# On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators

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## Abstract

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 \({\text {SCI}}=1\) in this case, as well.

## Keywords

Schrödinger operators Spectral approximation Computational complexity## Mathematics Subject Classification

Primary 35P99 Secondary 35Q40## 1 Introduction

*V*are given by the eigenvalues of the corresponding Schrödinger operator \(-\Delta +V\). 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 amount of information about the operator. One might hope that by increasing the dimension of the approximation (or decreasing the step size of the discretisation), one will eventually obtain a reasonable approximation of the spectrum. Hence, it is a legitimate question to ask:

It turns out that the answer to the above question is not always in the affirmative. Indeed, it has been shown in [2] that if \(\Omega =L(\mathcal {H})\) (the space of bounded operators on a separable Hilbert space \(\mathcal {H}\)), then for any sequence of algorithms there exists \(T\in \Omega \) whose spectrum is not approximated by that sequence. This observation has led to the wider definition of the so-calledGiven a class of operators \(\Omega \), does there exist a sequence of algorithms \(\Gamma _n\) such that \(\Gamma _n(T)\rightarrow \sigma (T)\) (in an appropriate sense) for all \(T\in \Omega \)?

*Solvability Complexity Index*(SCI), introduced in [9], of which we will now give a brief review.

### Definition 1.1

*computational problem*is a quadruple \((\Omega ,\Lambda ,\Xi ,{\mathcal {M}})\), where

- (i)
\(\Omega \) is a set, called the

*primary set*, - (ii)
\(\Lambda \) is a set of complex valued functions on \(\Omega \), called the

*evaluation set*, - (iii)
\({\mathcal {M}}\) is a metric space,

- (iv)
\(\Xi :\Omega \rightarrow M\) is a map, called the

*problem function*.

In the above definition, \(\Omega \) is the set of objects that give rise to the computational problem, \(\Lambda \) plays the role of providing the information accessible to the algorithm, and \(\Xi :\Omega \rightarrow {\mathcal {M}}\) 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 Hilbert space \(\mathcal {H}\) with orthonormal basis \(\{e_i\}\), one can choose \(\Omega =L(\mathcal {H})\), \({\mathcal {M}} = \{\text {compact subsets of }{\mathbb {C}}\}\), equipped with the Hausdorff metric, and \(\Xi (T)=\sigma (T)\). For the evaluation set one could choose \(\Lambda :=\{f_{ij}\,|\,i,j\in {\mathbb {N}}\}\), where \(f_{ij}(T) = \langle Te_i,e_j\rangle \) give the matrix elements of an operator with respect to the basis \(\{e_i\}\).

### Definition 1.2

*arithmetic algorithm*is a map \(\Gamma :\Omega \rightarrow {\mathcal {M}}\) such that for each \(T\in \Omega \) there exists a finite subset \(\Lambda _\Gamma (T)\subset \Lambda \) such that

- (i)
the action of \(\Gamma \) on

*T*depends only on \(\{f(T)\}_{f\in \Lambda _\Gamma (T)}\), - (ii)
for every \(S\in \Omega \) with \(f(T)=f(S)\) for all \(f\in \Lambda _\Gamma (T)\) one has \(\Lambda _\Gamma (S)=\Lambda _\Gamma (T)\),

- (iii)
the action of \(\Gamma \) on

*T*consists of performing only finitely many arithmetic operations on \(\{f(T)\}_{f\in \Lambda _\Gamma (T)}\).

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 \(\Omega \) is the set of compact operators on a separable Hilbert space \(\mathcal {H}\), then there exists a sequence of algorithms \(\Gamma _n:\Omega \rightarrow {\mathbb {C}}\) such that \(\Gamma _n(T)\rightarrow \sigma (T)\) (in Hausdorff sense) for all \(T\in \Omega \), while for the set of bounded selfadjoint operators \(\Omega =\{T\in L(\mathcal {H})\,|\,T^*=T\}\) this is not possible.

