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On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators

  • Frank RöslerEmail author
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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

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 \(-\Delta +V\). 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:

Given 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 \(T\in \Omega \)?

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

Definition 1.1

A computational problem is a quadruple \((\Omega ,\Lambda ,\Xi ,{\mathcal {M}})\), where
  1. (i)

    \(\Omega \) is a set, called the primary set,

     
  2. (ii)

    \(\Lambda \) is a set of complex valued functions on \(\Omega \), called the evaluation set,

     
  3. (iii)

    \({\mathcal {M}}\) is a metric space,

     
  4. (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

Let \((\Omega ,\Lambda ,\Xi ,{\mathcal {M}})\) be a computational problem. An 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
  1. (i)

    the action of \(\Gamma \) on T depends only on \(\{f(T)\}_{f\in \Lambda _\Gamma (T)}\),

     
  2. (ii)

    for every \(S\in \Omega \) with \(f(T)=f(S)\) for all \(f\in \Lambda _\Gamma (T)\) one has \(\Lambda _\Gamma (S)=\Lambda _\Gamma (T)\),

     
  3. (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.

However, it turns out that there exists a family \(\Gamma _{mn}\) of algorithms such that
$$\begin{aligned} \lim _{n\rightarrow \infty }\lim _{m\rightarrow \infty }\Gamma _{mn}(T) = \sigma (T) \end{aligned}$$
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 \((\Omega ,\Lambda ,\Xi ,{\mathcal {M}})\) be a computational problem. A 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 \(T\in \Omega \)
$$\begin{aligned} \Xi (T) = \lim _{n_k\rightarrow \infty }\cdots \lim _{n_1\rightarrow \infty }\Gamma _{n_1,n_2, \ldots ,n_k}(T). \end{aligned}$$

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 this article we are mainly interested in the spectral problem and will therefore write \({\text {SCI}}(\Omega ,\Lambda )\) instead of \({\text {SCI}}(\Omega ,\Lambda ,\Xi ,{\mathcal {M}})\), where it is understood that \(\Xi (T)=\sigma (T)\) and \({\mathcal {M}}\) is the set of closed subsets of \({\mathbb {C}}\) equipped with the Attouch–Wets metric \(d_{\text {AW}}\) defined as
$$\begin{aligned} d_{\text {AW}}(A,B) = \sum _{i=1}^\infty 2^{-i}\min \left\{ 1 ,\,\sup _{|x|<i}\left| {\text {dist}}(x,A) - {\text {dist}}(x,B) \right| \right\} . \end{aligned}$$
(Note that if \(A,B\subset {\mathbb {C}}\) are bounded, then \(d_{\text {AW}}\) coincides with the Hausdorff distance.)

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
$$\begin{aligned} {\text {SCI}}(\Omega ,\sigma (\cdot ))&= 3\quad \text { if } \Omega = L(\mathcal {H}) \\ {\text {SCI}}(\Omega ,\sigma (\cdot ))&= 2\quad \text { if } \Omega = \{T\in L(\mathcal {H})\,|\,T\text { selfadjoint}\} \\ {\text {SCI}}(\Omega ,\sigma (\cdot ))&= 1\quad \text { if } \Omega = {\mathcal {K}}(\mathcal {H}), \end{aligned}$$
where \({\mathcal {K}}(\mathcal {H})\) denotes the set of compact operators. The last of the above bounds, \({\text {SCI}}({\mathcal {K}}(\mathcal {H}),\sigma (\cdot )) = 1\), 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
$$\begin{aligned} \Omega = \left\{ -\Delta +V\,|\,V \text { is sectorial and }|V(x)|\rightarrow \infty \text { as }|x|\rightarrow \infty \right\} , \end{aligned}$$
(1.1)
then \({\text {SCI}}(\Omega ,\sigma (\cdot ))=1\). 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 \(\Omega \) 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 2.13.1 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

Let \(\mathcal {H}\) be a separable Hilbert space and \(\mathcal {H}_n\subset \mathcal {H}\) be a sequence of finite dimensional subspaces such that \(\mathcal {H}_n\subset \mathcal {H}_{n+1}\) for all \(n\in {\mathbb {N}}\) and \(P_n\xrightarrow {s}I\), where \(P_n\) denotes the orthogonal projection onto \(\mathcal {H}_n\). Definewhere
$$\begin{aligned} {\widehat{\sigma }}_e(T)=\sigma _{e2}(T)\cup {\left\{ \begin{array}{ll} \{+\infty \}, &{}\text {if }T\text { unbounded above}\\ \{-\infty \}, &{}\text {if }T\text { unbounded below} \end{array}\right. } \end{aligned}$$
(2.2)
and
$$\begin{aligned} \sigma _{e2}(T)=\{\lambda \in {\mathbb {C}}\,|\, \exists (x_k)\subset {\text {dom}}(T) : \Vert x_k\Vert =1\,\forall k,\, x_k\rightharpoonup 0,\, \Vert (T-\lambda )x_k\Vert \rightarrow 0\}.\nonumber \\ \end{aligned}$$
(2.3)
Furthermore, for each \(n\in {\mathbb {N}}\), let \(\{e_1^{(n)}, \ldots ,e_{k_n}^{(n)}\}\) be an orthonormal basis of \(\mathcal {H}_n\) and define
$$\begin{aligned} \Lambda _1 := \big \{ f_{i,j,n} \,|\, 1\le i,j\le k_n,\,n\in {\mathbb {N}}\big \}, \end{aligned}$$
(2.4)
where \(f_{i,j,n}:T\mapsto \big \langle Te_i^{(n)},e_j^{(n)}\big \rangle \) are the evaluation functions producing the (ij)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

  1. (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 \(\sigma _e(T)\subset \{0\}\)).

     
  2. (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

In addition to (2.3) we will need another version of the essential spectrum, which is sometimes denoted \(\sigma _{e5}\). Let H be a closed, densely defined operator on \(\mathcal {H}\). Then
$$\begin{aligned} \sigma _{e5}(H):={\mathbb {C}}{\setminus }\Delta _5(H), \end{aligned}$$
(2.5)
where \(\Delta _5(H)\) denotes the union of all components of the set \(\{\lambda \in {\mathbb {C}}\,|\,H-\lambda \text { is semi-Fredholm}\}\) which intersect \(\rho (H)\). Note that the definitions (2.3) and (2.5) do 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

Let \(T\in \Omega _1\) and define the truncated operator
$$\begin{aligned} T_n:=P_nT|_{\mathcal {H}_n}. \end{aligned}$$
(2.6)
This operator can be represented by a finite dimensional (square) matrix with elements \((T_n)_{ij}=\big \langle Te_i^{(n)},e_j^{(n)}\big \rangle \). Moreover, let \(G_n^{\mathbb {R}}:=\frac{1}{n}{\mathbb {Z}}\cap (-n,n)\subset {\mathbb {R}}.\)

Lemma 2.4

Let \(\lambda \in G_n^{\mathbb {R}}\) and denote by \(s(\cdot )\) the smallest singular value of a matrix. Then
  1. (i)

    For all n and \(\lambda \), we have \(s(T_n-\lambda )=\Vert (T_n-\lambda )^{-1}\Vert ^{-1}_{L(\mathcal {H}_n)}\).

