Computing Scattering Resonances
aa r X i v : . [ m a t h . SP ] J un COMPUTING SCATTERING RESONANCES
JONATHAN BEN-ARTZI, MARCO MARLETTA, AND FRANK R ¨OSLER
Abstract.
The question of whether it is possible to compute scattering resonances of Schr¨odingeroperators – independently of the particular potential – is addressed. A positive answer is given,and it is shown that the only information required to be known a priori is the size of the support ofthe potential. The potential itself is merely required to be C . The proof is constructive, providinga universal algorithm which only needs to access the values of the potential at any requested point. Introduction and Main Result
This paper provides an affirmative answer to the following question:
Does there exist a universal algorithm for computing the resonances of Schr¨odingeroperators with complex potentials?
To the authors’ best knowledge this is the first time this question is addressed. Furthermore,the proof of existence provides an actual algorithm (that is, the proof is constructive). We test thisalgorithm on some standard examples, and compare to known results.The framework required for this analysis is furnished by the
Solvability Complexity Index (SCI),which is an abstract theory for the classification of the computational complexity of problems thatare infinite-dimensional. This framework has been developed over the last decade by Hansen andcollaborators (cf. [14, 4, 5]) and draws inspiration from the seminal result [10] on solving quinticequations via a tower of algorithms . We therefore emphasize that ours is an abstract resultin analysis, not in numerical analysis.
Quantum Resonances.
Let us first define what a quantum resonance is. Let q : R d → C becompactly supported, let H q := − ∆ + q be the associated Schr¨odinger operator in L ( R d ) and let χ : R d → R be some compactly supportedfunction with χ ≡ q ). It follows from the explicit form of the free fundamental solution(cf. eq. (1.1) below) that the map z I + q ( − ∆ − z ) − χ is an analytic operator-valued function on C \ { } . We define: Definition 1.1 (Resonance) . A resonance of H q is defined to be a pole of the meromorphic operator-valued function z ( I + q ( − ∆ − z ) − χ ) − . Date : June 8, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Scattering resonance, Solvability Complexity Index, Computational complexity.JBA and FR acknowledge support from an Engineering and Physical Sciences Research Council Fellowship(EP/N020154/1).
This definition is independent of the specific choice of χ (so long as χ ≡ q )), andcoincides with the poles of the scattering matrix of q , cf. [17, Prop. 8] and [15, III.5].Resonances can be regarded as states whose wave function disperses very slowly in time, andcan therefore be considered as “almost bound states”. In physics, such phenomena arise in thedescription of unstable particles and radioactive decay. Resonant states, just like eigenfunctions,can only exist at certain energies. The slow-dispersal-in-time approach to resonances motivatesone of the earlier definitions of resonances used in the computational physics literature, namelymaximization of the so-called time delay function – see, e.g., Le Roy and Liu [16] and Smith [22].This approach leads to real resonance energies for real-valued potentials and, in the one-dimensionalcase at least, is closely related to the concept of spectral concentration – see, e.g., Eastham [12],which describes one mechanism by which such concentrations may arise. For additional discussionwe refer to the review article [24] and the book [11].It is widely accepted that the reliable computation of resonances is a challenging task. This isnot usually due to the intrinsic ill-posedness of analytic continuation, since that step is usually doneexplicitly, but rather due to the fact that complex scaling changes resonance problems either intonon-selfadjoint spectral problems, for which the pseudospectra may be far from the spectrum [13],or into problems with a nonlinear dependence on the spectral parameter, for which sensitivity toperturbations may also be problematic. In this context we refer to [8] (including the references anddiscussion therein) where interval-arithmetic was used to compute resonances.We show that resonances can be computed as the limit of a sequence of approximations, each ofwhich can be computed precisely using finitely many arithmetic operations. The proof is construc-tive: we define an algorithm and prove its convergence. We emphasize that this single algorithmis valid for any Schr¨odinger operator H q as defined above, so long as diam(supp( q )) satisfies an apriori bound. We implement this algorithm in one-dimension and compare its performance to thatof Bindel and Zworski [6].1.2. The Solvability Complexity Index.
The Solvability Complexity Index (SCI) addressesquestions which are at the nexus of pure and applied mathematics, as well as computer science:
How do we compute objects that are “infinite” in nature if we can only handle afinite amount of information and perform finitely many mathematical operations?Indeed, what do we even mean by “computing” such an object?