*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

*tower of algorithms*of height

*k*is a family \(\Gamma _{n_1,n_2, \ldots ,n_k}:\Omega \rightarrow {\mathcal {M}}\) of arithmetic algorithms such that for all \(T\in \Omega \)

The examples above show that the number of limits required to compute the problem function \(\Xi \) is a measure for the numerical complexity of the underlying computational problem. This motivates the

### Definition 1.4

[2] A computational problem \((\Omega ,\Lambda ,\Xi ,{\mathcal {M}})\) 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 \(\Xi \).

If a computational problem has solvability complexity index *k*, we write \({\text {SCI}}(\Omega ,\Lambda ,\Xi ,{\mathcal {M}})=k\).

### Remark 1.5

In practice it is often important to have control of the approximation error \(d\big (\Gamma _{n_1, \ldots ,n_k}(T),\Xi (T)\big )\) for all \(T\in \Omega \). It is straightforward to show, however, that such an estimate is impossible to obtain as soon as \({\text {SCI}}(\Omega ,\Lambda ,\Xi ,{\mathcal {M}})>1\) (cf. [2, Thm. 6.1]). Indeed, it is easy to see that if for a tower of algorithms \(\Gamma _{n_1, \ldots ,n_k}\) there exist subsequences \(n_1(m), \ldots ,n_k(m)\) such that \(\Gamma _{n_1(m), \ldots ,n_k(m)}(T)\rightarrow 0\) for all \(T\in \Omega \), then \({{\tilde{\Gamma }}}_m:=\Gamma _{n_1(m), \ldots ,n_k(m)}\) is in fact a tower of height 1 for \(\Omega \) and hence \({\text {SCI}}(\Omega )=1\).

For this reason, it is of particular interest to find classes \(\Omega \) of operators for which \({\text {SCI}}(\Omega ,\Lambda ,\sigma (\cdot ))=1\) (with appropriately chosen \(\Lambda \)). The present 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 \({\text {SCI}}=1\). 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

**Bounded operators**Let \(\mathcal {H},\,\Lambda \) be as in the example above Definition 1.2. It was shown in [2, Th. 3.3, Th. 3.7] that then

**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

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 \(\Omega \) denotes the set of Schrödinegr operators on \({\mathbb {R}}^d\) with *V* bounded and of bounded variation, then \({\text {SCI}}(\Omega ,\sigma (\cdot ))\le 2\). 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 \(\Omega \) denotes the set of all Schrödinger operators \(-\Delta +V\) with \({\text {supp}}(V)\subset B_M(0)\) and \(|\nabla V|\le M\), then \({\text {SCI}}(\Omega ,\sigma (\cdot ))=1\) (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 \({\text {SCI}}=1\) remains an interesting open problem and will be addressed in future work.

## 2 Selfadjoint Operators

*i*,

*j*)th matrix elements. This is the set of information accessible to the algorithm.

### Theorem 2.1

We have \({\text {SCI}}(\Omega _1,\Lambda _1,\sigma (\cdot ))=1\).

### Remark 2.2

- (i)
Note that Theorem 2.1 in particular applies to bounded selfadjoint operators with convex essential spectrum. In this sense, Theorem 2.1 can be viewed as an extension of [2, Th. 3.7], where it was shown that \({\text {SCI}}=1\) for the set of all

*compact*operators (which naturally satisfy \(\sigma _e(T)\subset \{0\}\)). - (ii)
Theorem 2.1 is optimal in the sense that the selfadjointness assumption in (2.1) cannot be dropped. Indeed, counterexamples show that \({\text {SCI}}\ge 2\) for non selfadjoint bounded operators with convex essential spectrum (cf. [2, Proof of Th. 3.7, Step II] for an explicit construction).

### Remark 2.3

*H*be a closed, densely defined operator on \(\mathcal {H}\). Then

*not*agree in general. However, it can be shown that for

*selfadjoint*operators

*T*on \(\mathcal {H}\), one always has \(\sigma _{e2}(T)=\sigma _{e5}(T)\) (cf. [8, Th. IX.1.6]). For this reason, we will simply use the notation \(\sigma _e(T)\) to denote the essential spectrum, whenever the operators in question are selfadjoint.

