On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators

We study the solvability complexity index (SCI) for unbounded selfadjoint operators on separable Hilbert spaces and perturbations thereof. In particular, we show that if the extended essential spectrum of a selfadjoint operator is convex, then the SCI for computing its spectrum is equal to 1. This result is then extended to relatively compact perturbations of such operators and applied to Schrödinger operators with (complex valued) potentials decaying at infinity to obtain SCI=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {SCI}}=1$$\end{document} in this case, as well.


Introduction
The problem of computing spectra of partial differential operators is fundamental to many problems in physics with real world applications. Perhaps one of the most prominent examples of this is quantum mechanics, where the possible bound state energies of a particle subject to a force described by a potential function V are given by the eigenvalues of the corresponding Schrödinger operator −Δ + V . Generically, the spectral problem of such an operator cannot be solved explicitly and one has to resort to numerical methods. By practical constraints, any computer algorithm, which might be used to compute the spectrum, will only be able to handle a finite amount of information about the operator and perform a finite number of arithmetic operations on this information (in practice, this "finite amount of information" is usually given by some sort of discretisation of the domain, which approximates the infinite dimensional spectral problem by a finite dimensional one). In other words, any algorithm will always "ignore" an infinite In the above definition, Ω is the set of objects that give rise to the computational problem, Λ plays the role of providing the information accessible to the algorithm, and Ξ : Ω → M gives the quantity that one wishes to compute numerically.
An example of a computational problem in the sense of Definition 1.1 is given by the spectral problem discussed above. Indeed, given a separable We will refer to any arithmetic algorithm simply as an algorithm from now on. For more general concepts the reader may consult [2].
In [2] it has been shown that if Ω is the set of compact operators on a separable Hilbert space H, then there exists a sequence of algorithms Γ n : Ω → C such that Γ n (T ) → σ(T ) (in Hausdorff sense) for all T ∈ Ω, while for for all bounded selfadjoint operators. Hence, it is possible to compute the spectrum of non-compact operators using algorithms, but the number of limits required may increase (this general phenomenon has first been observed by Doyle and McMullen in the context of finding zeros of polynomials, cf. [7]). In order to capture this phenomenon, the following definition has been made Definition 1.3. [2] Let (Ω, Λ, Ξ, M) be a computational problem. A tower of algorithms of height k is a family Γ n1,n2,...,n k : Ω → M of arithmetic algorithms such that for all T ∈ Ω Ξ(T ) = lim The examples above show that the number of limits required to compute the problem function Ξ is a measure for the numerical complexity of the underlying computational problem. This motivates the Definition 1.4. [2] A computational problem (Ω, Λ, Ξ, M) is said to have Solvability Complexity Index k if k is the smallest integer for which there exists a tower of algorithms of height k that computes Ξ.
If a computational problem has solvability complexity index k, we write SCI(Ω, Λ, Ξ, M) = k. Remark 1.5. In this article we are mainly interested in the spectral problem and will therefore write SCI(Ω, Λ) instead of SCI(Ω, Λ, Ξ, M), where it is understood that Ξ(T ) = σ(T ) and M is the set of closed subsets of C equipped with the Attouch-Wets metric d AW defined as In practice it is often important to have control of the approximation error d Γ n1,...,n k (T ), Ξ(T ) for all T ∈ Ω. It is straightforward to show, however, that such an estimate is impossible to obtain as soon as SCI(Ω, Λ, Ξ, M) > 1 (cf. [2, Thm. 6.1]). Indeed, it is easy to see that if for a tower of algorithms Γ n1,...,n k there exist subsequences n 1 (m), . . . , n k (m) such that Γ n1(m),...,n k (m) (T ) → 0 for all T ∈ Ω, thenΓ m := Γ n1(m),...,n k (m) is in fact a tower of height 1 for Ω and hence SCI(Ω) = 1.
For this reason, it is of particular interest to find classes Ω of operators for which SCI(Ω, Λ, σ(·)) = 1 (with appropriately chosen Λ). The present 54 Page 4 of 23 F. Rösler IEOT article addresses precisely this question. In fact, we will show that for selfadjoint operators whose extended essential spectrum (see (2.2)) is convex, we have SCI = 1. This is done by explicitly constructing a sequence of arithmetic algorithms which computes the spectrum of any such operator. The result is then extended to certain relatively compact perturbations of such operators. We stress that the new aspect of our work is to consider the shape of the essential spectrum as a relevant criterion for reducing the numerical complexity of the spectral problem. As an application of this approach, we will show that our results apply to non-selfadjoint Schrödinger operators with certain well behaved potentials. The problem of determining the SCI for spectral problems has previously been studied in [2,9] for operator in abstract Hilbert spaces, as well as for partial differential operators. Previous results include where K(H) denotes the set of compact operators. The last of the above bounds, SCI(K(H), σ(·)) = 1, is related to the fact that compact operators can be approximated in operator norm by finite range operators.

