . Consider the perturbed operator H κ = L N + κ | ψ ih ψ | Let
O ⊂ R + an open interval such that λ = 1 ∈ O . To verify hypothesis b) of Theorem 4.2we shall find an interval I ⊂ R + such that the eigenvalue λ = 1 ∈ I and | U L N ψ ( λ ) | ˆ ρ ( λ ) is in C ,α ( I ) with 0 < α ≤ ESONANCES RANK ONE 25
The unitary operator U L N is given by the transform( U L N ψ )( λ ) = Z R + ω ( t, λ ) ψ ( t ) dt where ω ( t, λ ) is a solution of the eigenvalue problem − u ′′ + q ( x ) u = λu u (0 , λ ) = 0 , u ′ (0 , λ ) = 1for any λ ∈ C . The function ω ( t, λ ) is an entire function of λ .If we choose ψ to be of compact support then ( U L N ψ )( λ ) ∈ C ∞ ( O ). Since ˆ ρ ∈ C ∞ ( O ) then | ( U L N ψ )( λ ) | ˆ ρ ( λ ) ∈ C ∞ ( O ) and therefore there is an open interval I containing λ = 1 suchthat | ( U L N ψ )( λ ) | ˆ ρ ( λ ) ∈ C ,α ( I ), so condition b) in Theorem 4.2 is satisfied.To realize condition c) of Theorem 4.2 we can take for instance ψ ( t ) = χ ∆ ( t ) ω ( t,
1) where ∆is an open interval. So, | ( U L N ψ )(1) | = Z ∆ | ω ( t, | dt > | ( U L N ψ )(1) | ˆ ρ (1) > ρ (1) = π (1+ k ) >
0. Moreover h ϕ, ψ i H = h U L N ϕ, U L N ψ i L = Z R χ { } ( x )( U L N ψ )( x ) dρ N ( x )= ( U L N ψ )(1) ρ N ( { } ) = ( U L N ψ )(1) k = 0 . Boundary limit of Borel transforms
Let I ⊂ R be an open interval and consider a measure µ I : B → R + defined on the Borel sets B of R such that µ I restricted to I is absolutely continuous with respect to Lebesgue measure,i.e. there exists a measurable function f : R → R + such that for any Borel set ∆ ⊂ I , it holdsthat Z ∆ f ( x ) dx = µ (∆) . Assume moreover that f ∈ C ,α ( I ), that is, f ∈ C ( I ) with f ′ α − H¨older continuous on I ,0 < α ≤
1. We write F ( z ) for the corresponding Borel transform associated to the measure µ I , F ( z ) = Z dµ I ( x ) x − z , ℑ z = 0 . The following results assures the existence and the smoothness of the boundary values lim ǫ → + F ( λ + iǫ ), See [33], [7] [8],[11]. Theorem 6.1.
Let I ⊂ R be an open interval and µ I the measure defined above. Then F ( λ ) := lim ǫ → + F ( λ + iǫ ) exists and F ∈ C ( I ) . Moreover for any interval J such that ¯ J ⊂ I , the function F ′ is β -H¨oldercontinuous in J for all β < α ., that is, F ∈ C ,β ( J ) . Proof.