### 2.1 Definition of the Algorithm

### Lemma 2.4

- (i)
For all

*n*and \(\lambda \), we have \(s(T_n-\lambda )=\Vert (T_n-\lambda )^{-1}\Vert ^{-1}_{L(\mathcal {H}_n)}\). - (ii)
For any \(q>0\), testing whether \(s(T_n-\lambda )>q\) requires only finitely many arithmetic operations on the matrix elements of \(T_n\).

### Proof

Part (i) was proved in [9], while part (ii) follows by noting that \(s(T_n-\lambda )>q\) is equivalent to \((T_n-\lambda )^*(T_n-\lambda )-q^2I\) being positive definite; see [2, Prop. 10.1] for a full proof. \(\square \)

*n*, where \(\sigma _{\frac{1}{n}}(\cdot )\) denotes the \(\frac{1}{n}\)-pseudospectrum, i.e. \(\sigma _{\frac{1}{n}}(T_n)=\left\{ z\in {\mathbb {C}}\,|\,\Vert (T_n-z)^{-1}\Vert >n \right\} \). Next we prove a version of the second resolvent identity for our operator approximation.

### Lemma 2.5

Let \(T:{\text {dom}}(T)\rightarrow \mathcal {H}\) be selfadjoint, \(\bigcup _n\mathcal {H}_n\) form a core of *T* and \(T_n\) be defined as in (2.6). Then each \(T_n\) is selfadjoint on \(\mathcal {H}_n\) and \(T_n\rightarrow T\) in strong resolvent sense.

### Proof

*T*and \(P_n\) converges strongly to the identity. \(\square \)

### 2.2 General Results on Spectral and Pseudospectral Pollution

*limiting essential spectrum:*the

*limiting*\(\varepsilon \)-

*near spectrum*:the

*essential numerical range*

*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 \(\Lambda _{e,\varepsilon }\big ((H_n)_{n\in {\mathbb {N}}}\big )\) 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

*H*on \(\mathcal {H}\) of the form \(H=T+V\), where

*T*is selfadjoint and

*V*is bounded, and all \(\varepsilon <\varepsilon '\) one has

### Proof

### Lemma 2.7

- (i)For any closed, densely defined operator
*H*on \(\mathcal {H}\) one has$$\begin{aligned} \bigcap _{\varepsilon >0} \bigcup _{\delta \in (0,\varepsilon ]} \Lambda _{e,\delta }\big ((H_n)_{n\in {\mathbb {N}}}\big )\;\subset \; \sigma _e\big ((H_n)_{n\in {\mathbb {N}}}\big ). \end{aligned}$$ - (ii)
The above inclusion holds, if \(\bigcap _{\varepsilon >0} \bigcup _{\delta \in (0,\varepsilon ]}\) is replaced by \(\bigcap _{k} \bigcup _{\delta \in (0,\varepsilon _k]}\) for any sequence \((\varepsilon _k)\) with \(\varepsilon _k\rightarrow 0\).

### Proof

\(\Vert x_k\Vert =1\) for all

*k*\(x_k\rightharpoonup 0\) as \(k\rightarrow \infty \)

\(\Vert (H_{n_k}-\lambda )x_k\Vert \rightarrow \delta \).

*m*. Now, construct a diagonal sequence as follows. Since \(\mathcal {H}\) is separable, the weak topology is metrisable on the unit ball. Let

*d*denote a corresponding metric. Now, for any given \(m\in {\mathbb {N}}\), choose \(k_m\in {\mathbb {N}}\) large enough such that

*m*, \(d(y,0)\rightarrow 0\) and \(\Vert (H_{n_{k_m}(m)}-\lambda )y_m\Vert \rightarrow 0\) as \(m\rightarrow \infty \). Hence \(\lambda \in \sigma _e\big ((H_n)_{n\in {\mathbb {N}}}\big )\).