Schrödinger operators
In [2], the SCI for the spectral problem of Schrödinger operators with complex valued potentials V has been studied. It has been shown that if then SCI(Ω, σ(·)) = 1. The proof relies on the fact that operators as in (1.1) have compact resolvent.
In the case of bounded potentials, one lacks compact resolvent and the situation is somewhat more difficult. It has been shown in [2,Th. 4.2] that if Ω denotes the set of Schrödinegr operators on R d with V bounded and of bounded variation, then SCI(Ω, σ(·)) ≤ 2. It has since then been an open problem, whether without any additional information the SCI of this problem is equal to one or two.
The SCI of certain unbounded operators in separable Hilbert spaces, whose matrix representation is banded, has been studied in [9].
In this article, we will take a step towards closing this gap. We will prove that if M > 0 and Ω denotes the set of all Schrödinger operators −Δ + V with supp(V ) ⊂ B M (0) and |∇V | ≤ M , then SCI(Ω, σ(·)) = 1 (for the precise statement, see Sect. 4). This is done by first proving two abstract theorems about the SCI of selfadjoint operators which are of independent interest. The proofs of these abstract results rely on recent developments in the theory of essential numerical ranges for unbounded operators, cf. [3]. The main theorems of this article are Theorems 2.1, 3.1 and 4.3. The question as to wether the assumption on the decay of V is essential for having SCI = 1 remains an interesting open problem and will be addressed in future work.

Selfadjoint Operators
Let H be a separable Hilbert space and H n ⊂ H be a sequence of finite dimensional subspaces such that H n ⊂ H n+1 for all n ∈ N and P n s − → I, where P n denotes the orthogonal projection onto H n . Define kn } be an orthonormal basis of H n and define i , e (n) j are the evaluation functions producing the (i, j)th matrix elements. This is the set of information accessible to the algorithm. . For this reason, we will simply use the notation σ e (T ) to denote the essential spectrum, whenever the operators in question are selfadjoint.

Definition of the Algorithm
Let T ∈ Ω 1 and define the truncated operator This operator can be represented by a finite dimensional (square) matrix with with the convention that Proof. Part (i) was proved in [9], while part (ii) follows by noting that For n ∈ N we define a map Γ (1) Then, by the above lemma, each Γ (1) n is an arithmetic tower of height one in the sense of Definition 1.3. Clearly, Γ Next we prove a version of the second resolvent identity for our operator approximation.

Lemma 2.5.
Let T : dom(T ) → H be selfadjoint, n H n form a core of T and T n be defined as in (2.6). Then each T n is selfadjoint on H n and T n → T in strong resolvent sense. Proof. We start by showing that each T n is selfadjoint. First note that each T n is automatically bounded, since the H n are finite dimensional. Now let x, y ∈ H n . Then we have T n x, y = P n T x, y = T x, P n y = T x, y = x, T y = P n x, T y = x, P n T y = x, T n y . (2.7) and hence T n is selfadjoint. The claim now follows directly from [13,Satz 9.29], since n H n is a core for T and P n converges strongly to the identity.

General Results on Spectral and Pseudospectral Pollution
In this subsection we collect facts about spectral and pseudospectral pollution for closed, densely defined operators H, H n on H, which are not necessarily selfadjoint. These results will be used in later sections. The sets of spectral and pseudospectral pollution are defined, respectively, as The following definitions from [3], which are related to the essential spectrum, will be used frequently in the sequel. The limiting essential spectrum: the limiting ε-near spectrum: the essential numerical range and the limiting essential numerical range The essential limiting spectrum was originally introduced in [1] in the context of Galerkin approximation and later adapted to a more general framework in [4,5], where the set Λ e,ε (H n ) n∈N was introduced. The essential numerical range was originally introduced by Stampfli and Williams in [12] for bounded operators and recently extended to unbounded operators in [3]. It was shown there that the essential numerical range is a convenient tool when studying spectral and pseudospectral pollution of operator approximations. This fact will prove useful to our purpose as we shall see in the following. In order to prove the next lemma, we need a fact about closures of pseudospectra.

