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 complexityMathematics Subject Classification
Primary 35P99 Secondary 35Q401 Introduction
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.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 \(T\in \Omega \)?
Definition 1.1
- (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
- (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.
Definition 1.3
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
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
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
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. \(\square \)
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
2.2 General Results on Spectral and Pseudospectral Pollution
In order to prove the next lemma, we need a fact about closures of pseudospectra.
Lemma 2.6
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 \(k\rightarrow \infty \)
\(\Vert (H_{n_k}-\lambda )x_k\Vert \rightarrow \delta \).

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 \(\lambda _n\rightarrow \lambda \). \(\square \)
- (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
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
- (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).
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
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
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 \(W_e(H)=W_e(T)\).
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 \(\lambda _n\rightarrow \lambda \). \(\square \)
- (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
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
Lemma 4.2
Proof
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)
\(\left\| |\xi |^2(u_n-u)\right\| _{L^2({\mathbb {R}}^d)}\rightarrow 0\)
Sketch of function \(F_n\)
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.
Lemma 4.5
Proof
Lemma 4.6
Proof
Corollary 4.7
Proof
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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