     
  2. (ii)

    For any \(q>0\), testing whether \(s(T_n-\lambda )>q\) requires only finitely many arithmetic operations on the matrix elements of \(T_n\).

     
with the convention that \(\Vert (T_n-\lambda )^{-1}\Vert ^{-1}=0\) for \(\lambda \in \sigma (T_n)\).

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. \(\square \)

For \(n\in {\mathbb {N}}\) we define a map \(\Gamma _n^{(1)}:\Omega _1\rightarrow \{\text {closed subsets of }{\mathbb {C}}\}\) by
$$\begin{aligned} \Gamma _n^{(1)}(T):=\left\{ \lambda \in G_n^{\mathbb {R}}\,\Big |\,s(T_n-\lambda )\le \frac{1}{n}\right\} . \end{aligned}$$
Then, by the above lemma, each \(\Gamma _n^{(1)}\) is an arithmetic tower of height one in the sense of Definition 1.3. Clearly, \(\Gamma _n^{(1)}(T)\subset \sigma _{\frac{1}{n}}(T_n)\) for all 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

We start by showing that each \(T_n\) is selfadjoint. First note that each \(T_n\) is automatically bounded, since the \(\mathcal {H}_n\) are finite dimensional. Now let \(x,y\in \mathcal {H}_n\). Then we have
$$\begin{aligned} \begin{aligned} \langle T_n x,y\rangle&= \langle P_n T x,y\rangle = \langle Tx,P_n y\rangle = \langle Tx , y\rangle \\&= \langle x , Ty\rangle = \langle P_nx,Ty\rangle = \langle x,P_nTy\rangle = \langle x,T_ny\rangle . \end{aligned} \end{aligned}$$
(2.7)
and hence \(T_n\) is selfadjoint. The claim now follows directly from [13, Satz 9.29], since \(\bigcup _n\mathcal {H}_n\) is a core for T and \(P_n\) converges strongly to the identity. \(\square \)

2.2 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 \(\mathcal {H}\), which are not necessarily selfadjoint. These results will be used in later sections. The sets of spectral and pseudospectral pollution are defined, respectively, as
$$\begin{aligned} \sigma _{\text {poll}}(H)&= \{z\in {\mathbb {C}}{\setminus }\sigma (H)\,|\,\exists \text { sequence }\lambda _n\in \sigma (H_n):\lambda _n\rightarrow \lambda \}\\ \sigma _{\varepsilon ,\text {poll}}(H)&= \{z\in \mathbb C{\setminus }\sigma _\varepsilon (H)\,|\,\exists \text { sequence }\lambda _n\in \sigma _\varepsilon (H_n):\lambda _n\rightarrow \lambda \}. \end{aligned}$$
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 \(\varepsilon \)-near spectrum:the essential numerical range
$$\begin{aligned} W_e(H)&:= \{\lambda \in {\mathbb {C}}\,|\, \exists x_k\in {\text {dom}}(H) \text { s.t. } \Vert x_k\Vert =1\,\forall k,\, x_k\rightharpoonup 0,\, \langle Hx_k,x_k\rangle \rightarrow \lambda \} \end{aligned}$$
and the limiting essential numerical rangeThe 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

For all operators H on \(\mathcal {H}\) of the form \(H=T+V\), where T is selfadjoint and V is bounded, and all \(\varepsilon <\varepsilon '\) one has
$$\begin{aligned} \overline{\sigma _\varepsilon (H)}\subset \sigma _{\varepsilon '}(H). \end{aligned}$$

Proof

This follows from the fact that the resolvent norm of any such operator tends to 0 at \(\mathrm {i}\infty \) and hence cannot be constant on an open set (cf. [6, Th. 3.2]). Indeed, in this case we have\(\square \)

Lemma 2.7

  1. (i)
    For any closed, densely defined operator H on \(\mathcal {H}\) one has
    $$\begin{aligned} \bigcap _{\varepsilon >0} \bigcup _{\delta \in (0,\varepsilon ]} \Lambda _{e,\delta }\big ((H_n)_{n\in {\mathbb {N}}}\big )\;\subset \; \sigma _e\big ((H_n)_{n\in {\mathbb {N}}}\big ). \end{aligned}$$
     
  2. (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

We first prove (i). Let \(\lambda \in \bigcap _{\varepsilon >0}\bigcup _{\delta \in (0,\varepsilon ]} \Lambda _{e,\delta }\big ((H_n)_{n\in {\mathbb {N}}}\big )\). Then for all \(\varepsilon >0\) there exists \(\delta \in (0,\varepsilon ]\) and a sequence \((x_k)\) with \(x_k\in {\text {dom}}(H_{n_k})\) (for some subsequence \((n_k)\)) such that
  • \(\Vert x_k\Vert =1\) for all k

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

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

Hence, for every \(m\in {\mathbb {N}}\) there exists a sequence \((x_k^{(m)})_{k\in {\mathbb {N}}}\) with \(\bigl \Vert x_k^{(m)}\bigr \Vert =1\), Open image in new window and
$$\begin{aligned} \lim _{k\rightarrow \infty } \left\| (H_{n_k(m)}-\lambda )x_k^{(m)}\right\| <\frac{1}{m}. \end{aligned}$$
The notation \(n_k(m)\) indicates that the corresponding subsequence of \((H_n)\) depends on 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
$$\begin{aligned} d\big (x_{k_m}^{(m)},0\big )&<\frac{1}{m}\\ \left\| (H_{n_{k_m}(m)}-\lambda )x_{k_m}^{(m)}\right\|&<\frac{1}{m}. \end{aligned}$$
Then for the sequence \(y_m:=x^{(m)}_{k_m}\), one has \(\Vert y_m\Vert =1\) for all 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 \(\varepsilon \). \(\square \)

Finally, we prove the following characterisation of convergence of sets in the Attouch–Wets metric. We recall that \(d_{\mathrm {AW}}(X_n,X)\rightarrow 0\) if and only if \(d_K(X_n,X)\rightarrow 0\) for all \(K\subset {\mathbb {C}}\) compact, where
$$\begin{aligned} d_K(X,Y) := \max \left\{ \sup _{x\in X\cap K}{\text {dist}}(x,Y) ,\,\sup _{y\in Y\cap K}{\text {dist}}(y,X) \right\} . \end{aligned}$$

Proposition 2.8

Let \(X,X_n,\;n\in {\mathbb {N}}\) be closed subsets of \({\mathbb {C}}\). Assume that
  1. (a)

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

     
  2. (b)

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

     
Then one has \(d_{\mathrm {AW}}(X_n,X)\rightarrow 0\).

