## Abstract

The notion of second-order relative spectrum of a self-adjoint operator acting on a Hilbert space has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. In this paper we examine how the second-order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of the underlying operator. Our theoretical findings are supported by various numerical experiments on the computation of guaranteed eigenvalue inclusions via finite element bases.

## 1. Introduction

Let *A* be a self-adjoint operator acting on an infinite-dimensional Hilbert space and let *λ* be an isolated eigenvalue of *A*. For let , where *E*_{μ} is the spectral measure associated to *A*. The numerical estimation of *λ*, whenever constitutes a serious challenge in computational spectral theory. Indeed, it is well established that classical approaches, such as the Galerkin method, suffer from variational collapse under no further restrictions on the approximating space and therefore might lead to spectral pollution (see Rappaz *et al*. 1997; Dauge & Suri 2002; Lewin & Séré 2010; and references therein).

The notion of second-order relative spectrum, originated from Davies (1998), has recently allowed the formulation of a general pollution-free strategy for eigenvalue computation. This was proposed by Shargorodsky (2000) and subsequently examined by Levitin & Shargorodsky (2004), Boulton (2007), Boulton & Strauss (2007) and Strauss (in press). Numerical implementations of the general principle very much preserve the spirit of the Galerkin method and have presently been tested on applications from Stokes systems (Levitin & Shargorodsky 2004), solid-state physics (Boulton & Levitin 2007), magnetohydrodynamics (Strauss in press) and relativistic quantum mechanics (Boulton & Boussaid 2010).

In this paper we further examine the potential role of this pollution-free technique for robust computation of spectral inclusions. Our goal is twofold. On the one hand, we establish various abstract properties of limit sets of second-order relative spectra. On the other hand, we report on the outcomes of various numerical experiments. Both our theoretical and practical findings indicate that second-order spectra provide reliable information about the multiplicity of any isolated eigenvalue of *A*.

Section 2 is devoted to reformulating some of the concepts around the notion of second-order spectra in order to allow a more general setting. In this framework, we consider the natural notion of algebraic and geometric multiplicity of second-order spectral points (definition 2.2) and establish a ‘second-order’ spectral mapping theorem (lemma 2.6).

In §3 we pursue a detailed analysis of accumulation points of the second-order relative spectra on the real line. Our main contribution (theorem 3.4) is a significant improvement upon similar results previously found. It allows calculation of rigorous convergence rates when the test subspaces are generated by a non-orthogonal basis. Concrete applications include the important case of a finite-element basis, which was not covered by theorem 2.1 of Boulton (2007). Our present approach relies upon a homotopy argument, which yields a precise control on the multiplicity of the second-order spectral points. The argument is reminiscent of the method of proof of Goerisch theorem (see Plum 1990).

Theorem 3.4 also determines the precise manner in which second-order spectra encode information about the multiplicity of points in the spectrum of *A*. When an approximating space is ‘sufficiently close’ to the eigenspace corresponding to an eigenvalue of finite multiplicity, a finite set of conjugate pairs in the second-order spectrum becomes ‘isolated’ and clusters near the eigenvalue. It turns out that the total multiplicity of these conjugate pairs exactly matches the multiplicity of the eigenvalue. This indicates that second-order spectra capture, in a reliable manner, points to the discrete spectrum and their multiplicities, even under the above variational collapse condition. In §4 we examine the practical validity of this statement on benchmark differential operators for subspaces generated by a basis of finite elements.

A final appendix is aimed at finding the minimal region in the complex plane where the limit of second-order spectra is allowed to accumulate. It turns out that, modulo a subset of topological dimension zero, this minimal region is completely determined by the essential spectrum of *A*. This gives an insight on the difficulties involving the problem of finding conditions on the test subspaces to guarantee convergence to the essential spectrum.

### (a) Notation

Below we will denote by D(*A*) the domain of *A* and by Spec(*A*) its spectrum. We decompose Spec(*A*) in the standard manner as the union of essential and discrete spectrum (see Kato 1980), and denote . Recall that the discrete spectrum is the set of isolated eigenvalues of finite multiplicity of *A*.