These broad topics are addressed in the sequence of papers [14, 4, 5]. Let us summarize the maindefinitions and discuss how these relate to our problem of finding resonances:
Definition 1.2 (Computational problem) . A computational problem is a quadruple (Ω , Λ , Ξ , M ),where(i) Ω is a set, called the primary set ,(ii) Λ is a set of complex-valued functions on Ω, called the evaluation set ,(iii) M is a metric space,(iv) Ξ : Ω → M is a map, called the problem function . Definition 1.3 (Arithmetic algorithm) . Let (Ω , Λ , Ξ , M ) be a computational problem. An arith-metic algorithm is a map Γ : Ω → M such that for each T ∈ Ω there exists a finite subset Λ Γ ( T ) ⊂ Λsuch that(i) the action of Γ on T depends only on { f ( T ) } f ∈ Λ Γ ( T ) , OMPUTING SCATTERING RESONANCES 3 (ii) for every S ∈ Ω with f ( T ) = f ( S ) for all f ∈ Λ Γ ( T ) one has Λ Γ ( S ) = Λ Γ ( T ),(iii) the action of Γ on T consists of performing only finitely many arithmetic operations on { f ( T ) } f ∈ Λ Γ ( T ) . Definition 1.4 (Tower of arithmetic algorithms) . Let (Ω , Λ , Ξ , M ) be a computational problem. A tower of algorithms of height k for Ξ is a family Γ n ,n ,...,n k : Ω → M of arithmetic algorithms suchthat for all T ∈ Ω Ξ( T ) = lim n k →∞ · · · lim n →∞ Γ n ,n ,...,n k ( T ) . Definition 1.5 (SCI) . A computational problem (Ω , Λ , Ξ , M ) is said to have a Solvability Com-plexity Index (
SCI ) of k if k is the smallest integer for which there exists a tower of algorithms ofheight k for Ξ. If a computational problem has solvability complexity index k , we writeSCI(Ω , Λ , Ξ , M ) = k. In the present article, our computational problem is made up of the following elements (to bespecified more precisely in Section 1.3):(i) Ω is a class of Schr¨odinger operators H q with potentials q which have a common compactsupport and a uniform bound in C ,(ii) Λ is the set of all pointwise evaluations of q , as well as pointwise evaluations of the Green’sfunction associated with the Helmholtz operator − ∆ − z ,(iii) M is the space cl( C ) of all closed subsets of C equipped with the Attouch-Wets metric (whichis a generalization of the Hausdorff metric to the case of unbounded sets),(iv) Ξ : Ω → M is the map that associates to a particular Schr¨odinger operator its set of resonances,and we denote it by Res( H q ).We show that for this computational problem there exists a tower of height 1, i.e. there existsa family of algorithms { Γ n } n ∈ N such that Γ n ( H q ) → Res( H q ) as n → + ∞ for any H q ∈ Ω, wherethe convergence is in the sense of the Attouch-Wets metric [3], generated by the following distancefunction:
Definition 1.6 (Attouch-Wets distance) . Let
A, B be closed sets in C . The Attouch-Wets distance between them is defined as d AW ( A, B ) = ∞ X i =1 − i min ( , sup | x |
A, B ⊂ C are bounded, then d AW is equivalent to the Hausdorff distance. Remark . It can be shown (cf. [20, Prop. 2.8]) that a sequence of sets A n ⊂ C converges to A inAttouch-Wets metric, if the following two conditions are satisfied • If λ n ∈ A n and λ n → λ , then λ ∈ A . • If λ ∈ A , then there exist λ n ∈ A n with λ n → λ .1.3. Main Result.
Let d ∈ N , fix M > M denote the class of Schr¨odinger operators H q := − ∆ + q on L ( R d ) JONATHAN BEN-ARTZI, MARCO MARLETTA, AND FRANK R ¨OSLER with q ∈ C ( R d ; C ) and supp( q ) ⋐ Q M , where Q M denotes the cube of edge length M centered atthe origin. Moreover, for x ∈ R d and z ∈ C let G ( x, z ) := ı4 (cid:16) z π | x | (cid:17) d − H (1) d − (cid:0) z | x | (cid:1) , d ≥ , ı2 z e ı z | x | , d = 1 , (1.1)where H (1) ν denotes the Hankel function of the first kind. For Im( z ) > G ( x, z ) is the fundamentalsolution to the free Helmholtz operator − ∆ − z (cf. [21, Ch. 22]). We define the evaluation set Λto be Λ := { M } ∪ (cid:8) q q ( x ) | x ∈ R d (cid:9) ∪ (cid:8) G ( x, z ) | x ∈ R d , z ∈ C (cid:9) . (1.2)Then the quadruple (Ω M , Λ , Res( · ) , cl( C )) poses a computational problem in the sense of Definition1.2. The main result of the present article is the following. Theorem 1.8.
Resonances can be computed in one limit:
SCI(Ω M , Λ , Res( · ) , cl( C )) = 1 . We prove this theorem by explicitly constructing an algorithm which computes the set of reso-nances in one limit. This algorithm can be implemented numerically; some numerical experimentsare provided in Section 5.
Remark . (i) We note that there exist examples of computational spectral problems for whichSCI ≥
2, even in the selfadjoint case (cf. [4, Th. 6.5]). Whenever SCI ≥ q ) ⋐ Q M from the definition of Ω, i.e. if the size of supp( q ) isnot assumed to be known a priori , then Theorem 1.8 implies SCI ≤
2, but it is no longer clearthat SCI = 1. Indeed, let us provide a sketch of the proof:
Sketch of proof:
Choose a sequence 0 ≤ M k → + ∞ and for each fixed k let { Γ k,n } n ∈ N be a sequence of algorithms as in Theorem 1.8, which computes the resonances of Ω M k inone limit. Given a compactly supported potential function q , whose support is not known,choose a smooth cutoff function ρ k with Q M k − ⊂ supp( ρ k ) ⋐ Q M k . Then from Theorem1.8 we know that { Γ k,n } n ∈ N computes the set of resonances of H ρ k q in one limit. As soon assupp( q ) ⋐ Q M k − , one has ρ k q ≡ q and the sequence { lim n →∞ Γ nk ( H ρ k q ) } k ∈ N will be constantin k , and hence convergent. This proves that lim k →∞ lim n →∞ Γ k,n ( q ) = Res( H q ), that is, theset of resonances can be computed in two limits.The proof of Theorem 1.8 is divided into several steps. First, we obtain quantitative resolventnorm estimates for the operator K ( z ) := q ( H q − z ) − χ from Definition 1.1. These are then usedto bound the error between K ( z ) itself and a discretized version K n ( z ), obtained by replacing thepotential q by a piecewise constant approximation. Finally, the poles of ( I + K ( z )) − are identifiedthrough a thresholding of the discretized operator function ( I + K n ( z )) − . Organization of the paper.