The proof of claim (ii) is now immediate, because the sequence of sets \(\bigcup _{\delta \in (0,\varepsilon ]} \Lambda _{e,\delta }\big ((H_n)_{n\in {\mathbb {N}}}\big )\) is shrinking with \(\varepsilon \). \(\square \)

### Proposition 2.8

- (a)
If \(\lambda _n\in X_n\) and \(\lambda _n\rightarrow \lambda \), then \(\lambda \in X\).

- (b)
If \(\lambda \in X\), then there exist \(\lambda _n\in X_n\) with \(\lambda _n\rightarrow \lambda \).

### Proof

Let \(K\subset {\mathbb {C}}\) be compact. We will show that if (a), (b) hold, then both distances \(\sup _{z\in X_n\cap K}{\text {dist}}(z,X)\) and \(\sup _{w\in X\cap K}{\text {dist}}\big (w,X_n\big )\) converge to zero. We begin with the latter.

Let \(\varepsilon >0\). For all \(w\in X\cap K\), the ball \(B_\varepsilon (w)\) contains infinitely many elements \(z_n\in X_n\), by (b). The collection \(\{B_\varepsilon (w)\,|\,w\in X\cap K\}\) forms an open cover of the compact set \(X\cap K\). Hence, there exist finitely many \(w_1, \ldots ,w_k\in X\cap K\) such that \(B_\varepsilon (w_1), \ldots ,B_\varepsilon (w_k)\) cover \(X\cap K\). Now, any \(w\in X\cap K\) is contained in some \(B_\varepsilon (w_i)\) and hence \({\text {dist}}(w,X_n)<\varepsilon \) for any \(w\in X\cap K\), as soon as \(n=n(i)\) is large enough. But since there are only finitely many \(B_\varepsilon (w_i)\), one will have \({\text {dist}}(w,X_{n_0})<2\varepsilon \) for all \(w\in X\cap K\) for \(n_0 = \max \{n_i\,|\,i=1, \ldots ,k\}\).

### 2.3 Proof of Theorem 2.1

Next, we prove convergence of the algorithm \(\Gamma _n^{(1)}\). By the conditions in (2.1) and Lemma 2.5, we have \(T_n\rightarrow T\) in strong resolvent sense for all \(T\in \Omega \).

### Lemma 2.9

### Proof

### Lemma 2.10

For every \(\lambda \in \sigma (T)\) there exist \(\lambda _n\in \Gamma _n^{(1)}(T)\) such that \(\lambda _n\rightarrow \lambda \).

### Proof

Let \(\lambda \in \sigma (T)\). A simple adaption of the proof of [11, Th. VIII.24] shows that there exists a sequence \((\mu _n)\) with \(\mu _n\in \sigma (T_n)\) and \(\mu _n\rightarrow \lambda \).

For each *n*, there exists \(\lambda _n\in G_n^{\mathbb {R}}\) such that \(|\mu _n-\lambda _n|<\frac{1}{n}\) and hence \(\Vert (T_n-\lambda _n)^{-1}\Vert _{L(\mathcal {H}_n)}\ge n\) which implies \(\lambda _n\in \Gamma _n^{(1)}(T)\). Since \(|\mu _n-\lambda _n|\rightarrow 0\) and \(\mu _n\rightarrow \lambda \), it follows that \(\lambda _n\rightarrow \lambda \). \(\square \)

**Conclusion**We have shown that

- (a)
If \(\lambda _n\in \Gamma _n^{(1)}(T)\) and \(\lambda _n\rightarrow \lambda \), then \(\lambda \in \sigma (T)\).

- (b)
If \(\lambda \in \sigma (T)\), then there exist \(\lambda _n\in \Gamma _n^{(1)}(T)\) with \(\lambda _n\rightarrow \lambda \).

## 3 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

### Theorem 3.1

*i*,

*j*)th matrix elements (see (2.4)). Then one has \({\text {SCI}}(\Omega _2,\Lambda _2,\sigma (\cdot ))=1\).