Lemma 2.6. For all operators H on H of the form
Proof. This follows from the fact that the resolvent norm of any such operator tends to 0 at i∞ and hence cannot be constant on an open set (cf. [6,Th. 3.2]). Indeed, in this case we have for any sequence (ε k ) with ε k → 0.

Lemma 2.7. (i) For any closed, densely defined operator H on H one has
Proof. We first prove (i). Let λ ∈ ε>0 δ∈(0,ε] Λ e,δ (H n ) n∈N . Then for all ε > 0 there exists δ ∈ (0, ε] and a sequence (x k ) with x k ∈ dom(H n k ) (for some subsequence (n k )) such that Hence, for every m ∈ N there exists a sequence (x The notation n k (m) indicates that the corresponding subsequence of (H n ) depends on m. Now, construct a diagonal sequence as follows. Since H is separable, the weak topology is metrisable on the unit ball. Let d denote a corresponding metric. Now, for any given m ∈ N, choose k m ∈ N large enough such that Then for the sequence y m := x The proof of claim (ii) is now immediate, because the sequence of sets δ∈(0,ε] Λ e,δ (H n ) n∈N is shrinking with ε. Finally, we prove the following characterisation of convergence of sets in the Attouch-Wets metric. We recall that d AW (X n , X) → 0 if and only if d K (X n , X) → 0 for all K ⊂ C compact, where Proposition 2.8. Let X, X n , n ∈ N be closed subsets of C. Assume that (a) If λ n ∈ X n and λ n → λ, then λ ∈ X.
(b) If λ ∈ X, then there exist λ n ∈ X n with λ n → λ. Then one has d AW (X n , X) → 0.
Proof. Let K ⊂ C be compact. We will show that if (a), (b) hold, then both distances sup z∈Xn∩K dist(z, X) and sup w∈X∩K dist w, X n converge to zero. We begin with the latter. Let ε > 0. For all w ∈ X ∩ K, the ball B ε (w) contains infinitely many elements z n ∈ X n , by (b). The collection {B ε (w) | w ∈ X ∩ K} forms an open cover of the compact set X ∩K. Hence, there exist finitely many w 1 , . . . , is contained in some B ε (w i ) and hence dist(w, X n ) < ε for any w ∈ X ∩ K, as soon as n = n(i) is large enough. But since there are only finitely many B ε (w i ), one will have dist(w, X n0 ) < 2ε for all w ∈ X ∩ K for n 0 = max{n i | i = 1, . . . , k}.
To show that sup z∈Xn∩K dist(z, X) → 0 as n → ∞, note that since all sets X n ∩ K are compact, we can choose a sequence z n ∈ X n ∩ K such that sup z∈Xn∩K dist(z, X) = dist(z n , X).
Since the sequence (z n ) is obviously bounded, we can extract a convergent subsequence z nj → z 0 ∈ K. Now use assertion (a) from above to conclude that in fact z 0 ∈ X ∩ K. This readily implies Since the same reasoning can be applied to every subsequence of the sequence we conclude that the whole sequence converges to zero.

Proof of Theorem 2.1
Next, we prove convergence of the algorithm Γ  Proof. By definition of Γ (1) n , one has that 1 n ≥ (λ n − T n ) −1 −1 ≥ dist(λ n , σ(T n )) for all n ∈ N. Hence, there exists a sequence z n ∈ σ(T n ) such that |z n − λ n | → 0 and consequently z n → λ. We conclude from [5, Th.
To conclude, we apply [3, Th. 6.1] to show that spectral pollution is in fact absent for T ∈ Ω 1 . Indeed, let λ n ∈ Γ It remains to prove spectral inclusion, i.e. nothing is missed by Γ

Conclusion
We have shown that (a) If λ n ∈ Γ

Relatively Compact Perturbations
In this section we show that Theorem 2.1 remains true for certain relatively compact, bounded perturbations of selfadjoint operators. More precisely, we have where f i,j,n are the evaluation functions producing the (i, j)th matrix elements (see (2.4)). Then one has SCI(Ω 2 , Λ 2 , σ(·)) = 1.