Let . Below will be the open disc in the complex plane with centre (*a*+*b*)/2 and radius (*b*−*a*)/2, and . We allow or in the obvious way to denote half-planes or the whole of . For *b*=*a*, and .

Let be a Hilbert space. Let be an arbitrary subset and let be a finite subset. We will often writeHere, we include the possibility of and write .

Let . We endow D(*A*^{p}) with the graph inner product defined for all *u*,*v*∈D(*A*^{p}) as . By construction, (D(*A*^{p}),〈·,·〉_{p}) is a Hilbert space with the associated norm denoted by ∥·∥_{p}. Below we will consider sequences of finite-dimensional subspaces growing towards D(*A*^{p}) with a given density property determined as follows:Note that if , then *Λ*_{0}=*Λ*_{p} for any . If and , then is dense norm ∥·∥_{p}.

For a family of closed subsets , the limit set of this family is defined asThe limit set of a family of closed subsets is always closed.

Let be a basis for a subspace . Without further mention, we will identify with a corresponding via: and , where is the basis conjugate to {*b*_{j}}. If the vectors *b*_{j} are mutually orthogonal and ∥*b*_{j}∥=1, then . Below the orthogonal projection onto is denoted by . When referring to sequences of subspaces we will use *P*_{n} instead. Clearly iff *P*_{n}→*I* strongly.

### (b) Toy models of spectral pollution in the Galerkin method

We now present a series of examples which will later illustrate our main results. We follow theorem 2.1 of Levitin & Shargorodsky (2004) and remark 2.5 of Lewin & Séré (2010). Here and below we denote |*v*〉〈*w*|(*u*)=〈*u*,*w*〉*v*.

### Example 1.1

Let . Let be an orthonormal set of vectors and let . Let defined in its maximal domain, where . Then, and so . Note that the resolvent of *A* is compact.

### Example 1.2

If *A* is strongly indefinite and it has a compact resolvent, then there exists a sequence for all , such that Indeed, let where the eigenvalues are repeated according to multiplicity with and . Let be eigenvectors associated to and assume that is an orthonormal set. Let be an ordering of . For each , choose with *λ*^{−}_{k}<*γ*_{j}<*λ*^{+}_{k} for 1≤*j*≤*n*. Let *θ*_{j}∈(−*π*,*π*] be such that for 1≤*j*≤*n* and define The desired conclusion holds by taking

### Example 1.3

Let and be as in example 1.1. Let For *r*>0, define in its maximal domain. Then Spec_{ess}(*A*)={−1} and . It is not difficult to see that now where −1 is an eigenvalue of multiplicity *n*. Then, if *r*=2, the origin is an accumulation point of . Note that the resolvent of *A* is not compact and *A* is now semi-bounded below.

## 2. The second-order spectra of a self-adjoint operator

### Definition 2.1

**(see Levitin & Shargorodsky 2004).** Given a non-trivial subspace , the second-order spectrum of *A* relative to is the set

If , thenTypically, contains non-real points. From the definition, it follows that if and only if .

### (a) Algebraic and geometric multiplicity

Let where the *b*_{j} are linearly independent. Let be matrices with entries given by2.1and *Q*(*z*)=*B*−2*zL*+*z*^{2}*M*. It is readily seen that if and only if2.2By definition 2.1, is independent of the basis chosen. Since , it follows that consists of at most 2*n* points.

The quadratic eigenvalue problem (2.2) may be solved via suitable linearizations (see Gohberg *et al*. 2005). One such linearization is given by the matricesClearly, if and only if for some . Equivalently, one can consider the matrix where2.3so that . LetA straightforward calculation yields , so *Q*(*z*)⊕(−*I*) and are equivalent in the sense of matrix polynomials.