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 \(n_0 = \max \{n_i\,|\,i=1, \ldots ,k\}\).

To show that \(\sup _{z\in X_n\cap K}{\text {dist}}(z,X)\rightarrow 0\) as \(n\rightarrow \infty \), note that since all sets \(X_n\cap K\) are compact, we can choose a sequence \(z_n\in X_n\cap K\) such that
$$\begin{aligned} \sup _{z\in X_n\cap K}{\text {dist}}(z,X) = {\text {dist}}(z_n,X). \end{aligned}$$
Since the sequence \((z_n)\) is obviously bounded, we can extract a convergent subsequence \(z_{n_j}\rightarrow z_0\in K\). Now use assertion (a) from above to conclude that in fact \(z_0\in X\cap K\). This readily implies
$$\begin{aligned} \sup _{z\in X_{n_j}\cap K}{\text {dist}}(z,X)&= {\text {dist}}(z_{n_j},X)\rightarrow 0. \end{aligned}$$
Since the same reasoning can be applied to every subsequence of the sequence
$$\begin{aligned} \left( \sup _{z\in X_n\cap K}{\text {dist}}(z,X)\right) _{n\in {\mathbb {N}}}, \end{aligned}$$
we conclude that the whole sequence converges to zero. \(\square \)

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

If \(\lambda _n\in \Gamma _n^{(1)}(T),\;n\in {\mathbb {N}}\) and \(\lambda _n\rightarrow \lambda \), then
$$\begin{aligned} \lambda \in \sigma (T)\cup \sigma _e\big ((T_n)_{n\in {\mathbb {N}}}\big ). \end{aligned}$$

Proof

By definition of \(\Gamma _n^{(1)}\), one has that
$$\begin{aligned} \frac{1}{n} \ge \Vert (\lambda _n-T_n)^{-1}\Vert ^{-1} \ge {\text {dist}}(\lambda _n,\sigma (T_n)) \end{aligned}$$
for all \(n\in {\mathbb {N}}\). Hence, there exists a sequence \(z_n\in \sigma (T_n)\) such that \(|z_n-\lambda _n|\rightarrow 0\) and consequently \(z_n\rightarrow \lambda \). We conclude from [5, Th. 2.3] that \(\lambda \in \sigma (T)\cup \sigma _e\big ((T_n)_{n\in {\mathbb {N}}}\big )\). \(\square \)
To conclude, we apply [3, Th. 6.1] to show that spectral pollution is in fact absent for \(T\in \Omega _1\). Indeed, let \(\lambda _n\in \Gamma _n^{(1)}(T)\) with \(\lambda _n\rightarrow \lambda \). Then by Lemma 2.9 and [3, Prop. 5.6, Th. 6.1], we get
$$\begin{aligned} \lambda&\in \sigma (T)\cup \sigma _e\big ((T_n)_{n\in {\mathbb {N}}}\big )\\&\subset \sigma (T)\cup W_e(T)\\&= \sigma (T)\cup {\text {conv}}({\widehat{\sigma }}_e(T)){\setminus }\{\pm \infty \}\\&= \sigma (T)\cup \sigma _e(T)\\&= \sigma (T). \end{aligned}$$
It remains to prove spectral inclusion, i.e. nothing is missed by \(\Gamma _n^{(1)}(T)\).

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
  1. (a)

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

     
  2. (b)

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

     
By Proposition 2.8, this implies Attouch–Wets convergence.

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

Define a computational problem byFor every \(H\in \Omega _2\), choose a decomposition \(H=T+V\) as in the definition of \(\Omega _2\) and define the maps \(s_T(H):=T\) and \(s_V(H):=V\). Then let
$$\begin{aligned} \Lambda _2&:= \left\{ f_{i,j,n}\circ s_T\,|\,1\le i,j\le n,\;n\in {\mathbb {N}}\right\} \cup \left\{ f_{i,j,n}\circ s_V\,|\,1\le i,j\le n,\;n\in {\mathbb {N}}\right\} , \end{aligned}$$
where \(f_{i,j,n}\) are the evaluation functions producing the (ij)th matrix elements (see (2.4)). Then one has \({\text {SCI}}(\Omega _2,\Lambda _2,\sigma (\cdot ))=1\).

Remark 3.2

  1. (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.

     
  2. (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

[11, XIII.4, Cor. 2] Let T be a selfadjoint operator on \(\mathcal {H}\) and V relatively compact w.r.t. T. Then
  1. (i)

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

     
  2. (ii)

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

     
From Theorems 3.3 and 3.4 we immediately conclude that for all \(H\in \Omega _2\) the spectrum of \(\mathcal {H}\) is of the form
$$\begin{aligned} \sigma (H) = \sigma (T)\cup \{\lambda _1, \lambda _2, \ldots \}, \end{aligned}$$
with isolated eigenvalues \(\lambda _i\in {\mathbb {C}}\).

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

For \(V_n\) defined as above, we have the following
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     

Proof

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

To see assertion (ii), let \(u\in \mathcal {H}\) and note that then \(P_nu\rightarrow u\) strongly. By continuity of V, it immediately follows that \(VP_nu\rightarrow Vu\) in \(\mathcal {H}\). Hence, from the definition of \(V_n\) we conclude that
$$\begin{aligned} V_nP_nu&= P_nV|_{\mathcal {H}_n}P_nu = \underbrace{P_n}_{\rightarrow I\text { strongly}}\underbrace{VP_nu}_{\rightarrow Vu} \,\rightarrow \, Vu. \end{aligned}$$
Assertion (iii) now immediately follows by combining (i) and (ii). \(\square \)

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

This follows from [4, Cor. 3.5], since
$$\begin{aligned} \Vert (T-z)^{-1}\Vert ^{-1},\,\Vert (T_n-z)^{-1}\Vert ^{-1} \ge {\text {dist}}(z,{\mathbb {R}}), \end{aligned}$$
which tends to \(\infty \) as \(z\rightarrow \mathrm {i}\infty \), and \(\Vert V\Vert ,\Vert V_n\Vert \) are uniformly bounded. \(\square \)
The algorithm The algorithm for \(\Omega _2,\Lambda _2\) is defined analogously to that in Sect. 2. Namely, we define \(G_n^{\mathbb {C}}:=\frac{1}{n}({\mathbb {Z}}+\mathrm {i}{\mathbb {Z}})\cap B_n(0)\subset {\mathbb {C}}.\)
$$\begin{aligned} \Gamma _n^{(2)}(H):=\left\{ \lambda \in G_n^{\mathbb {C}}\,\bigg |\, \min \left\{ s(H_n{-}\lambda ),s(H_n^*{-}{\overline{\lambda }})\right\} \le \frac{1}{n} \right\} \,\cup \,\Gamma _n^{(1)}(T). \end{aligned}$$
(3.1)
Note that we have \(\min \{s(M-\lambda ),s(M^*-{\overline{\lambda }})\}=\Vert (M-\lambda )^{-1}\Vert ^{-1}\) for any \(n\times n\) matrix 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
$$\begin{aligned} {{\tilde{\Gamma }}}_n(H):=\left\{ \lambda \in G_n^{\mathbb {C}}\,\bigg |\, \min \left\{ s(H_n-\lambda ),s(H_n^*-{\overline{\lambda }})\right\} \le \frac{1}{n} \right\} \end{aligned}$$
This will be done in the remainder of this section.