### Remark 3.2

- (i)
Note that the information provided to the algorithm in \(\Lambda _2\) includes the decomposition of \(H\in \Omega _2\) 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 \(\Omega _2\) imply that \(\sigma (T)\) is convex. Indeed, for any selfadjoint operator with purely essential spectrum, \({\widehat{\sigma }}_e(T)\) is convex if and only if \(\sigma (T)\) is convex.

Note the additional assumption \(\sigma (T)=\sigma _{e}(T)\) in the selfadjoint part *T*. This will be needed later in order to exclude spectral pollution of the algorithm.

### 3.1 Proof of Theorem 3.1

**Spectrum of** \({\varvec{H}}\). The proof of Theorem 3.1 is via perturbation theory. We first focus on the spectrum of an operator \(H\in \Omega _2\). Recall the definitions of the essential spectra \(\sigma _{e2},\,\sigma _{e5}\) from Sect. 2. In the proof, we will need the following results, which are classical.

### Theorem 3.3

[8, Th. IX.1.5] For any closed, densely defined operator *H* on \(\mathcal {H}\), one has \(\lambda \notin \sigma _{e5}(H)\) if and only if \(H-\lambda \) is Fredholm with \({\text {ind}}(H-\lambda )=0\) and a deleted neighbourhood of \(\lambda \) lies in \(\rho (H)\).

In other words, if \(\lambda \notin \sigma _{e5}(H)\), then \(\lambda \) is an isolated eigenvalue of finite multiplicity. Furthermore, the following perturbation result is known.

### Theorem 3.4

*T*be a selfadjoint operator on \(\mathcal {H}\) and

*V*relatively compact w.r.t.

*T*. Then

- (i)
\(H:=T+V\) is closed on \({\text {dom}}(T)\) and

- (ii)
\(\sigma _{e5}(H)=\sigma _{e}(T)\).

**Strong resolvent convergence** Let \(P_n:\mathcal {H}\rightarrow \mathcal {H}_n\) be defined as in Sect. 2 and set \(V_n:=P_nV|_{\mathcal {H}_n}\).

### Lemma 3.5

- (i)
\((V_n)^* = (V^*)_n\) (i.e. compression to \(\mathcal {H}_n\) commutes with taking the adjoint) and

- (ii)
\(V_nP_n\rightarrow V\) strongly in \(\mathcal {H}\).

- (iii)
\(V_n^*P_n\rightarrow V^*\) strongly in \(\mathcal {H}\).

### Proof

Assertion (i) is easily shown by an analogous calculation to (2.7).

*V*, it immediately follows that \(VP_nu\rightarrow Vu\) in \(\mathcal {H}\). Hence, from the definition of \(V_n\) we conclude that

The next lemma shows that even the perturbed operators \(H_n\) converge in strong resolvent sense.

### Lemma 3.6

For \(H\in \Omega _2\) and \(H_n=P_nH|_{\mathcal {H}_n}\), one has \(H_n\rightarrow H\) and \(H_n^*\rightarrow H^*\) in strong resolvent sense.

### Proof

**The algorithm**The algorithm for \(\Omega _2,\Lambda _2\) is defined analogously to that in Sect. 2. Namely, we define \(G_n^{\mathbb {C}}:=\frac{1}{n}({\mathbb {Z}}+\mathrm {i}{\mathbb {Z}})\cap B_n(0)\subset {\mathbb {C}}.\)

*M*(cf. [9]). Since we have already shown that \(\Gamma _n^{(1)}\) approximates \(\sigma (T)\) correctly and that \(\sigma (T)=\sigma _{e}(T)=\sigma _{e5}(H)\), we know that \(\Gamma _n^{(2)}\) will not miss anything in \(\sigma _{e5}(H)\). Thus, it only remains to prove absence of spectral pollution and spectral inclusion for the discrete set \(\sigma (H){\setminus }\sigma _{e5}(H)\) for the algorithm