Every matrix polynomial is known to be equivalent to a diagonal matrix polynomial diag[*i*_{1}(*z*),*i*_{2}(*z*), … ,*i*_{r}(*z*),0, … ,0], where the diagonal entries *i*_{j}(*z*) are polynomials of the form *Π*_{k}(*z*−*z*_{jk})^{βjk} with the property that *i*_{j}(*z*) is divisible by *i*_{j−1}(*z*). The factors (*z*−*z*_{jk})^{βjk} are called elementary divisors and the *z*_{jk} are the eigenvalues. The degrees of the elementary divisors associated to a particular eigenvalue *z*_{0} are called the partial multiplicities at *z*_{0} (see Gohberg *et al*. 2005). For a linear matrix polynomial, such as , the degrees of the elementary divisors associated to a particular eigenvalue *z*_{0} coincide with the sizes of the Jordan blocks associated to *z*_{0} as an eigenvalue of . The notion of multiplicity of an eigenvalue now follows from the fact that two *n*×*n* matrix polynomials are equivalent if and only if they have the same collection of elementary divisors (see Gohberg *et al*. 2005, theorem A.6.2).

### Definition 2.2

The geometric and algebraic multiplicities of are defined to be the geometric and algebraic multiplicity of *z* as an eigenvalue of .

Note that definition 2.2 is independent of the basis used to assemble the matrices *B*,*L*,*M* and . Indeed, let *B*,*L*,*M* be assembled with respect to a basis and with respect to a basis , where . Then,where *N*_{ij}=〈*c*_{j},*b*^{*}_{i}〉, and therefore and are equivalent.

### (b) The approximate spectral distance

Suppose that the given basis of is orthonormal so that *M*=*I* in (2.1) and *Q*(*z*)=*B*−2*zL*+*z*^{2}*I*. For , we define . The right-hand side of this expression is independent of the orthonormal basis chosen for . If , then and for we have . Furthermore, if and only if . The map is a Lipschitz subharmonic function (Davies 1998; Boulton 2007). Therefore, is completely characterized via the maximum principle.

### (c) The spectrum and the second-order spectra

Let and . For all ,2.4(see Shargorodsky 2000; Strauss in press). Thus, is a subset of for any sequence .

A growing interest in the second-order relative spectrum and corresponding limit sets has been stimulated by the following property: if , then2.5(see Levitin & Shargorodsky 2004; also lemma 2.3 below). Thus,2.6Therefore, intervals of inclusion for points in the spectrum of *A* can be obtained from Re *z* with a two-sided explicit residual given by |Im *z*|.

In fact, the order of magnitude of the residue in the approximation of Spec(*A*) by projecting into can be improved to |Im *z*|^{2}, if some information on the localization of Spec(*A*) is at hand. Indeed, if and with , then2.7(see Boulton & Levitin 2007; Strauss in press).

The following lemma is an improvement upon theorem 5.2 of Shargorodsky (2000). It immediately implies equation (2.5).

### Lemma 2.3

*Let* *be such that* *. If* *, then* *where*2.8

### Proof.

Without loss of generality assume that Im *z*≥0, and consider the triangle *Δazb*. Let *θ* be the angle of *Δazb* at *z*. Since , then *π*/2<*θ*≤*π*. Let *θ*_{1} and *θ*_{2} be the angles at *b* and *a*. Then *θ*+*θ*_{1}+*θ*_{2}=*π*. The region *F*_{z}:={*μ*−*z*:*μ*∈Spec(*A*)} is contained in two sectors, one between the real line and the ray *r*e^{−iθ1} (*r*≥0), the other between the real line and r*e*^{i(π+θ2)}, where 0≤*θ*_{1}+*θ*_{2}<*π*/2. We have , and by the cosine rule, . Now is contained in a sector between the rays r*e*^{−i2θ1} and r*e*^{i2θ2}. ThereforeBy virtue of the spectral theorem,The result follows from the Cauchy–Schwarz inequality. ■

If , it follows from equation (2.5) that does not intersect . Therefore, . We now consider two examples where . The first one has a compact resolvent while the second one has −1 in the essential spectrum.

### Example 2.4

Suppose that , and *A* are as in example 1.1. It is readily seen that .

### Example 2.5

Suppose that , and *A* be as in example 1.3. It is readily seen that , where and . The geometric multiplicity of the second-order spectral point −1 is *n*−1 and its algebraic multiplicity is 2(*n*−1). As , the non-real point *α*_{n}±i*γ*_{n}→*β*, where *β*=−1 for 0<*r*<1, *β*=−1±i for *r*=1 and for *r*>1. Hence, for *r*≠1.