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

We note that the choice \(\frac{1}{n}\) as an upper bound for \(s(H_n-\lambda )\) in (3.1) is arbitrary. The proof below will show that one could equally well have chosen
$$\begin{aligned} \Xi _n(H)&:=\left\{ \lambda \in G_n^{\mathbb {C}}\,\bigg |\, \min \left\{ s(H_n-\lambda ),s(H_n^*-{\overline{\lambda }})\right\} \le \frac{3}{n} \right\} \cup \Gamma _n^{(1)}(T) \end{aligned}$$
instead of \(\Gamma ^{(2)}_n(H)\). This fact will be used in Sect. 4.
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
$$\begin{aligned} \sigma _e\big ( (H_n)_{n\in {\mathbb {N}}} \big )\cup \sigma _e\big ( (H_n^*)_{n\in {\mathbb {N}}} \big )^*\cup \bigcup _{\delta \in (0,\varepsilon ]}\Lambda _{e,\delta }\big ( (H_n)_{n\in {\mathbb {N}}} \big ). \end{aligned}$$
Hence, for any sequence \(\lambda _n\in {{\tilde{\Gamma }}}_n(H)\) with \(\lambda _n\rightarrow \lambda \in {\mathbb {C}}\) we have
$$\begin{aligned} \lambda \in \bigcap _{\varepsilon >0}\left( \sigma _\varepsilon (H)\cup \sigma _e(H_n)_{n\in {\mathbb {N}}}\cup \sigma _e\big ( (H_n^*)_{n\in {\mathbb {N}}} \big )^*\cup \bigcup _{\delta \in (0,\varepsilon ]}\Lambda _{e,\delta }\big ( (H_n)_{n\in {\mathbb {N}}} \big )\right) . \nonumber \\ \end{aligned}$$
(3.2)
We conclude with the following

Lemma 3.8

It follows from (3.2) that
$$\begin{aligned} \lambda \in \sigma (H)\cup \sigma _e\big ((H_n)_{n\in {\mathbb {N}}}\big )\cup \sigma _e\big ( (H_n^*)_{n\in {\mathbb {N}}} \big )^*. \end{aligned}$$

Proof

Let (3.2) hold. Then
  • 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.

In the first case, it follows that
$$\begin{aligned} \lambda&\in \bigcap _{\varepsilon>0}\Big (\sigma _\varepsilon (H)\cup \sigma _e\big ((H_n)_{n\in {\mathbb {N}}}\big )\cup \sigma _e\big ( (H_n^*)_{n\in {\mathbb {N}}} \big )^*\Big ) \\&= \left( \bigcap _{\varepsilon >0}\sigma _\varepsilon (H)\right) \cup \sigma _e\big ((H_n)_{n\in {\mathbb {N}}}\big )\cup \sigma _e\big ( (H_n^*)_{n\in {\mathbb {N}}} \big )^* \\&= \sigma (H)\cup \sigma _e\big ((H_n)_{n\in {\mathbb {N}}}\big )\cup \sigma _e\big ( (H_n^*)_{n\in {\mathbb {N}}} \big )^*. \end{aligned}$$
In the second case, we have
$$\begin{aligned} \lambda&\in \bigcap _{k\in {\mathbb {N}}}\bigcup _{\delta \in (0,\varepsilon _k]}\Lambda _{e,\delta }\big ((H_n)_{n\in {\mathbb {N}}}\big )\\&\subset \sigma _e\big ((H_n)_{n\in {\mathbb {N}}}\big ), \end{aligned}$$
by Lemma 2.7 (ii). \(\square \)

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 \(W_e(H)=W_e(T)\).

But now by our assumptions on T, we can see from [3, Th. 3.8] that
$$\begin{aligned} W_e(T)={\text {conv}}\big ({\widehat{\sigma }}_e(T)\big ){\setminus }\{\pm \infty \} = \sigma _e(T) = \sigma _e(H). \end{aligned}$$
\(\square \)

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 \(\lambda _n\rightarrow \lambda \).

By Proposition 2.8 this implies \(d_{\mathrm {AW}}\big (\Gamma _n^{(2)}(H),\sigma (H)\big )\rightarrow 0\).

4 Application to Schrödinger Operators

In this section we will apply the results of Sects. 2 and 3 to Schrödinger operators on \(L^2({\mathbb {R}}^d)\). More specifically, fix a continuous, monotone decreasing function \(g:[0,\infty )\rightarrow [0,\infty )\) with \(g(t)\rightarrow 0\) as \(t\rightarrow \infty \) and let \(M>0\). We define
$$\begin{aligned} \Omega _3&:= \left\{ -\Delta +V\,\big |\,V\in {\mathcal {C}}^1({\mathbb {R}},{\mathbb {C}}),\, \Vert \nabla V\Vert _{\infty }\le M,\, |V(x)|\le g(|x|)\right\} .\qquad \quad \end{aligned}$$
(4.1)
By the above definition, every \(H\in \Omega _3\) is a relatively compact perturbation of the free Laplacian with domain \(H^2({\mathbb {R}}^d)\) (cf. e.g. [10, Ch. V, Lemma 5.8]). In fact, our assumptions on 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.
In order to define the computational problem, we choose a finite lattice in \({\mathbb {R}}^d\)
$$\begin{aligned} L_n := \left\{ \frac{i}{n} \;\bigg |\;i\in {\mathbb {Z}}^d, |i|<n \right\} . \end{aligned}$$
Moreover, let \(\mathcal {H}_n\) denote the subspace of \(L^2({\mathbb {R}}^d)\) spanned by all characteristic functions of cubes of edge length \(\frac{1}{n}\) with centres inside a ball of radius n:
$$\begin{aligned} {\widehat{\mathcal {H}}}_n:={\text {span}}\left\{ \chi _{i+[0,\frac{1}{n})^d}\,\Big |\,i\in L_n \right\} \end{aligned}$$
It is easily seen by smooth approximation that \(P_{{\widehat{\mathcal {H}}}_n}\rightarrow I\) strongly in \(L^2({\mathbb {R}}^d)\). However, none of the basis functions \(\chi _{i+[0,\frac{1}{n})^d}\) are contained in the domain of \(-\Delta \). In order to circumvent this, the space we will actually work with will be
$$\begin{aligned} \mathcal {H}_n&:={\text {span}}\left\{ {\widehat{\chi }}_{i+[0,\frac{1}{n})^d}\,\Big |\, i\in L_n \right\} , \end{aligned}$$
(4.2)
where the hat denotes the Fourier transform in \(L^2({\mathbb {R}}^d)\). For any enumeration \(i_k\) of the set \(L_n\), we define
$$\begin{aligned} e_k^{(n)}:=n^{\frac{d}{2}}\cdot {\widehat{\chi }}_{i_k+[0,\frac{1}{n})^d}, \end{aligned}$$
where the normalisation constant \(n^{\frac{d}{2}}\) is chosen such that \(\big \Vert e_k^{(n)}\big \Vert _{L^2({\mathbb {R}}^d)}=1\) for all \(n\in {\mathbb {N}}\). These are smooth functions in \(L^2({\mathbb {R}}^d)\) and it is easily checked that their first and second derivatives are again in \(L^2({\mathbb {R}}^d)\).