Section 2 contains a short discussion of Definition 1.1 and meromorphiccontinuation. In Section 3 we prove some estimates for convergence of finite-dimensional approxima-tions of linear operators, which are then used in Section 4 to construct an explicit algorithm whichcomputes resonances in one limit, thereby proving Theorem 1.8. Section 5 is dedicated to numericalexperiments. In Appendix A we review some properties of the Green’s function G ( x, z ) introducedin (1.1). OMPUTING SCATTERING RESONANCES 5 Analytic Continuation
We use this section for a more detailed discussion of Definition 1.1 and to fix some notations andconventions. For the sake of self containedness, we prove the existence of z ( I + q ( − ∆ − z ) − χ ) − as a meromorphic operator-valued function on the domain C ext := C if d is odd , logarithmic cover of C if d is even . This result follows from the classical Analytic Fredholm Theorem (cf. e.g. [18, Sec. VI.5])
Theorem 2.1 (Analytic Fredholm Theorem) . Let D ⊂ C be open and connected and let F : D → L ( H ) be an analytic operator-valued function such that F ( z ) is compact for all z ∈ D . Then, either(i) ( I + F ( z )) − exists for no z ∈ D , or(ii) ( I + F ( z )) − exists for all z ∈ D \ S , where S is a discrete subset of D . In this case, z ( I + F ( z )) − is meromorphic in D , analytic in D \ S , the residues at the poles are finite rankoperators, and if z ∈ S then ker( I + F ( z )) = { } . Next, recall that Q M denotes the cube of edge length M in R d centered at the origin. Let χ := χ Q M be the indicator function of Q M . Note that the operator-valued function z q ( − ∆ − z ) − χ is an analytic function on C ext \ { } . This follows from the explicit representation of the freefundamental solution (1.1) (cf. Remark A.2). Lemma 2.2.
The function C + ∋ z (cid:0) I + q ( − ∆ − z ) − χ (cid:1) − has a meromorphic continuation to C ext . Moreover, the residues at the poles are finite rank operators.Proof. The operator q ( − ∆ − z ) − χ is compact by the Fr´echet-Kolmogorov theorem and the inverse (cid:0) I + q ( − ∆ − z ) − χ (cid:1) − exists for Im( z ) > (cid:3) The above observations lead us to study the spectrum of the compact operator K ( z ) := q ( − ∆ − z ) − χ, z ∈ C ext . (2.1)Since the integral kernel for the free resolvent is given explicitly by (1.1) as an analytic function of z ∈ C ext \ { } , we have an explicit representation of (2.1) as an integral operator on L ( R d ): (cid:0) q ( − ∆ − z ) − χ f (cid:1) ( x ) = q ( x ) Z R d G ( x − y, z ) χ ( y ) f ( y ) dy, z ∈ C ext \ { } . (2.2) 3. Abstract Error Estimates
We recall that the resonances of H q = − ∆ + q are defined to be the poles of C ext ∋ z (cid:0) I + K ( z ) (cid:1) − where K ( z ) = q ( − ∆ − z ) − χ is a compact operator. In this section we prove general,abstract, estimates for approximations of families of linear operators. These are largely independentof the rest of this paper and will be applied in the proof of Theorem 1.8. Abusing notation, ourgeneric abstract analytic operator family is denoted K ( z ).Let H be a separable Hilbert space and denote by L ( H ) the space of bounded operators on H . Let H n ⊂ H be a finite-dimensional subspace, P n : H → H n the orthogonal projection and K : C ext → L ( H ) continuous in operator norm. Moreover, let K n : C ext → L ( H n ) be analytic forevery n ∈ N . Assume that for any compact subset B ⊂ C ext there exist a sequence a n ↓ C > k K ( z ) − K n ( z ) P n k L ( H ) ≤ Ca n , (3.1) JONATHAN BEN-ARTZI, MARCO MARLETTA, AND FRANK R ¨OSLER k P n K ( z ) | H n − K n ( z ) k L ( H n ) ≤ Ca n , (3.2) k K ( z ) − P n K ( z ) P n k L ( H ) ≤ Ca n , (3.3)for all z ∈ B .3.1. Error Estimates.Lemma 3.1. If z ∈ C ext is such that − / ∈ σ ( K ( z )) , then (cid:0) − Ca n k ( I + K ( z )) − k L ( H ) (cid:1) (cid:13)(cid:13) ( I + K n ( z )) − (cid:13)(cid:13) L ( H n ) ≤ (cid:13)(cid:13) ( I + K ( z )) − (cid:13)(cid:13) L ( H ) , where we use the convention that k ( I + K n ( z )) − k L ( H ) = + ∞ if − ∈ σ ( K n ( z )) .Proof. Whenever the left hand side is non-positive the assertion is trivially true, so we may assumew.l.o.g. that 1 − Ca n k ( I + K ( z )) − k L ( H ) >
0. In this case, the assertion follows by a Neumannseries argument, as follows. We have(3.4) I + K n ( z ) P n = I + K ( z ) + ( K n ( z ) P n − K ( z ))= ( I + K ( z )) (cid:2) I + ( I + K ( z )) − ( K n ( z ) P n − K ( z )) (cid:3) Because Ca n < k ( I + K ( z )) − k , the second factor in (3.4) is invertible by the Neumann series and (cid:2) I + ( I + K ( z )) − ( K n ( z ) P n − K ( z )) (cid:3) − = ∞ X j =0 (cid:16) ( I + K ( z )) − ( K n ( z ) P n − K ( z )) (cid:17) j . Hence, (cid:13)(cid:13) ( I + K n ( z ) P n ) − (cid:13)(cid:13) L ( H ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =0 (cid:16) ( I + K ( z )) − ( K n ( z ) − K ( z )) (cid:17) j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ) (cid:13)(cid:13) ( I + K ( z )) − (cid:13)(cid:13) L ( H ) ≤ ∞ X j =0 (cid:13)(cid:13) ( I + K ( z )) − (cid:13)(cid:13) j +1 L ( H ) k K n ( z ) P n − K ( z ) k jL ( H ) ≤ ∞ X j =0 (cid:13)(cid:13) ( I + K ( z )) − (cid:13)(cid:13) j +1 L ( H ) ( Ca n ) j = (cid:13)(cid:13) ( I + K ( z )) − (cid:13)(cid:13) L ( H ) ∞ X j =0 (cid:13)(cid:13) ( I + K ( z )) − (cid:13)(cid:13) jL ( H ) ( Ca n ) j = k ( I + K ( z )) − k L ( H ) − k ( I + K ( z )) − k L ( H ) Ca n for any n ∈ N . It remains to replace the L ( H ) norm on the left hand side by the L ( H n ) norm. Thisfollows from Claim 3.2. This completes the proof. (cid:3) Claim 3.2.