Remark 3.2.
(i) Note that the information provided to the algorithm in Λ 2 includes the decomposition of H ∈ Ω 2 into a selfadjoint part T and a perturbation V . This means, that the algorithm does not have to compute this decomposition. It gets it for free. This is a reasonable assumption in many applications as we will see in Sect. 4. (ii) In fact, the assumptions in the definition of Ω 2 imply that σ(T ) is convex. Indeed, for any selfadjoint operator with purely essential spectrum, σ e (T ) is convex if and only if σ(T ) is convex.
Note the additional assumption σ(T ) = σ e (T ) in the selfadjoint part T . This will be needed later in order to exclude spectral pollution of the algorithm.

Proof of Theorem 3.1
Spectrum of H. The proof of Theorem 3.1 is via perturbation theory. We first focus on the spectrum of an operator H ∈ Ω 2 . Recall the definitions of the essential spectra σ e2 , σ e5 from Sect. 2. In the proof, we will need the following results, which are classical.

H) if and only if H − λ is Fredholm with ind(H − λ) = 0 and a deleted neighbourhood of λ lies in ρ(H).
In other words, if λ / ∈ σ e5 (H), then λ is an isolated eigenvalue of finite multiplicity. Furthermore, the following perturbation result is known. To see assertion (ii), let u ∈ H and note that then P n u → u strongly. By continuity of V , it immediately follows that V P n u → V u in H. Hence, from the definition of V n we conclude that V n P n u = P n V | Hn P n u = P n →I strongly Assertion (iii) now immediately follows by combining (i) and (ii).
The next lemma shows that even the perturbed operators H n converge in strong resolvent sense. Proof. This follows from [4,Cor. 3.5], since which tends to ∞ as z → i∞, and V , V n are uniformly bounded.

The algorithm
The algorithm for Ω 2 , Λ 2 is defined analogously to that in Sect. 2. Namely, we define G C n := 1 n (Z + iZ) ∩ B n (0) ⊂ C.  [9]). Since we have already shown that Γ (1) n approximates σ(T ) correctly and that σ(T ) = σ e (T ) = σ e5 (H), we know that Γ (2) n will not miss anything in σ e5 (H). Thus, it only remains to prove absence of spectral pollution and spectral inclusion for the discrete set σ(H)\σ e5 (H) for the algorithmΓ 1 n This will be done in the remainder of this section.
However, let us first take a moment to assure that Γ (2) n defines a reasonable algorithm. Clearly, each Γ n is an admissible algorithm as well.
Remark 3.7. We note that the choice 1 n as an upper bound for s(H n − λ) in (3.1) is arbitrary. The proof below will show that one could equally well have chosen n (H). This fact will be used in Sect. 4.

Spectral pollution
Let us prove that the approximation Γ (2) n (H) does not have spectral pollution for H ∈ Ω 2 . To this end, note that againΓ n (H) ⊂ σ ε (H n ) for ε > 0 fixed and n large enough. According to [5,Th. 3.6 ii)], ε-pseudospectral pollution of the approximation H n → H is confined to Hence, for any sequence λ n ∈Γ n (H) with λ n → λ ∈ C we have We conclude with the following -there exists a sequence ε k with ε k 0 and λ ∈ δ∈(0,ε k ] Λ e,δ (H n ) n∈N for all k.
In the first case, it follows that In the second case, we have  Note that the previous lemma is the only place in which we need the semiboundedness assumption in the definition of Ω 2 . Overall we have shown that for any sequence λ n ∈Γ n (H) which converges to some λ ∈ C we necessarily have λ ∈ σ(H), in other words, spectral pollution does not exist.

Spectral inclusion
It remains to show that the approximation (Γ (2) n (H)) is spectrally inclusive, i.e. that for any λ ∈ σ(H) there exists a sequence λ n ∈ Γ (2) n (H) such that λ n → λ. As explained above, the existence of such a sequence is already guaranteed for all λ ∈ σ e5 (H). Proof. First note that by Theorem 3.3 λ is an isolated point. Moreover, we have seen in the proof of Lemma 3.9 that σ e (H n ) n∈N ∪ σ e (H * n ) n∈N * ⊂ σ e (H) and hence λ does not belong to this set either. From Lemma 3.6 and [5, Th. 2.3 i)] we conclude that there exists a sequence μ n ∈ σ(H n ) with μ n → λ. Now, by definition of G C n , for each n there exists λ n ∈ G C n such that |μ n −λ n | < 1 n and hence (H n −λ n ) −1 L(Hn) ≥ n which implies λ n ∈Γ n (H). Since |μ n − λ n | → 0 and μ n → λ, it follows that λ n → λ.