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

This follows immediately from the unitarity of the Fourier transform and the equality
$$\begin{aligned}&\bigg \Vert \!\sum _k\left\langle f, n^{\frac{d}{2}}\chi _{i_k+[0,\frac{1}{n})^d}\right\rangle \! n^{\frac{d}{2}}\chi _{i_k+[0,\frac{1}{n})^d} - f\bigg \Vert _{L^2({\mathbb {R}}^d)} \!\!\!\\&\quad = \bigg \Vert \!\sum _k\left\langle {\hat{f}}, e_k^{(n)}\right\rangle \! e_k^{(n)} \!-\! {\hat{f}}\bigg \Vert _{L^2({\mathbb {R}}^d)} \end{aligned}$$
\(\square \)
We note that the functions \(e_k^{(n)}\) can be calculated explicitly. Indeed, one has
$$\begin{aligned} e_k^{(n)}(\xi ) = \left( \frac{n}{2\pi }\right) ^{\frac{d}{2}}\prod _{j=1}^d\frac{e^{\text {i}\xi _j((i_k)_j+\frac{1}{n})} - e^{\text {i}\xi _j (i_k)_j}}{\xi _j}, \end{aligned}$$
where \((i_k)_j\) denotes the 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

For each \(n\in {\mathbb {N}}\) one has
$$\begin{aligned} \big \Vert e_k^{(n)}\big \Vert _\infty , \,\big \Vert \nabla e_k^{(n)}\big \Vert _\infty \le (2\pi )^{-\frac{d}{2}}dn^{3-\frac{d}{2}} \end{aligned}$$
for all \(k\in \{1, \ldots ,n\}\).

Proof

From the definition of \(e_k^{(n)}\) it follows by direct calculation that
$$\begin{aligned} \big \Vert e_k^{(n)}\big \Vert _\infty&< (2\pi n)^{-\frac{d}{2}},\\ \big \Vert \partial _j e_k^{(n)}\big \Vert _\infty&< (2\pi )^{-\frac{d}{2}}\frac{n^{-\frac{d}{2}+1}}{2}\left( \left( (i_k)_j+\tfrac{1}{n}\right) ^2 - (i_k)_j^2 \right) \end{aligned}$$
from which the assertion follows. Note that the bound in the second equation can be made independent of k, because \(i_k\in L_n\subset B_n(0)\) for all k. \(\square \)
The information accessible to the algorithm will be the set
$$\begin{aligned} \Lambda _3 := \Lambda _3^{(1)}\cup \Lambda _3^{(2)}\cup \Lambda _3^{(3)}\cup \Lambda _3^{(4)}, \end{aligned}$$
(4.3)
with
$$\begin{aligned} \Lambda _3^{(1)}&= \left\{ \rho _x \,|\,x\in {\mathbb {R}}^d\right\} \\ \Lambda _3^{(2)}&= \left\{ e_k^{(n)}(i)\,\Big |\,i\in l^{-1}{\mathbb {Z}}^d,\, l\in {\mathbb {N}},\;k\in \{1, \ldots ,n\},\;n\in {\mathbb {N}}\right\} \\ \Lambda _3^{(3)}&= \bigg \{\!\tfrac{n\delta _{mk}}{3}\!\sum _{j=1}^d\left( \left( (i)_j+\tfrac{1}{n}\right) ^3 - (i)_j^3 \right) \Big |\, i\in L_n,\, m,k\in \{1, \ldots ,\# L_n\} ,\, n\in {\mathbb {N}}\bigg \} \\ \Lambda _3^{(4)}&= \left\{ g\big (l^{\frac{1}{2d}}\big )\,\Big |\,l\in {\mathbb {N}} \right\} \end{aligned}$$
where \(\rho _x(V)=V(x)\) are the evaluation functionals and \(e_k^{(n)}(i)\) denote constant functions that map 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

By means of the Fourier transform the assertion is equivalent to the space \(\bigcup _{n\in {\mathbb {N}}}{\widehat{\mathcal {H}}}_n\) being a core for the multiplication operator \(u\mapsto |\xi |^2 u\) in \(L^2({\mathbb {R}}^d)\). To verify this, we have to show that for every \(u\in {\text {dom}}(|\xi ^2|)\) there exists a sequence \(u_n\in \mathcal {H}_n\) such that
  1. (i)

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

     
  2. (ii)