We have k ( I + K n ( z )) − k L ( H n ) ≤ k ( I + K n ( z ) P n ) − k L ( H ) for all z for which bothoperators are boundedly invertible.Proof. For x ∈ H n we have ( I + K n P n ) − x = ( I + K n ) − x , because if u ∈ H n solves ( I + K n ) u = x ,then ( I + K n P n ) u = x and by invertibility it follows that u = ( I + K n P n ) − x . We conclude thatsup x ∈H n , k x k =1 k ( I + K n P n ) − x k H = sup x ∈H n , k x k =1 k ( I + K n ) − x k H n and therefore sup x ∈H , k x k =1 k ( I + K n P n ) − x k H ≥ sup x ∈H n , k x k =1 k ( I + K n ) − x k H n . OMPUTING SCATTERING RESONANCES 7 (cid:3)
Lemma 3.3. If z ∈ C ext is such that either − ∈ σ ( K ( z )) or (cid:13)(cid:13) ( I + K ( z )) − (cid:13)(cid:13) L ( H ) ≥ Ca n , theneither − ∈ σ ( P n K ( z ) P n ) or (cid:13)(cid:13) ( I + P n K ( z ) P n ) − (cid:13)(cid:13) L ( H ) ≥ Ca n . Proof. If − ∈ σ ( K ( z )), then unless − ∈ σ ( P n K ( z ) P n ), we have I + K ( z ) = I + P n K ( z ) P n + ( K ( z ) − P n K ( z ) P n )= ( I + P n K ( z ) P n ) (cid:2) I + ( I + P n K ( z ) P n ) − ( K ( z ) − P n K ( z ) P n ) (cid:3) We now argue by contradiction. If we had k ( I + P n K ( z ) P n ) − k L ( H ) < Ca n , then we wouldhave k ( I + P n K ( z ) P n ) − ( K ( z ) − P n K ( z ) P n ) k L ( H ) < I + K ( z ) would be invertible by theNeumann series contradicting our assumption that − ∈ σ ( K ( z )). Thus we must have k ( I + P n K ( z ) P n ) − k L ( H ) ≥ Ca n .Now let us turn to the case where − / ∈ σ ( K ( z )) and (cid:13)(cid:13) ( I + K ( z )) − (cid:13)(cid:13) L ( H ) ≥ Ca n . The samecalculation as in the proof of Lemma 3.1 shows that (cid:16) − Ca n (cid:13)(cid:13) ( I + P n K ( z ) P n ) − (cid:13)(cid:13) L ( H ) (cid:17) (cid:13)(cid:13) ( I + K ( z )) − (cid:13)(cid:13) L ( H ) ≤ (cid:13)(cid:13) ( I + P n K ( z ) P n ) − (cid:13)(cid:13) L ( H ) from which it follows easily that Ca n ≤ (cid:13)(cid:13) ( I + P n K ( z ) P n ) − (cid:13)(cid:13) L ( H ) . (cid:3) An Abstract Algorithm For Computing Poles.
We now demonstrate how the the assump-tions (3.1)-(3.3) allow us to construct an abstract algorithm that computes the poles of (cid:0) I + K ( z ) (cid:1) − .By an abstract algorithm we mean a sequence of subsets of C ext , which is constructed from K n andwhich converges in Attouch-Wets metric to { z ∈ C ext | − ∈ σ ( K ( z )) } . Note that this is not yetan arithmetic algorithm in the sense of Definition 1.3, since the sets are not computed from a finiteamount of information in finitely many steps.Let B ⊂ C ext be compact and define the exponentially fine lattice L n := e − an ( Z + ı Z ) ∩ B . Sincewe assume that a n is explicitly known and K n ( z ) can be computed in finitely many steps, we candefine the set Θ Bn ( K ) = (cid:26) z ∈ L n (cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13) ( I + K n ( z )) − (cid:13)(cid:13) L ( H n ) ≥ √ a n (cid:27) . Moreover, note that by [4, Prop. 10.1], determining whether (cid:13)(cid:13) ( I + K n ( z )) − (cid:13)(cid:13) L ( H n ) ≥ √ a n can bedone with finitely many arithmetic operations on the matrix elements of K n ( z ) for each z ∈ L n . Lemma 3.4.
The assumptions (3.1) - (3.3) imply the convergence Θ Bn ( K ) → { z ∈ B | − ∈ σ ( K ( z )) } in Attouch-Wets metric.Proof. I. Excluding spectral pollution. Assume that z n ∈ Θ Bn ( K ) with z n → z for some z ∈ B .Then for each n we have k ( I + K n ( z n )) − k L ( H n ) ≥ √ a n and hence by Lemma 3.1 (cid:13)(cid:13) ( I + K ( z n )) − (cid:13)(cid:13) L ( H ) ≥ (cid:16) − Ca n (cid:13)(cid:13) ( I + K ( z n )) − (cid:13)(cid:13) L ( H ) (cid:17) a − n . (with the convention that k ( I + K ( z n )) − k L ( H ) = + ∞ if − ∈ σ ( K ( z n ))). Whenever √ a n ≤ C thisleads to (cid:13)(cid:13) ( I + K ( z n )) − (cid:13)(cid:13) L ( H ) ≥ a − n C √ a n ≥ a − n . JONATHAN BEN-ARTZI, MARCO MARLETTA, AND FRANK R ¨OSLER
It follows that k ( I + K ( z n )) − k L ( H ) → + ∞ as n → + ∞ and hence I + K ( z ) is not invertible (thisfollows by yet another Neumann series argument, together with norm continuity of K ). Hence z isa pole. II. Spectral inclusion.