However, let us first take a moment to assure that \(\Gamma _n^{(2)}\) defines a reasonable algorithm. Clearly, each \(\Gamma _n^{(2)}\) depends only on the matrix elements \(\big \langle Te_i^{(n)},e_j^{(n)}\big \rangle \) and \(\big \langle Ve_i^{(n)},e_j^{(n)}\big \rangle \), \(1\le i,j\le k_n\). Moreover, by Lemma 2.4 it only requires finitely many algebraic operations on these numbers to determine whether \(\lambda \in G_n^{\mathbb {C}}\) belongs to the set \(\left\{ \lambda \,|\, \min \left\{ s(H_n-\lambda ),s(H_n^*-{\overline{\lambda }})\right\} \le \frac{1}{n} \right\} \). Finally, since \(\Lambda _2\) contains all matrix elements \(\big \langle Te_i^{(n)},e_j^{(n)}\big \rangle \), it follows from the comments made in Sect. 2 that \(\Gamma _n^{(1)}\) is an admissible algorithm as well.

### Remark 3.7

**Spectral pollution**Let us prove that the approximation \(\Gamma _n^{(2)}(H)\) does not have spectral pollution for \(H\in \Omega _2\). To this end, note that again \(\tilde{\Gamma }_n(H)\subset \sigma _\varepsilon (H_n)\) for \(\varepsilon >0\) fixed and

*n*large enough. According to [5, Th. 3.6 ii)], \(\varepsilon \)-pseudospectral pollution of the approximation \(H_n\rightarrow H\) is confined to

### Lemma 3.8

### Proof

Either there exists \(\varepsilon _0>0\) such that \(\lambda \in \sigma _\varepsilon (T)\cup \sigma _e\big ((H_n)_{n\in {\mathbb {N}}}\big )\cup \sigma _e\big ( (H_n^*)_{n\in {\mathbb {N}}} \big )^*\) for all \(\varepsilon \in (0,\varepsilon _0)\), or

there exists a sequence \(\varepsilon _k\) with \(\varepsilon _k\searrow 0\) and \(\lambda \in \bigcup _{\delta \in (0,\varepsilon _k]}\Lambda _{e,\delta }\big ((H_n)_{n\in {\mathbb {N}}}\big )\) for all

*k*.

Next, by [3, Th. 6.1] we have \(\sigma _e\big ( (H_n)_{n\in {\mathbb {N}}} \big )\cup \sigma _e\big ( (H_n^*)_{n\in {\mathbb {N}}} \big )^*\subset W_e(H)\) and hence \(\lambda \in \sigma (H)\cup W_e(H)\). In order to exclude spectral pollution it only remains to prove \(W_e(H)\subset \sigma (H)\).

### Lemma 3.9

For \(H=T+V\in \Omega _2\) one has \(W_e(H)\subset \sigma _e(H)\).

### Proof

Let \(H=T+V\) with *T* selfadjoint, semibounded and \(V\in L(\mathcal {H})\) such that \(V,\,V^*\) are *T*-compact. Then denoting \({\text {Re}}(V):=\frac{1}{2}(V+V^*)\) and \({\text {Im}}(V):=\frac{1}{2i}(V-V^*)\) we have that \(V={\text {Re}}(V)+\mathrm {i}{\text {Im}}(V)\) with \({\text {Re}}(V),\,{\text {Im}}(V)\) relatively compact w.r.t. *T*. Applying [3, Th. 4.5] we conclude that \(W_e(H)=W_e(T)\).

*T*, we can see from [3, Th. 3.8] that

Note that the previous lemma is the only place in which we need the semiboundedness assumption in the definition of \(\Omega _2\). Overall we have shown that for any sequence \(\lambda _n\in {\tilde{\Gamma }}_n(H)\) which converges to some \(\lambda \in {\mathbb {C}}\) we necessarily have \(\lambda \in \sigma (H)\), in other words, spectral pollution does not exist.