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

     
Point (i) is easily shown by choosing
$$\begin{aligned} u_n:=\sum _{i\in L_n}\left\langle u , n^{\frac{d}{2}}\chi _{i+[0,\frac{1}{n})^d}\right\rangle n^{\frac{d}{2}}\chi _{i+[0,\frac{1}{n})^d}. \end{aligned}$$
(4.4)
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
$$\begin{aligned} \left\| |\xi |^2(u_n-u)\right\| _{L^2({\mathbb {R}}^d)}^2&= \int _{B_R} \left| |\xi |^2(u_n-u)\right| ^2\,d\xi + \int _{{\mathbb {R}}^d{\setminus } B_R} \left| |\xi |^2(u_n-u)\right| ^2\,d\xi , \nonumber \\ \end{aligned}$$
(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 \(\chi _i:=n^{\frac{d}{2}}\chi _{i+[0,\frac{1}{n})^d}\). On the whole space we have
$$\begin{aligned} \left\| |\xi |^2u_n \right\| _{L^2({\mathbb {R}}^d{\setminus } B_R)}^2&= \left\| |\xi |^2\sum _{i\in L_n}\langle u,\chi _i\rangle \chi _i \right\| _{L^2({\mathbb {R}}^d{\setminus } B_R)}^2\\&\le \sum _{i\in L_n{\setminus } B_R} |\langle u,\chi _i\rangle |^2 \left\| |\xi |^2\chi _i\right\| _{L^2({\mathbb {R}}^d)}^2\\&\le \sum _{i\in L_n{\setminus } B_R}\Vert u\Vert _{L^2(i+[0,\frac{1}{n})^d)}^2\left\| n^{\frac{d}{2}}|\xi |^2\right\| _{L^2(i+[0,\frac{1}{n})^d)}^2, \end{aligned}$$
where we have used the fact that \({\text {supp}}(\chi _i)\cap {\text {supp}}(\chi _j)=\emptyset \) for \(i\ne j\). The factor \(\big \Vert n^{\frac{d}{2}}|\xi |^2\big \Vert _{L^2(i+[0,\frac{1}{n})^d)}\) on the right hand side is clearly bounded by \(\sup _{\xi \in i+[0,\frac{1}{n})^d}|\xi |^2\). Thus, if we define a function \(F_n\) by
$$\begin{aligned} F_n(\xi ):=\sum _{i\in \frac{1}{n}{\mathbb {Z}}^d}\left( \sup _{\eta \in i+[0,\frac{1}{n})^d}|\eta |^2\right) \chi _{i}, \end{aligned}$$
then we will have (note that \(F_n\) is constant on each of the cubes \(i+[0,\frac{1}{n})^d\))
$$\begin{aligned} \left\| |\xi |^2u_n \right\| _{L^2({\mathbb {R}}^d{\setminus } B_R)}^2&\le \sum _{i\in L_n{\setminus } B_R}\Vert u\Vert _{L^2(i+[0,\frac{1}{n})^d)}^2 F_n(\xi )^2\\&= \sum _{i\in L_n{\setminus } B_R}\left\| F_n(\xi )u\right\| _{L^2(i+[0,\frac{1}{n})^d)}^2\\&\le \left\| F_n(\xi )u\right\| _{L^2\big ({\mathbb {R}}^d{\setminus } B_{R-\frac{\sqrt{d}}{n}}\big )}^2 \end{aligned}$$
Next, we note that it is easy to see that there exist constants \(a,b>0\) such that \(F_n(\xi )\le a|\xi |^2+b\) uniformly in n (see Fig. 1).
Fig. 1

Sketch of function \(F_n\)

Overall we conclude that
$$\begin{aligned} \left\| |\xi |^2u_n \right\| _{L^2({\mathbb {R}}^d{\setminus } B_R)}^2&\le \left\| F_n(\xi )u\right\| _{L^2\big ({\mathbb {R}}^d{\setminus } B_{R-\frac{\sqrt{d}}{n}}\big )}^2\\&\le \left\| (a|\xi |^2+b)u\right\| _{L^2({\mathbb {R}}^d{\setminus } B_{R-1})}^2, \end{aligned}$$
where the last term on the right hand side is finite because by assumption \(u\in {\text {dom}}(|\xi |^2)\). In fact, from this last inequality we can see immediately that
$$\begin{aligned} \left\| |\xi |^2u_n \right\| _{L^2({\mathbb {R}}^d{\setminus } B_R)}^2\rightarrow 0 \end{aligned}$$
as \(R\rightarrow \infty \) uniformly in n. Estimating the second term on the right hand side of Eq. (4.5) is now straightforward. We get
$$\begin{aligned} \int _{{\mathbb {R}}^d{\setminus } B_R} \left| |\xi |^2(u_n-u)\right| ^2\,d\xi&\le \left\| |\xi |^2u_n\right\| ^2_{L^2({\mathbb {R}}^d{\setminus } B_R)} + \left\| |\xi |^2u\right\| ^2_{L^2({\mathbb {R}}^d{\setminus } B_R)}\\&\le \left\| (a|\xi |^2+b)u\right\| _{L^2({\mathbb {R}}^d{\setminus } B_{R-1})}^2 + \left\| |\xi |^2u\right\| ^2_{L^2({\mathbb {R}}^d{\setminus } B_R)}. \end{aligned}$$
Now let \(\varepsilon >0\) and choose 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
$$\begin{aligned} \limsup _{n\rightarrow \infty }\left\| |\xi |^2(u_n-u)\right\| _{L^2({\mathbb {R}}^d)}^2&\le \limsup _{n\rightarrow \infty }\int _{B_R} \left| |\xi |^2(u_n-u)\right| ^2\,d\xi +\varepsilon \\&\le \limsup _{n\rightarrow \infty }R^2\int _{B_R} \left| u_n-u\right| ^2\,d\xi +\varepsilon \\&= \varepsilon , \end{aligned}$$
because \(u_n\rightarrow u\) in \(L^2({\mathbb {R}}^d)\). Since \(\varepsilon \) was arbitrary, it follows that
$$\begin{aligned} \limsup _{n\rightarrow \infty }\left\| |\xi |^2(u_n-u)\right\| _{L^2({\mathbb {R}}^d)}^2=0, \end{aligned}$$
\(\square \)

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
$$\begin{aligned} \left\langle -\Delta e_k^{(n)},e_m^{(n)}\right\rangle&= \left\langle |\xi |^2n^{\frac{d}{2}}\chi _{i_k+[0,\frac{1}{n})^d} ,\, n^{\frac{d}{2}}\chi _{i_m+[0,\frac{1}{n})^d}\right\rangle \\&= n^d\delta _{mk}\int _{i_k+[0,\frac{1}{n})^d} |\xi |^2\,d\xi \\&= \frac{n\delta _{mk}}{3}\sum _{j=1}^d\left( \left( (i_k)_j+\frac{1}{n}\right) ^3 - (i_k)_j^3 \right) . \end{aligned}$$
Note that these are precisely the terms in the third factor in Eq. (4.3).
Next, we will compute the matrix elements \(\langle Ve_k^{(n)},e_m^{(n)}\rangle \). Since any algorithm can only use finitely many values of 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
$$\begin{aligned} P_l:= \frac{1}{l}{\mathbb {Z}}^d\cap Q_l, \end{aligned}$$
where \(Q_l\) denotes the cube of edge length \(l^{\frac{1}{2d}}\) centered at 0. Next, let
$$\begin{aligned} V_l(x) := \sum _{i\in P_l} V(i)\chi _{i+[0,\frac{1}{l})^d}. \end{aligned}$$

Lemma 4.5

For any function \(f\in C^1({\mathbb {R}}^d)\) one has
$$\begin{aligned} \bigg \Vert f-\sum _{i\in P_l} f(i)\chi _{i+[0,\frac{1}{l})^d}\bigg \Vert _{L^\infty (Q_l)}\le \frac{\Vert \nabla f\Vert _\infty }{l}, \end{aligned}$$