Assume now that z is a pole, i.e. − ∈ σ ( K ( z )). Our reasoning will havethe structure − ∈ σ ( K ( z )) ⇓∃ z n ∈ L n : k ( I + K ( z n )) − k L ( H ) large ⇓k ( I + P n K ( z n ) P n ) − k L ( H ) large ⇓k ( I + P n K ( z n ) | H n ) − k L ( H n ) large ⇓k ( I + K n ( z n )) − k L ( H n ) large,with a quantitative estimate in each step. To this end, note first that if − ∈ σ ( K ( z )) for some z ∈ B , then there exist ν, c, ε > n ) such that for all ζ in a ε -neighborhood of z , k ( I + K ( ζ )) − k L ( H ) ≥ c | z − ζ | − ν . (3.5)Indeed, since all singularities of ( I + K ( z )) − are of finite order by the analytic Fredholm theorem,this follows from the Laurent expansion of meromorphic operator valued functions.It follows from (3.5) that for any z n such that | z − z n | ≤ e − an one will have k ( I + K ( z n )) − k L ( H ) ≥ c | z − z n | − ν ≥ ce νan ≥ Ca n for all large enough n . We conclude that for any pole z there exists a sequence z n ∈ L n such that z n → z as n → + ∞ and k ( I + K ( z n )) − k L ( H ) > Ca n for all but finitely many n ∈ N .Next, Lemma 3.3 shows that k ( I + P n K ( z n ) P n ) − k L ( H ) > Ca n . Studying this norm further, wehave ( I H + P n K ( z n ) P n ) − = (cid:0) I H n + P n K ( z n ) | H n (cid:1) − ⊕ I H ⊥ n and thus (cid:13)(cid:13) ( I H + P n K ( z n ) P n ) − (cid:13)(cid:13) L ( H ) = max n(cid:13)(cid:13)(cid:0) I H n + P n K ( z n ) | H n (cid:1) − (cid:13)(cid:13) L ( H n ) , o . Hence, as soon as a n < C , we have k ( I + P n K ( z n ) P n ) − k L ( H ) = k ( I + P n K ( z n ) | H n ) − k L ( H n ) . Weconclude that if z is a pole, then there exists z n ∈ L n such that k ( I + P n K ( z n ) | H n ) − k L ( H n ) > Ca n (3.6)( n large enough). A similar reasoning as in Lemma 3.1 (using (3.2)) shows that now (cid:0) − Ca n k ( I + K n ( z n )) − k L ( H n ) (cid:1) k ( I + P n K ( z n ) | H n ) − k L ( H n ) ≤ k ( I + K n ( z n )) − k L ( H n ) , and rearranging terms, together with (3.6), gives k ( I + K n ( z n )) − k L ( H n ) ≥ Ca n OMPUTING SCATTERING RESONANCES 9 and therefore z n ∈ Θ Bn ( K ) for large enough n . The assertion about Attouch-Wets convergence nowfollows from Remark 1.7. (cid:3) Definition of the Algorithm
Error Estimates.
In this section, we will apply the abstract results of Section 3 to our reso-nance problem. To this end, define K ( z ) = q ( − ∆ − z ) − χ . We write K for the integral kernel of K ( z ) to simplify notation. Recall that K was given by (2.2) and supp( K ) ⊂ Q M × Q M . We willconstruct an operator approximation K n of K , which satisfies (3.1)-(3.3) and in addition(H1) The matrix elements of K n can be computed in finitely many steps from a finite subsetΛ n ⊂ Λ (cf. eq. (1.2) and Def. 1.3);(H2) The convergence rate a n is explicitly known (i.e. the sequence a n can be used to define thealgorithm).To this end, let us define H n , P n as follows. R d = [ i ∈ n Z d S n,i := [ i ∈ n Z d (cid:16)(cid:2) , n (cid:1) d + i (cid:17) (4.1) H n = (cid:8) f ∈ L ( Q M ) (cid:12)(cid:12) f | S n,i constant ∀ i ∈ n Z d ∩ Q M (cid:9) P n f ( x ) = X i ∈ n Z d ∩ Q M (cid:18) n d Z S n,i f ( t ) dt (cid:19) χ S n,i ( x )(4.2)Furthermore, we have to make a concrete choice for the approximation K n . An obvious choice isthe integral kernel K n ( x, y ) = X i,j ∈ n − Z d ∩ Q M K ( i, j ) χ S n,i ( x ) χ S n,j ( y ) , i.e. a piecewise constant approximation of K ( · , · ) which can be computed from the values of K onthe lattice n − Z d (in dimensions larger than one, the fundamental solution G has a singularity at x = y . Hence, we put K n := 0 for i = j in this case).We will now show that the operators K, K n satisfy eqs. (3.1)-(3.3). To streamline the presenta-tion, we will restrict ourselves to d ≥ d ≤ n will be denoted C and their valuemay change from line to line.Proof of (3.3). Using the definitions (4.1)-(4.2), we have Kf ( x ) − P n KP n f ( x ) = Z R d K ( x, y ) f ( y ) dy − Z R d P xn K ( x, y ) P n f ( y ) dy, where P xn K ( x, y ) means ( P n K ( · , y ))( x ). Using L -selfadjointness of P n , we conclude that Kf ( x ) − P n KP n f ( x ) = Z R d K ( x, y ) f ( y ) dy − Z R d P yn P xn K ( x, y ) f ( y ) dy = Z R d (cid:0) K ( x, y ) − P yn P xn K ( x, y ) (cid:1) f ( y ) dy, Note that P yn P xn K ( x, y ) simply yields a step function approximation of K ( x, y ) like (4.2), but indimension 2 d . We conclude by applying Young’s inequality [23, Th. 0.3.1], that k Kf − P n KP n f k L ( R d ) ≤ η n k f k L ( R d ) , where η n = max ( sup x ∈ R d Z R d | K ( x, y ) − P yn P xn K ( x, y ) | dy , sup y ∈ R d Z R d | K ( x, y ) − P yn P xn K ( x, y ) | dx ) (4.3)Thus, all we have to do is estimate the L ∞ - L difference between K and its projection onto stepfunctions. To this end, fix x ∈ Q M , let ε > n and decompose the integrals as follows Z R d | K ( x, y ) − P yn P xn K ( x, y ) | dy = Z Q M | K ( x, y ) − P yn P xn K ( x, y ) | dy = Z Q M \ B ε ( x ) | K ( x, y ) − P yn P xn K ( x, y ) | dy + Z B ε ( x ) | K ( x, y ) − P yn P xn K ( x, y ) | dy (4.4)The integral over B ε ( x ) can be estimated by R B ε ( x ) | K ( x, y ) | dy , while for the remaining integral wecan use the fact that the derivative of K is bounded, as follows. Let j ∈ n Z d be such that x ∈ S n,j ,see Figure 1. Let i ∈ n Z d be such that | i − j | > ε . Then: x B ε ( x ) j B ε ( x ) n Figure 1.