Application to Schrödinger Operators
In this section we will apply the results of Sects. 2 and 3 to Schrödinger operators on L 2 (R d ). More specifically, fix a continuous, monotone decreasing function g : [0, ∞) → [0, ∞) with g(t) → 0 as t → ∞ and let M > 0. We define Moreover, let H n denote the subspace of L 2 (R d ) spanned by all characteristic functions of cubes of edge length 1 n with centres inside a ball of radius n: H n := span χ i+[0, 1 n ) d i ∈ L n It is easily seen by smooth approximation that P Hn → I strongly in L 2 (R d ). However, none of the basis functions χ i+[0, 1 n ) d are contained in the domain of −Δ. In order to circumvent this, the space we will actually work with will be where the hat denotes the Fourier transform in L 2 (R d ). For any enumeration i k of the set L n , we define e (n)  Proof. This follows immediately from the unitarity of the Fourier transform and the equality We note that the functions e (n) k can be calculated explicitly. Indeed, one has e (n) where (i k ) j denotes the j'th component of the vector i j and ξ = (ξ 1 , . . . , ξ d ) ∈ R d . Using this explicit representation, it can be easily seen that we have the following.

Lemma 4.2. For each n ∈ N one has
Proof. From the definition of e (n) k it follows by direct calculation that from which the assertion follows. Note that the bound in the second equation can be made independent of k, because i k ∈ L n ⊂ B n (0) for all k.
The information accessible to the algorithm will be the set where ρ x (V ) = V (x) are the evaluation functionals and e (n) k (i) denote constant functions that map V to the number e (n) k (i). The meaning of the constants nδ mk 3 d j=1 j will become clear later on. Together, Ω 3 and Λ 3 define a computational problem (Ω 3 , Λ 3 , σ(·)). The main result of this section is the following. The proof of Theorem 4.3 will be by reduction to Theorem 3.1. In order to accomplish this, we need to be able to compute the matrix elements (−Δ + V )e i , e j by performing only a finite number of algebraic operations on a finite number of values of V . This will be the main difficulty.

Proof of Theorem 4.3
We first show that the spaces H n defined in (4.2) are indeed a reasonable choice for the problem at hand. More precisely, we have Proof. By means of the Fourier transform the assertion is equivalent to the space n∈N H n being a core for the multiplication operator u → |ξ| 2 u in L 2 (R d ). To verify this, we have to show that for every u ∈ dom(|ξ 2 |) there exists a sequence u n ∈ H n such that Indeed, for smooth u the L 2 -convergence of u n to u is standard, while the general case follows by a density argument. We omit the technical details. To show point (ii), let R > 0 and decompose the norm in (ii) as |ξ| 2 (u n − u) 2 L 2 (R d ) = BR |ξ| 2 (u n − u) 2 dξ + R d \BR |ξ| 2 (u n − u) 2 dξ, (4.5) where B R denotes the ball of radius R centered at 0. We first estimate the second term on the right hand side. To this end, we let u n be defined by (4.4) and employ the shorthand notation χ i := n d 2 χ i+[0, 1 n ) d . On the whole space we have where we have used the fact that supp(χ i ) ∩ supp(χ j ) = ∅ for i = j. The factor n Next, we note that it is easy to see that there exist constants a, b > 0 such that F n (ξ) ≤ a|ξ| 2 + b uniformly in n (see Fig. 1).
Overall we conclude that where the last term on the right hand side is finite because by assumption u ∈ dom(|ξ| 2 ). In fact, from this last inequality we can see immediately that Now let ε > 0 and choose R so large that (a|ξ| 2 + b)u 2 L 2 (R d \BR−1) + |ξ| 2 u 2 L 2 (R d \BR) < ε. From Eq. (4.5) we then see that lim sup n→∞ |ξ| 2 (u n − u) because u n → u in L 2 (R d ). Since ε was arbitrary, it follows that lim sup Our strategy for proving Theorem 4.3 is as follows. By the assumptions on V stated in the definition of Ω 3 and Lemma 4.4 we know that we have Ω 3 ⊂ Ω 2 , if we choose H = L 2 (R d ) and H n as in (4.2). Hence, we already know from Theorem 3.1 that Γ