**Spectral inclusion** It remains to show that the approximation \((\Gamma _n^{(2)}(H))\) is spectrally inclusive, i.e. that for any \(\lambda \in \sigma (H)\) there exists a sequence \(\lambda _n\in \Gamma _n^{(2)}(H)\) such that \(\lambda _n\rightarrow \lambda \). As explained above, the existence of such a sequence is already guaranteed for all \(\lambda \in \sigma _{e5}(H)\).

### Lemma 3.10

For every \(\lambda \in \sigma (H){\setminus }\sigma _{e5}(H)\) there exists a sequence \(\lambda _n\in {\tilde{\Gamma }}(H)\) with \(\lambda _n\rightarrow \lambda \).

### Proof

First note that by Theorem 3.3 \(\lambda \) is an isolated point. Moreover, we have seen in the proof of Lemma 3.9 that \(\sigma _e\big ( (H_n)_{n\in {\mathbb {N}}} \big )\cup \sigma _e\big ( (H_n^*)_{n\in {\mathbb {N}}} \big )^*\subset \sigma _e(H)\) and hence \(\lambda \) does not belong to this set either. From Lemma 3.6 and [5, Th. 2.3 i)] we conclude that there exists a sequence \(\mu _n\in \sigma (H_n)\) with \(\mu _n\rightarrow \lambda \).

Now, by definition of \(G_n^{\mathbb {C}}\), for each *n* there exists \(\lambda _n\in G_n^{\mathbb {C}}\) such that \(|\mu _n-\lambda _n|<\frac{1}{n}\) and hence \(\Vert (H_n-\lambda _n)^{-1}\Vert _{L(\mathcal {H}_n)}\ge n\) which implies \(\lambda _n\in {\tilde{\Gamma }}_n(H)\). Since \(|\mu _n-\lambda _n|\rightarrow 0\) and \(\mu _n\rightarrow \lambda \), it follows that \(\lambda _n\rightarrow \lambda \). \(\square \)

**Conclusion**Overall we have shown that

- (a\('\))
If \(\lambda _n\in \Gamma _n^{(2)}(H)\) and \(\lambda _n\rightarrow \lambda \), then \(\lambda \in \sigma (H)\).

- (b\('\))
If \(\lambda \in \sigma (H)\), then there exist \(\lambda _n\in \Gamma _n^{(2)}(H)\) with \(\lambda _n\rightarrow \lambda \).

## 4 Application to Schrödinger Operators

*V*have been chosen such that every \(H\in \Omega _3\) even satisfies all conditions formulated in the set \(\Omega _2\) in Theorem 3.1.

*n*:

### Lemma 4.1

We have \(P_{\mathcal {H}_n}\rightarrow I\) strongly in \(L^2({\mathbb {R}}^d)\) and for any \(n\in {\mathbb {N}}\) the set \(\{e_k^{(n)}\}_{k=1}^{\# L_n}\) form an orthonormal basis of \(\mathcal {H}_n\).

### Proof

*j*’th component of the vector \(i_j\) and \(\xi =(\xi _1, \ldots ,\xi _d)\in {\mathbb {R}}^d\). Using this explicit representation, it can be easily seen that we have the following.

### Lemma 4.2

### Proof

*k*, because \(i_k\in L_n\subset B_n(0)\) for all

*k*. \(\square \)

*V*to the number \(e_k^{(n)}(i)\). The meaning of the constants \(\tfrac{n\delta _{mk}}{3}\sum _{j=1}^d\big (\big ((i)_j+\tfrac{1}{n}\big )^3 - (i)_j^3 \big )\) will become clear later on.

Together, \(\Omega _3\) and \(\Lambda _3\) define a computational problem \((\Omega _3,\Lambda _3,\sigma (\cdot ))\). The main result of this section is the following.

### Theorem 4.3

For \(\Omega _3\) and \(\Lambda _3\) defined as above, we have \({\text {SCI}}\big (\Omega _3,\Lambda _3,\sigma (\cdot )\big )=1\).

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 \(\left\langle (-\Delta +V)e_i,e_j \right\rangle \) *by performing only a finite number of algebraic operations on a finite number of values of V*. This will be the main difficulty.