Proof

This follows immediately from the identity
$$\begin{aligned} f(x)-f(i) = \int _{[i,x]}\nabla f(t)\cdot dt, \end{aligned}$$
where \(i\in P_l\) and [ix] denotes a line segment connecting i to \(x\in i+[0,\frac{1}{l})^d\). \(\square \)
In order to define our approximation of \(\big \langle Ve_k^{(n)},e_m^{(n)}\big \rangle \), we additionally introduce the step function approximation
$$\begin{aligned} E_{k,l}(x) := \sum _{i\in P_l} e_k^{(n)}(l)\chi _{i+[0,\frac{1}{l})^d}. \end{aligned}$$

Lemma 4.6

For \(-\Delta +V\in \Omega _3\) one has
$$\begin{aligned} \left| \left\langle Ve_k^{(n)},e_m^{(n)}\right\rangle - \left\langle V_l E_{k,l},E_{m,l}\right\rangle \right| \le \frac{3}{l^{\frac{1}{2}}}(M+g(0))(2\pi )^{-\frac{d}{2}}n^{3-\frac{d}{2}}d + g\big (l^{\frac{1}{2d}}\big ), \end{aligned}$$
for all \(l\in {\mathbb {N}}\), where M is as in Eq. (4.1).

Proof

We calculate the error
$$\begin{aligned}&\Big |\left\langle Ve_k^{(n)},e_m^{(n)}\right\rangle - \left\langle V_l E_{k,l},E_{m,l}\right\rangle \Big |\\&\quad = \Bigg |\!\sum _{i\in P_l}\int _{i+[0,\frac{1}{l})} \!\! Ve_k^{(n)}e_m^{(n)}\,dx -\! \sum _{i\in P_l}\int _{i+[0,\frac{1}{l})} \!\! V_lE_{k,l}E_{m,l}\,dx +\! \int _{{\mathbb {R}}^d{\setminus } Q_l}\!\! Ve_k^{(n)}e_m^{(n)}\,dx\Bigg | \\&\quad \le \sum _{i\in P_l}\,\int _{i+[0,\frac{1}{l})}\left| Ve_k^{(n)}e_m^{(n)} - V_lE_{k,l}E_{m,l}\right| dx + \int _{{\mathbb {R}}^d{\setminus } Q_l} \big |Ve_k^{(n)}e_m^{(n)}\big |\,dx\\&\quad \le \sum _{i\in P_l}\int _{i+[0,\frac{1}{l})} l^{-1}\left\| \nabla \big (Ve_k^{(n)}e_m^{(n)}\big )\right\| _\infty \, dx + g\big (l^{\frac{1}{2d}}\big )\int _{{\mathbb {R}}^d{\setminus } Q_l} \big |e_k^{(n)}e_m^{(n)}\big |\,dx \\&\quad = l^{-1} |Q_l| \left\| \nabla \big (Ve_k^{(n)}e_m^{(n)}\big )\right\| _\infty + g\big (l^{\frac{1}{2d}}\big )\big \Vert e_k^{(n)}\big \Vert _{L^2({\mathbb {R}}^d)}\big \Vert e_m^{(n)}\big \Vert _{L^2({\mathbb {R}}^d)} \\&\quad \le l^{-1}l^{\frac{1}{2}}\! \left( \left\| \nabla Ve_k^{(n)}e_m^{(n)}\right\| _\infty \!\! +\left\| V\nabla e_k^{(n)}e_m^{(n)}\right\| _\infty \!\! + \left\| Ve_k^{(n)}\nabla e_m^{(n)}\right\| _\infty \right) +g\big (l^{\frac{1}{2d}}\big )\\&\quad \le \frac{3}{l^{\frac{1}{2}}}(M+g(0))(2\pi )^{-\frac{d}{2}}n^{3-\frac{d}{2}}d + g\big (l^{\frac{1}{2d}}\big ), \end{aligned}$$
where we have used Lemma 4.5 in the third line and Lemma 4.2 and the fact that \(\Vert V\Vert _{\mathcal C^1}\le M\) in the last line. \(\square \)

Corollary 4.7

If we denote \(H_n:=P_n(-\Delta +V)|_{\mathcal {H}_n}\) and \(H_n^l:= P_n(-\Delta )|_{\mathcal {H}_n}+W^l\), where \(W^l\) denotes the operator on \(\mathcal {H}_n\) defined by the \(n\times n\) matrix with elements \((W^l)_{km}=\left\langle V_l E_{k,l},E_{m,l}\right\rangle \) in the basis \(\{e_k^{(n)}\}_{k=1}^{\#L_n}\), then
$$\begin{aligned} \left\| H_n - H_n^l \right\| _{L(\mathcal {H}_n)} \le \frac{3d(M+g(0))(2\pi n^{4})^{-\frac{d}{2}}}{l^{\frac{1}{2}}} + ng\big (l^{\frac{1}{2d}}\big ). \end{aligned}$$