Let K be an open subinterval such that ¯ J K ¯ K I and let us define η ∈ C ∞ ( R , [0 , η ( x ) = 0 if x / ∈ K and η ( x ) = 1 if x ∈ ¯ J . For any z ∈ C with ℑ z = 0, F ( z ) = Z K f ( x ) η ( x ) x − z dx + Z R − ¯ J (1 − η ( x )) x − z dµ ( x ) . If Re z ∈ J the second integral in the above equality represents an holomorphic function, there-fore this term is C ( I ) and α -H¨older continuous in any subinterval J with ¯ J ⊂ I . (Recall thatcontinuously differentiable function on compact sets of the real line are γ -H¨older continuouswith 0 < γ ≤ z = λ + iǫ , it follows that Z K f ( x ) η ( x )( x − λ ) − iǫ dx = R ( λ, ǫ ) + i I ( λ, ǫ )where R ( λ, ǫ ) := Z R f ( x ) η ( x )( x − λ )( x − λ ) + ǫ dx , I ( λ, ǫ ) := Z R ǫf ( x ) η ( x )( x − λ ) + ǫ dx . Since f η ∈ C ( R ) we obtain that I ( λ ) := lim ǫ → + I ( λ, ǫ ) = πf ( λ ) η ( λ ), see [26] Lemma 2.3 i)p.41, so I ( λ ) ∈ C ( R ).Moreover, the derivative I ′ ( λ ) = π ( f ( λ ) η ( λ )) ′ is α -H¨older and thus β -H¨older in J with β < α .This follows because ( f η ) ′ = f ′ η + f η ′ and f η ′ ∈ C ( I ) and therefore α -H¨older in J and | f ′ ( x ) η ( x ) − f ′ ( y ) η ( y ) | ≤ | η ( x )( f ′ ( x ) − f ′ ( y )) | + | ( η ( x ) − η ( y )) f ′ ( y ) |≤ M | x − y | α because f ′ is α -H¨older and η ∈ C ∞ .Now let us consider R ( λ ) := lim ǫ → + R ( λ, ǫ ). This limit exists because f η ∈ C and therefore f η is α -H¨older continuous with compact support, see Lemma 10 in[7]. Moreover, R ( λ ) = P.V. Z R f ( x ) η ( x ) x − λ := H [ f η ]( λ )where P.V. means the integral in the sense of principal value and H stands for the Hilberttransform, see [26] Lemma 2.5, p.51and [11]. Since f η ∈ C and f η ∈ L p one deducesthat ( H [ f η ]( λ )) ′ = H [( f η ) ′ ]( λ ), see [25] Theorem 1 and formula 3.24. Using again Privaloff-Korn’s Theorem or Lemma 10 in[7] and the α -H¨older continuity of ( f η ) ′ we finally obtain that H [( f η ) ′ ]( λ ) is β -H¨older continuous and thus ( R ( λ )) ′ is β -H¨older continuous. (cid:3) Acknowledgments.
R. del Rio would like to thank the warm hospitality of the PontificiaUniversidad Cat´olica de Chile. Also, C. Fern´andez expresses his gratitude to UNAM, M´exico,for the warm hospitality. R. del Rio thanks M. Ballesteros for useful comments.
References [1] V.Bach. J. Fr¨ohlich and I.M. Sigal, Renormalization group analysis of spectral problems in quantumfield theory. Adv. Math. 137 (1998), no. 2, 205-298.[2] V.Bach. J. Fr¨ohlich and I.M. Sigal, Quantum electrodynamics of confined nonrelativistic particles.Adv. Math. 137 (1998), no. 2, 299-395.
ESONANCES RANK ONE 27 [3] D. Hasler, I. Herbst and M. Huber, On the lifetime of quasi-stationary states in non-relativisticQED. Ann. Henri Poincar 9 (2008), no. 5, 1005-1028.[4] J. Fr¨ohlich, M. Griesemer and I.M. Sigal, Israel Michael Spectral renormalization group and localdecay in the standard model of non-relativistic quantum electrodynamics. Rev. Math. Phys. 23(2011), no. 2, 179-209.[5] N. I. Akhiezer and I. M. Glazman.
Theory of linear operators in Hilbert space . Dover Publications,Inc., New York, 1993.[6] M. A. Astaburuaga, P. Covian and C. Fern´andez,
Behaviour of the survival probability in someone-dimensional problems , J. Math. Phys. 43, 4571 (2002); doi: 10.1063/1.1500426.[7] J. Bellissard and H. Schulz-Baldes, Scattering theory for lattice operators in dimension 3. Rev. Math.Phys. 24 (2012), no. 8, 1250020, 51 pp.[8] M. Ben-Artzi., Smooth spectral calculus. In
Partial differential equations and spectral theory , volume211 of
Oper. Theory Adv. Appl. , pages 119–182. Birkh¨auser/Springer Basel AG, Basel, 2011.[9] M. S. Birman and M. Z. Solomjak.