Sketch of the geometry in the calculation leading to (4.5). The sum over i includes all cells whose nodes are outside the dashed ball centered at j . Z S n,i | K ( x, y ) − P yn P xn K ( x, y ) | dy = Z S n,i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ( x, y ) − − Z S n,i × S n,j K ( s, t ) dsdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy ≤ Z S n,i − Z S n,i × S n,j | K ( x, y ) − K ( s, t ) | dsdt dy = Z S n,i − Z S n,i × S n,j (cid:12)(cid:12)(cid:12)(cid:12)Z ∇ K (cid:0) τ (cid:0) xy (cid:1) + (1 − τ ) (cid:0) st (cid:1)(cid:1) · (cid:0)(cid:0) xy (cid:1) − (cid:0) st (cid:1)(cid:1) dτ (cid:12)(cid:12)(cid:12)(cid:12) dsdt dy ≤ Z S n,i − Z S n,i × S n,j Z (cid:12)(cid:12) ∇ K (cid:0) τ (cid:0) xy (cid:1) + (1 − τ ) (cid:0) st (cid:1)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) xy (cid:1) − (cid:0) st (cid:1)(cid:12)(cid:12) dτ dsdt dy ≤ Z S n,i − Z S n,i × S n,j Z (cid:13)(cid:13) ∇ K (cid:13)(cid:13) L ∞ ( S n,i × S n,j ) √ dn dτ dsdt dy Summing over i , we finally obtain (cf. Figure 1) Z R d \ B ε ( x ) | K ( x, y ) − P yn P xn K ( x, y ) | dy ≤ X i : | i − j | > ε Z S n,i | K ( x, y ) − P yn P xn K ( x, y ) | dy OMPUTING SCATTERING RESONANCES 11 ≤ X i : | i − j | > ε Z S n,i − Z S n,i × S n,j Z (cid:13)(cid:13) ∇ K (cid:13)(cid:13) L ∞ ( S n,i × S n,j ) √ dn dτ dsdt dy ≤ √ dn (cid:13)(cid:13) ∇ K (cid:13)(cid:13) L ∞ ( Q M \ B ε ( x )) Z Q M \ B ε ( x ) dy = | Q M | √ dn (cid:13)(cid:13) ∇ K (cid:13)(cid:13) L ∞ ( Q M \ B ε ( x )) ≤ | Q M | √ dn k q k C C (cid:16) ε (cid:17) − d ≤ C | Q M | n ε − d , (4.5)where the fifth line follows from (A.2), in the appendix, and the bound k q k C ≤ + ∞ . Using (4.5) in(4.4), we conclude that Z R d | K ( x, y ) − P yn P xn K ( x, y ) | dy ≤ C | Q M | n ε − d + Z B ε ( x ) | K ( x, y ) | dy ⇒ sup x ∈ R d Z R d | K ( x, y ) − P yn P xn K ( x, y ) | dy ≤ C | Q M | n ε − d + C ′ ε , where in the last line we have used (A.1) and the boundedness of q again.With an analogous calculation for sup y ∈ R d R R d | K ( x, y ) − P yn P xn K ( x, y ) | dx (which we omit here),and recalling that η n was defined by (4.3), we conclude that for all ε > η n ≤ n Cε − d + C ′ ε . Choosing ε := n − d +1 , we conclude that k Kf − P n KP n f k L ( R d ) ≤ C + C ′ n d +1 k f k L ( R d ) (4.6)and hence k K − P n KP n k L ( L ( R d )) → n → + ∞ with rate (at least) a n = n − d +1 ≤ n − d . Remark . Note that the constants
C, C ′ all depend on the spectral parameter z , but are boundedfor z in compact subsets of C ext , because K depends continuously on z .Proof of (3.2) and (H1). An orthonormal basis of H n is given by the functions e i := n d χ S n,i , i ∈ n Z d ∩ Q M , so that P n f = X j ∈ n Z ∩ Q M h f, e j i L e j in this basis. It is then easily seen that in this basis K n has the matrix elements( K n ) ij = n − d K ( i, j ) . Note that this proves (H1): The matrix elements of K n can be calculated in finitely many arithmeticoperations from the finite set Λ n := { K ( i, j ) | i, j ∈ n Z ∩ Q M } ⊂ Λ. Similarly, it can be seen thatthe matrix elements of P n K | H n in this basis are given by( P n K ) ij = n d Z S n,i Z S n,j K ( x, y ) dxdy =: n − d h K i ij , where we have introduced the notation h·i ij for the mean value on S n,i × S n,j . Let f = P j f j e j ∈ H n .From the above, and Young’s inequality, we conclude that k ( P n K − K n ) f k L = X i ∈ n Z d ∩ Q M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ n Z d ∩ Q M n − d (cid:0) K ( i, j ) − h K i ij (cid:1) f j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˜ η n k f k L , where˜ η n := max ( sup i ∈ n Z d ∩ Q M X j ∈ n Z d ∩ Q M n − d | K ( i, j ) − h K i ij | , sup j ∈ n Z d ∩ Q M X i ∈ n Z d ∩ Q M n − d | K ( i, j ) − h K i ij | ) . Hence, we have reduced