Proof

By Lemma 4.6 the matrix elements of \(H_n\) and \(H_n^l\) satisfy \(\big |(H_n)_{km} - (H_n^l)_{km}\big |\le \frac{3}{l^{\frac{1}{2}}}(M+g(0))(2\pi )^{-\frac{d}{2}}n^{3-\frac{d}{2}}d + g\big (l^{\frac{1}{2d}}\big )\). Now note that for any two matrices \(A=(A_{km})\) and \(B=(B_{km})\) one has
$$\begin{aligned} \left\| (A-B)x\right\| ^2_{\mathcal {H}_n}&= \sum _{k=1}^n\left| \sum _{m=1}^n (A_{km}-B_{km})x_m\right| ^2\\&\le \left( \sup _{k,m}|A_{km}-B_{km}|\right) \sum _{k,m=1}^n|x_m|^2\\&= n\left( \sup _{k,m}|A_{km}-B_{km}|\right) \Vert x\Vert _{\mathcal {H}_n}^2. \end{aligned}$$
This immediately implies the assertion. \(\square \)
We are finally ready to define our algorithm. Let \(n\in {\mathbb {N}}\) and choose \(l(n)\in {\mathbb {N}}\) large enough such that \(\frac{3}{l(n)^{\frac{1}{2}}}(M+g(0))(2\pi )^{-\frac{d}{2}}n^{3-\frac{d}{2}}d + g\big (l(n)^{\frac{1}{2d}}\big )<\frac{1}{2n}\) (note that this can be done by a computer in finite time). Define for \(H=-\Delta +V\in \Omega _3\)
$$\begin{aligned} \Lambda _{\Gamma _n^{(3)}}(H)&:= \Lambda _{\Gamma _n^{(3)}}^{(1)}\cup \Lambda _{\Gamma _n^{(3)}}^{(2)}\cup \Lambda _{\Gamma _n^{(3)}}^{(3)}\cup \Lambda _{\Gamma _n^{(3)}}^{(4)} \end{aligned}$$
where
$$\begin{aligned} \Lambda _{\Gamma _n^{(3)}}^{(1)}&= \left\{ \rho _i \,|\,i\in P_{l(n)}\right\} \\ \Lambda _{\Gamma _n^{(3)}}^{(2)}&= \left\{ e_k^{(n)}(i)\,\Big |\,i\in P_{l(n)},\;k\in \{1, \ldots ,n\}\right\} \\ \Lambda _{\Gamma _n^{(3)}}^{(3)}&= \bigg \{\tfrac{n\delta _{mk}}{3}\sum _{j=1}^d\left( \left( (i)_j+\tfrac{1}{n}\right) ^3 - (i)_j^3 \right) \,\bigg |\, i\in L_n,\; m,k\in \{1, \ldots ,\# L_n\} \bigg \} \\ \Lambda _{\Gamma _n^{(3)}}^{(4)}&= \left\{ g\big (l^{\frac{1}{2d}}\big )\,\Big |\,l=0\dots l(n) \right\} \end{aligned}$$
and let
$$\begin{aligned} \Gamma _n^{(3)}(H)&:= \left\{ \lambda \in G_n^{\mathbb {C}}\,\bigg |\, \left\| (H_n^{l(n)}-\lambda )^{-1}\right\| ^{-1} \le \frac{2}{n} \right\} \cup \Gamma _n^{(1)}(-\Delta ) \end{aligned}$$
with the convention that \(\big \Vert (H_n^{l(n)}-\lambda )^{-1}\big \Vert ^{-1}=0\) when \(\lambda \in \sigma \big (H_n^{l(n)}\big )\). Note that \(\Lambda _{\Gamma _n^{(3)}}(H)\) is a finite set for each \(H\in \Omega _3\) and by Lemma 2.4 determining whether \(\big \Vert (H_n^{l(n)}-\lambda )^{-1}\big \Vert ^{-1} \le \frac{2}{n}\) requires only finitely many algebraic operations on the matrix elements of \(H^{l(n)}_n\) (which are contained in \(\Lambda _{\Gamma _n^{(3)}}(H)\)). Moreover, since \(\Lambda _3\) contains all matrix elements of the Laplacian, we conclude that computing \(\Gamma _n^{(1)}(-\Delta )\) can also be done by performing a finite amount of algebraic operations on the information provided. Overall, we conclude that each \(\Gamma _n^{(3)}\) is an arithmetic algorithm in the sense of Definition 1.2.
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
$$\begin{aligned} (H_n^{l(n)}-\lambda )^{-1} - (H_n-\lambda )^{-1} = (H_n^{l(n)}-\lambda )^{-1}(H_n-H_n^{l(n)})(H_n-\lambda )^{-1}.\nonumber \\ \end{aligned}$$
(4.6)
From (4.6) we conclude that
$$\begin{aligned} \left| \Vert (H_n^{l(n)}-\lambda )^{-1}\Vert ^{-1} - \Vert (H_n-\lambda )^{-1}\Vert ^{-1}\right| \le \big \Vert H_n-H_n^{l(n)}\big \Vert . \end{aligned}$$
(4.7)
Indeed if \(\lambda \in \rho (H_n)\cap \rho \big (H_n^{l(n)}\big )\), this just follows by taking norms on both sides and using the reverse triangle inequality, while for \(\lambda \in \sigma (H_n)\cup \sigma \big (H_n^{l(n)}\big )\) it can be seen by the following argument. W.l.o.g. assume that \(\lambda \in \sigma (H_n){\setminus }\sigma \big (H_n^{l(n)}\big )\). Then \(\Vert (H_n-\lambda )^{-1}\Vert ^{-1}=0\) and \(\Vert (H_n^{l(n)}-\lambda )^{-1}\Vert ^{-1}>0\). Assume for contradiction that (4.7) is false, i.e.
$$\begin{aligned} \big \Vert H_n-H_n^{l(n)}\big \Vert < \Vert (H_n^{l(n)}-\lambda )^{-1}\Vert ^{-1}. \end{aligned}$$
Then by a standard Neumann series argument (cf. [10, Sec. I.4.4]) it follows that \(H_n\) is boundedly invertible, which contradicts the assumption \(\lambda \in \sigma (H_n)\). This argument is obviously symmetric in \(H_n\) and \(H_n^{l(n)}\).
Going back to (4.7) and recalling our specific choice of l(n), we conclude that for all \(\lambda \in G_n^{{\mathbb {C}}}\) one has
$$\begin{aligned} \Big |\Vert (H_n^{l(n)}-\lambda )^{-1}\Vert ^{-1} - \Vert (H_n-\lambda )&^{-1}\Vert ^{-1}\Big | \\&\le \Vert H_n-H_n^{l(n)}\Vert \\&\le \frac{3d(M+g(0))(2\pi n^{4})^{-\frac{d}{2}}}{l(n)^{\frac{1}{2}}} + ng\big (l(n)^{\frac{1}{2d}}\big )\\&\le \frac{1}{2n}, \end{aligned}$$
Now, if \(\lambda \in \Gamma _n^{(3)}(H)\) the above inequality implies that
$$\begin{aligned} \Vert (H_n-\lambda )^{-1}\Vert ^{-1}&\le \Vert (H_n^{l(n)}-\lambda )^{-1}\Vert ^{-1}+\frac{1}{2n}\\&\le \frac{2}{n} +\frac{1}{2n}\\&\le \frac{3}{n} \end{aligned}$$
and hence \(\lambda \in \Xi _n(H)\) (cf. Remark 3.7). Similarly, if \(\lambda \in \Gamma ^{(2)}_n(H)\) then
$$\begin{aligned} \Vert (H_n^{l(n)}-\lambda )^{-1}\Vert ^{-1}&\le \Vert (H_n-\lambda )^{-1}\Vert ^{-1} + \frac{1}{n}\\&\le \frac{1}{n} +\frac{1}{2n}\\&\le \frac{2}{n} \end{aligned}$$
and hence \(\lambda \in \Gamma _n^{(3)}(H)\). Thus, we have the inclusions
$$\begin{aligned} \Gamma ^{(2)}_n(H) \subset \Gamma _n^{(3)}(H) \subset \Xi _n(H). \end{aligned}$$
Since \(\Gamma ^{(2)}_n(H)\rightarrow \sigma (H)\) and \(\Xi _n(H)\rightarrow \sigma (H)\) by Theorem 3.1 and Remark 3.7, we conclude that \( \Gamma _n^{(3)}(H)\rightarrow \sigma (H)\) as well. This completes the proof of Theorem 4.3.

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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Authors and Affiliations

  1. 1.School of MathematicsCardiff UniversityCardiffUK

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