### 4.1 Proof of Theorem 4.3

We first show that the spaces \(\mathcal {H}_n\) defined in (4.2) are indeed a reasonable choice for the problem at hand. More precisely, we have

### Lemma 4.4

The union \(\bigcup _{n\in {\mathbb {N}}}\mathcal {H}_n\) is a core for \(-\Delta \).

### Proof

- (i)
\(\Vert u_n-u\Vert _{L^2({\mathbb {R}}^d)}\rightarrow 0,\)

- (ii)
\(\left\| |\xi |^2(u_n-u)\right\| _{L^2({\mathbb {R}}^d)}\rightarrow 0\)

*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

*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 \(\chi _i:=n^{\frac{d}{2}}\chi _{i+[0,\frac{1}{n})^d}\). On the whole space we have

*n*(see Fig. 1).

*n*. Estimating the second term on the right hand side of Eq. (4.5) is now straightforward. We get

*R*so large that \(\big \Vert (a|\xi |^2+b)u\big \Vert _{L^2({\mathbb {R}}^d{\setminus } B_{R-1})}^2 + \big \Vert |\xi |^2u\big \Vert ^2_{L^2({\mathbb {R}}^d{\setminus } B_R)}<\varepsilon \). From Eq. (4.5) we then see that

Our strategy for proving Theorem 4.3 is as follows. By the assumptions on *V* stated in the definition of \(\Omega _3\) and Lemma 4.4 we know that we have \(\Omega _3\subset \Omega _2\), if we choose \(\mathcal {H}=L^2({\mathbb {R}}^d)\) and \(\mathcal {H}_n\) as in (4.2). Hence, we already know from Theorem 3.1 that \(\Gamma ^{(2)}_n(H)\rightarrow \sigma (H)\) for all \(H\in \Omega _3\). However, \(\Gamma ^{(2)}_n\) uses the matrix elements \(\big \langle H e_k^{(n)}, e_j^{(n)}\big \rangle \), which we are not allowed to access in Theorem 4.3. Therefore, we will define a new algorithm \(\Gamma ^{(3)}_n\) which only accesses the information provided in \(\Lambda _3\) and which satisfies \(\Gamma ^{(3)}_n(H)\approx \Gamma ^{(2)}_n(H)\) for \(H\in \Omega _3\) in an appropriate sense.

**The algorithm**As described above, we need to approximate the matrix elements \(\langle -\Delta e_k^{(n)},e_m^{(n)}\rangle \) and \(\langle Ve_k^{(n)},e_m^{(n)}\rangle \) using only a finite amount of information provided in the set \(\Lambda _3\). We start with the Laplacian, which is the simpler case. Indeed, we have

*V*, we will have to perform an approximation procedure. To this end, let \(l\in {\mathbb {N}}\) and define a lattice \(P_l\subset {\mathbb {R}}^d\) by

### Lemma 4.5

### Proof

*i*,

*x*] denotes a line segment connecting

*i*to \(x\in i+[0,\frac{1}{l})^d\). \(\square \)

### Lemma 4.6

*M*is as in Eq. (4.1).

### Proof

### Corollary 4.7

### Proof

**Convergence**It remains to prove that \(\Gamma _n^{(3)}(H)\rightarrow \sigma (H)\) in the Attouch–Wets metric. To this end, let \(\lambda \in G_n^{\mathbb {C}}\) and note that by the second resolvent identity we have

*l*(

*n*), we conclude that for all \(\lambda \in G_n^{{\mathbb {C}}}\) one has

## Notes

### Acknowledgements

The author would like to thank J. Ben-Artzi, A. Hansen and M. Marletta for helpful and inspiring discussions. Moreover, I would like to thank the anonymous reviewer for their helpful comments and corrections. This work was supported by the Engineering and Physical Sciences Research Council (UK): Grant EP/N020154/1 “QUEST: Quantitative Estimates in Spectral Theory and Their Complexity”.

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