On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators

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 \(\{e_i\}\).

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.

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

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

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.

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 by

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

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.

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.

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).

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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 [i, x] 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.