the problem to estimating these ℓ ∞ - ℓ differences. This can be done similarlyto (4.4), by separating ( Q M × Q M ) ∩ ( n Z × n Z ) into an ε -region around i = j and the rest: X j ∈ n Z d ∩ Q M n − d | K ( i, j ) − h K i ij | = X | j − i | >ε n − d | K ( i, j ) − h K i ij | + X | j − i |≤ ε n − d | K ( i, j ) − h K i ij |≤ Cn − X | j − i | >ε n − d k∇ K k L ∞ ( {| x − y | >ε } ) + X | j − i |≤ ε n − d | K ( i, j ) − h K i ij |≤ Cn − ε − d +1 + X | j − i |≤ ε n − d | K ( i, j ) − h K i ij | , (4.7)where we have used (A.2) and the C -boundedness of q in the last line. To estimate the last termon the right hand side, note that | K ( i, j ) − h K i ij | ≤ C | j − i | − ( d − near i = j (cf. eq. (A.1)).Next, note that the sum n − d P j : | j − i |≤ ε | j − i | d − can be interpreted as an integral over a piecewiseconstant function, which approximates ( x, y )
7→ | x − y | − d . But this function is dominated by( x, y )
7→ | x − y | − d when | x − y | is small, and therefore we have n − d X j : | j − i |≤ ε | j − i | d − ≤ C Z B ε ( x ) | x − y | − d dy = C Z ε r − d ω d r d − dr = 2 Cω d ε (4.8)where ω d denotes the volume of the unit sphere in R d . Note that the above calculation is uniformin i , because q is bounded. Plugging (4.8) into (4.7), we arrive at X j ∈ n Z d ∩ Q M n − d | K ( i, j ) − h K i ij | ≤ Cn − ε − d +1 + 2 Cω d ε. Choosing ε = n − d yields X j ∈ n Z d ∩ Q M n − d | K ( i, j ) − h K i ij | ≤ C ′ n − d . (4.9)Finally, swapping i and j will give an analogous estimate and we can conclude that ˜ η n → a n = n − d . Remark . Note again that the constants
C, C ′ depend on z , but are bounded for z in compactsubsets of C ext , since K depends continuously on z . OMPUTING SCATTERING RESONANCES 13
Proof of (3.1) and (H2). Estimate (3.1) in fact follows from (3.3) and (3.2). Indeed, writing K n and K as block operator matrices w.r.t. the decomposition H = H n ⊕ H ⊥ n , we have K = P n K | H n D D D ! , with some operators D , D , D . Estimate (3.3) shows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) D D D !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ) < Ca n , (4.10)whereas estimate (3.2) shows that k P n K | H n − K n k L ( H n ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P n K | H n − K n
00 0 !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ) < Ca n . (4.11)Together, eqs. (4.10) and (4.11) imply that k K ( z ) − K n ( z ) P n k L ( H ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P n K | H n − K n D D D !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( H ) < Ca n . The explicit rates obtained in (4.6) and (4.9) prove that our approximation scheme satisfies (H2).4.2.
The Algorithm.
It remains to extend the algorithm Θ Bn from a single compact set B ⊂ C ext to the entire complex plane. This is done via a diagonal-type argument.4.2.1. Odd Dimensions.
We choose a tiling of C , where we start with a square B = (cid:8) z ∈ C (cid:12)(cid:12) | Re( z ) | ≤ , − ≤ | Im( z ) | ≤ (cid:9) and then add squares in a counterclockwise spiral manner as shown in Figure2. Re z Im zB B B B B B B ... Figure 2.
Tiling of the complex plane
Next, we define our algorithm as follows. We letΓ ( q ) := Θ B ( q )Γ ( q ) := Θ B ( q ) ∪ Θ B ( q )Γ ( q ) := Θ B ( q ) ∪ Θ B ( q ) ∪ Θ B ( q )...Γ n ( q ) := n [ j =1 Θ B j n ( q ) . Lemma 3.4 ensures that each Θ B k n converges to Res( q ) ∩ B k for fixed k and since the { B k } form atiling of C , it follows that Γ n ( q ) → Res( q ) in Attouch-Wets metric.Sheet 1 B (1)1 n = 1 : Sheet 2 Sheet 3 · · · n = 2 : Sheet 1 B (1)1 B (1)2 Sheet 2 B (2)1 Sheet 3 · · · n = 3 : Sheet 1 B (1)1 B (1)2 B (1)3 Sheet 2 B (2)1 B (2)2 Sheet 3 B (3)1 · · · Figure 3.
Tiling of the logarithmic Riemann surface
Even Dimensions.