### (d) Mapping of second-order spectra

### Lemma 2.6

*For* *, let f(w)=aw+b, g(w)=cw+d and F(w)=f(w)g(w)*^{−1}*. If ad≠cb and* *, then*2.9*Moreover, the algebraic and geometric multiplicities of z and F(z) coincide.*

### Proof.

Since *g*(*A*) preserves linear independence, a set of vectors is a basis for if and only if the corresponding set is a basis for . Let . It is readily seen that . Let , and . Then, for any , obtaining the left side from and the right side from {*b*_{j}}. Let *B*, *L*, *M* be as in (2.1) for *A* and {*b*_{j}}. Let be as in (2.1) for *F*(*A*) and . Let . ThenHere are arbitrary, so we deduce that2.10Thus, , and moreover, since the function (*ad*−*cb*)^{−2}*g*(*w*)^{2} is analytical and non-zero at *z*, the algebraic and geometric multiplicities of *z* and *F*(*z*) are the same (see Gohberg *et al*. 2005, theorem A.6.6). ■

## 3. Multiple points of the second-order spectra

### (a) Neighbourhoods of the discrete spectrum

Throughout this section we assume that3.1Let 1_{j}=1_{(λj−ε,λj+ε)}(*A*), where *ε*>0 is small enough so that *m*_{j}:=Rank(1_{j}) is equal to the multiplicity of *λ*_{j}. Let be the total multiplicity of the group of eigenvalues *λ*_{1}, … ,*λ*_{s}. We will denote by the eigenspace associated to the group of eigenvalues in (*a*,*b*), that is, . We fix orthonormal bases of Rank 1_{j} and let be a corresponding basis of .

### Lemma 3.1

*Let* *. Let*3.2*If the subspace* *is such that**then* *.*

### Proof.

Let . Define where . Then is self-adjoint and . Letthen *K*(*z*) is a finite rank operator with range and . Evidently, is invertible and for all we havewhere .

Let . For each eigenvector we assign a such that for *p*=0,1,2. In general ; however, . The hypothesis on *δ* ensures thatwhere and .

Note that satisfies the hypothesis of lemma 2.3. Let *α*(*z*) be given by equation (2.8). Thenwhere *β*_{3}(*z*)=*α*(*z*)^{−1}|*c*−*z*|^{−2}[*β*_{2}(*z*)+2|*c*−*z*|^{2}+*α*(*z*)]. ThusHence, for any ,

LetThen . ThusTherefore ■

### Lemma 3.2

*Let λ*∈Spec_{dis}*(A) be an eigenvalue of multiplicity m. Denote by d*_{λ}*=dist(λ,Spec(A)\{λ}). If* *, then*3.3*and λ as member of* *has geometric multiplicity m and algebraic multiplicity 2m.*

### Proof.

Assume that and *z*≠*λ*. Then, there exists a normalized , such that for all . In particular 〈*u*,*u*^{k}〉=0 for each . If we take *w*=*u* above, thenThus, |Im *z*|=∥(*A*−Re *z*)*u*∥≠0 and 〈(*A*−Re *z*)*u*,*u*〉=0, so thatSince ,however, the right-hand side is strictly less that *d*_{λ}, and equation (3.3) follows from the contradiction.

Let be a basis for and let *B*, *L* and *M* be the matrices (2.1) associated to this basis. Let *v* be can arbitrary member of . Thenso that . Hence,Since , are eigenvectors of corresponding to the eigenvalue *λ*. For *k*≠*j* we have , so the vectors are linearly independent. There is a one-to-one correspondence between the eigenvectors of associated to *λ* and the eigenvectors of *A* associated to *λ*. Thus, the geometric multiplicity of is equal to *m*.