Spectral theory of selfadjoint operators in Hilbert space . Mathe-matics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1987.[10] L. Cattaneo, G. M. Graf, and W. Hunziker. A general resonance theory based on Mourre’s inequality.
Ann. Henri Poincar´e , 7(3):583–601, 2006.[11] R. Courant and D. Hilbert.
Methods of mathematical physics. Vol. II . Wiley Classics Library. JohnWiley & Sons, Inc., New York, 1989.[12] E.B. Davies.
Resonances, spectral concentration and exponential decay . Letters in MathematicalPhysics, Vol.1, Issue1, pp 31-35, 1075.[13] Donoghue, William F., Jr. On the perturbation of spectra. Comm. Pure Appl. Math. 18 1965559–579[14] W.M. Greenlee
Spectral Concentration near Embedded Eigenvalues . Journal of Math. Analysis andApp. 151, 2@27 (1990)[15] A. Jensen and G. Nenciu. A unified approach to resolvent expansions at thresholds.
Rev. Math.Phys. , 13(6):717–754, 2001.[16] Jensen A., Lecture Notes on Schroedinger Operators Resonances Arising from a Perturbed Eigen-value. (2010), Aalborg University.[17] P. D. Hislop, I.M. Sigal
Introduction to Spectral Theory
Applied Mathematical Sciences 113. Springer-Verlag. 1996.[18] J. Howland, The Livsic matrix in perturbation theory. J. Math. Anal. Appl. 50 (1975), 415-437[19] T. Kato, Perturbation theory for linear operators. Springer-Verlag, Berlin, 1995.[20] C. King, Exponential decay near resonances, Letters in math. Phys. , (1991), 215–222.[21] R. Lavine, Exponential Decay , Diff. Eq. and Math. Phys., Proceedings of the Int. Conference U. ofAlabama at Birmingham, 132–142, 1995.[22] R. Lavine,
Spectral density and sojourn time , Atomic Scattering theory (J. Nutall ed.), U. of WesternOntario, London, Ontario, 1978.[23] B. M. Levitan.
Inverse Sturm-Liouville problems . VSP, Zeist, 1987.[24] M. A. Naimark.
Linear differential operators. Part I: Elementary theory of linear differential opera-tors . Frederick Ungar Publishing Co., New York, 1967.[25] J. N. Pandey.
The Hilbert transform of Schwartz distributions and applications . Pure and AppliedMathematics (New York). John Wiley & Sons, Inc., New York, 1996.[26] D. B. Pearson.
Quantum scattering and spectral theory , volume 9 of
Techniques of Physics . AcademicPress, Inc., London, 1988.[27] M. Reed and B. Simon.,
Methods of modern mathematical physics. I . Academic Press, Inc. [HarcourtBrace Jovanovich, Publishers], New York, 1980. [28] B. Simon.
Spectral Analysis of rank one perturbations and applications , Lectures at the VancouverSummer School in Mahematical Physics , 1993.[29] B. Simon.
Resonances and complex scaling: A rigorous overview , Int. J. Quantum Chemistry, ,(1978), 529–542.[30] B. Simon. On a theorem of Kac and Gilbert , Journal of Functional Analysis 223 (2005) 109 115.[31] J. Weidmann.
Linear operators in Hilbert spaces , volume 68 of
Graduate Texts in Mathematics .Springer-Verlag, New York-Berlin, 1980.[32] J. Weidmann.
Spectral theory of ordinary differential operators , volume 1258 of
Lecture Notes inMathematics . Springer-Verlag, Berlin, 1987.[33] D. Yafaev, Scattering theory: some old and new problems. Lecture Notes in Mathematics, 1735.Springer-Verlag, Berlin, 2000. xvi+169 pp. ISBN: 3-540-67587-6 *Pontificia Universidad Cat´olica de Chile. Facultad de Matem´aticas**IIMAS-UNAM, Department of Mathematical Physics, 04510 CDMX, Mexico
E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected] E-mail address ::