In even dimensions we have to cover not only the complex plane C , butits logarithmic covering space, which is equivalent to covering infinitely many copies of the complexplane. A similar strategy as in the odd dimensional case, together with a diagonal-type argumentdoes the job in this case. Indeed, we can construct a cover by boxes B n as follows (cf. Figure 3).(1) Start with box B (defined as in the odd dimensional case) on the first Riemann sheet;(2) Add a box B below B on sheet number 1 and add a box B on sheet number 2;(3) Add a box B on sheet number 1, add a box B on sheet number 2 and a box B on sheetnumber 3;(4) . . .Next, define againΓ ( q ) := Θ B (1)1 ( q )Γ ( q ) := Θ B (1)1 ( q ) ∪ Θ B (1)2 ( q ) ∪ Θ B (2)1 ( q )Γ ( q ) := Θ B (1)1 ( q ) ∪ Θ B (1)2 ( q ) ∪ Θ B (1)3 ( q ) ∪ Θ B (2)1 ( q ) ∪ Θ B (2)2 ( q ) ∪ Θ B (1)3 ( q )...Γ n ( q ) := n [ k =1 n − k +1 [ j =1 Θ B ( k ) j n ( q ) . OMPUTING SCATTERING RESONANCES 15
Lemma 3.4 ensures that each Θ B ( k ) j n converges to Res( q ) ∩ B ( k ) j for fixed k and since the { B ( k ) j } forma tiling of C ext , it follows that Γ n ( q ) → Res( q ) in Attouch-Wets metric. The proof of Theorem 1.8is complete. 5. Numerical Results
Software to compute resonances has been in existence for decades [19, 9, 1]. The authors of [7]recently proposed a collection of MATLAB codes to compute resonance poles and scattering of planewaves efficiently (“MatScat”, cf. [6]). In this section we compare the results of our algorithm tothat of MatScat.In order to study the actual numerical performance of our algorithm, we coded a MATLABroutine for the one-dimensional case with supp( q ) ⊂ [ a, b ] (for some known a < b ), which computesthe set n z ∈ L n ∩ B (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:16) n × n + (cid:0) K ( i, j ) (cid:1) i,j ∈ b − an Z ∩ [ a,b ] (cid:17) − (cid:13)(cid:13)(cid:13) > C o , where the region B in the complex plane, the lattice distance of L n and cutoff threshold C weretreated as independent parameters.Comparison of results. Figures 4 and 5 show the output of MatScat (black dots) versus the outputof our algorithm (blue regions) for a Gaussian well and trapping potential, respectively. As the plotsshow, there is agreement between the two. − − . − . − . − . . . . . − − Potential − − − − − − − − ReIm
Resonances
Our algorithm MatScat
Figure 4.
Comparison of the result of [6] (black) and our algorithm (blue) for a Gaussianwell supported between − n = 100; thresholdfor resolvent norm: C = 200; number of lattice points in the shown region of the complexplane: M × M = 1000 × Limitations. As mentioned before, MatScat has been developed with the goal to create an efficientalgorithm to compute resonances fast. Indeed, the computation of the black dots in Figure 4 takesless than a second, while computing the regions with our algorithm takes several hours on a personalcomputer. We stress that our MATLAB code was written mainly for illustration purposes and that − . − − . − . − . − . . . . . . Potential − − − − − − − − ReIm
Resonances
Our algorithm MatScat
Figure 5.
Comparison of the result of [6] (black) and our algorithm (blue) for a smoothtrapping potential supported between − . .
2. The chosen parameter values are: n = 100; threshold for resolvent norm: C = 200; number of lattice points in the shownregion of the complex plane: M × M = 1000 × there is considerable room for improvement in numerical efficiency. Moreover, our algorithm canonly yield reliable results in a certain region, as the following heuristic calculations make clear. • Imaginary part of z : Since the fundamental solution G ( x, z ) = iz e ı z | x | grows exponentiallywith − Im( z ) and x ∈ [ − a, a ], a limit is reached when | Im( z ) | ∼ log(2 M )2 a , where M is thelargest number the machine can store with adequate precision (for the interval [ − a, a ] =[ − ,
1] and M = 10 this bound yields Im( z ) & − . • Real part of z : Similarly, a natural bound on Re( z ) is reached when the period of e ı z | x | isless than twice the lattice spacing n , i.e. when | Re( z ) | . πn (for n = 30 this bound yields | Re( z ) | . z ) is fixed by the machine precision, while the bound on | Re( z ) | can be raised by increasing n . Remark . We note that our algorithm is not restricted to one-dimension or real-valued potentials.Indeed, the algorithm Γ n only uses the bound supp( q ) ⊂ Q M , and higher dimensional implementa-tions of Γ n can be coded similarly to the one-dimensional one. Appendix A. Fundamental Solution
In this appendix we gather some well-known results about the fundamental solution for theHelmholtz equation. These facts are used to show that the abstract framework of Section 3 holds inthe context of our algorithm as defined in Section 4, namely that eqs. (3.1)-(3.3) hold. We adopt thenotation of [2] and write f ( ζ ) ∼ ζ ν if f and ζ ν are asymptotically equal , i.e. | f ( ζ ) − ζ ν | = O ( | ζ | ν +1 )as | ζ | → OMPUTING SCATTERING RESONANCES 1710 20 30 40 50 60 70 80 90 100 − − ReIm n = 15 : Algorithm output
10 20 30 40 50 60 70 80 90 100 − − ReIm n = 30 : Algorithm output
Figure 6.
Numerical artefacts for large real part of z . Top: Output of our algorithmfor Gaussian well potential on the interval [ − ,
1] with n = 15. Bottom: Output for thesame problem with n = 30. The locations of the spurious peaks agree with the bound | Re( z ) | ∼ πn in each case. Remark
A.1 . By the asymptotic expansion of the Hankel functions H (1) ν ( ζ ) ∼ − Γ( ν ) π (cid:16) ζ (cid:17) − ν , ν > , π log( ζ ) , ν = 0 , where Γ denotes the Gamma function and log denotes the principal branch of the logarithm (cf. [2,Ch. 9.1.9]), we find that the fundamental solution (1.1) satisfies the small | x | asymptotics G ( x, z ) ∼ − ıΓ( d − ) π (cid:18) z | x | (cid:19) − d − ı4 (cid:18) z π | x | (cid:19) d − = Γ( d − )4 π n | x | d − , as | x | → , for n ≥
3, and G ( x, z ) ∼ − π log( z | x | ) , as | x | → , for n = 2. Hence | G ( x, z ) | ≤ C z · | x | d − , n ≥ , log( | x | ) , n = 2 , (A.1)where C z > z in a compact subset of C . Similar formulas hold for thederivatives of G . Indeed, identities for Hankel functions (cf. [2, Ch. 9.1.30]) show that |∇ G ( x, z ) | ≤ C z | x | d − , for d ≥ . (A.2) Remark
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