We now compute the algebraic multiplicity. Let and . Ifthen and . Thereforesowhere . Ifthen and . Therefore, . Thus, so that . This ensures that the spectral subspace associated with *λ* as eigenvalue of is given by ■

### Lemma 3.3

*Let* *be a Hilbert space and* *be a set of orthogonal vectors with* *for 1≤j≤m. Let* *be an n-dimensional subspace of* *where n≥m. Let* *, where* *is the orthogonal projection associated to* *. Let w*_{j}*(t)=tw*_{j}*+(1−t)u*_{j} *for t∈[0,1]. If* *for each 1≤j≤m, then there exist vectors {w*_{m+1}*, … ,w*_{n}*}, such that {w*_{1}*(t), … ,w*_{m}*(t),w*_{m+1}*, … ,w*_{n}*} is linearly independent for all t∈[0,1].*

### Proof.

If *a*_{1}*w*_{1}+⋯+*a*_{m}*w*_{m}=0, where not all of the *a*_{j}=0, we would havewhich contradicts Hölder’s inequality. Hence, necessarily {*w*_{1}, … ,*w*_{m}} is linearly independent. Let {*w*_{m+1}, … ,*w*_{n}} be any completion of {*w*_{1}, … ,*w*_{m}} to a basis of . Let *t*∈[0,1]. Suppose now that *a*_{1}*w*_{1}(*t*)+⋯+*a*_{m}*w*_{m}(*t*)+*a*_{m+1}*w*_{m+1}+⋯+*a*_{n}*w*_{n} vanishes. ThenThe two terms on the right-hand side are orthogonal and therefore each must vanish. As the set {*w*_{1}, … ,*w*_{n}} is linearly independent, all *a*_{j}=0. ■

### (b) The main result

### Theorem 3.4

*Let a,b*∉Spec(*A*). *Assume that condition* (3.1) *holds for a group of eigenvalues* {*λ*_{1}< … <*λ*_{s}} *with corresponding multiplicities m*_{j} *and their total multiplicity is* . *Let* . *Let* *κ*=*d*^{2}/*γ* *where γ is defined by equation* (3.2). Let*If* *is such that* *for an orthonormal set of eigenfunctions* *associated to* , *then*3.4*and the total algebraic multiplicity of* *is 2m*_{j}.

### Proof.

Let and , thenIn order to find the right-hand side, we may assume without loss of generality that *a*=−*r* and *b*=*r*. If *z*=*x*+*iy* where *x*^{2}+*y*^{2}=*r*, thenWe also assume that *x*≥0 as the case where *x*<0 can be treated analogously. A straightforward calculation yieldshence lemma 3.1 applied to the interval ensures (3.4).

We now prove the second part of the theorem. Let , and define subspaces for *t*∈[0,1] (see lemma 3.3). Note that is also an orthogonal set in D(*A*^{2}) and . Since *κε*^{2}<*m*^{−1/2}, we get for all *t*∈[0,1]. Now,thus , and by virtue of lemma 3.1 we obtainDenote by the linearization matrix defined as in equation (2.3) for . According to lemma 3.2, the algebraic multiplicities of is 2*m*_{j}. If we let *π*_{j}(*t*) be the spectral projection associated to thenThus, *π*_{j}(*t*)→*π*_{j}(1) as *t*→1. By virtue of lemma 4.10 of §1.4.6 in Kato (1980), it is guaranteed that Rank *π*_{j}(0)=Rank *π*_{j}(1). ■

An immediate consequence of theorem 3.4 is the fact that, if a sequence of test subspaces approximates a normalized eigenfunction *u* associated to a simple eigenvalue *λ* at a given rate in the graph norm of *A*^{2}, ∥*u*−*u*_{n}∥_{D(A2)}=*δ*(*n*)→0 for , then there exist a conjugate pair such that |*λ*−*z*_{n}|=*O*(*δ*^{1/2}). Simple examples show that this estimate is sub-optimal.

### Example 3.5

Let . Let , where and *β*_{n}→0. Then where *γ*_{n}∼*inβ*_{n} for . On the other hand, ∥*A*^{p}((*α*_{n}−1)*e*_{0}+*β*_{n}*e*_{n})∥∼*n*^{p}*β*_{n} for *p*=0,1,2. Thus, |*γ*_{n}|∼*nβ*_{n} while . Moreover, we observe that in general might diverge as and still we might be able to recover |*γ*_{n}|→0; take for instance *β*_{n}=1/*n*^{3/2}.

An application of lemma 2.6 yields convergence to eigenvalues under the weaker assumption .

### Corollary 3.6

*Let* *be such that*. *If* , *then* .

### Proof.

Pick *ε*>0 such that . Let *μ*∈(*a*,*a*+*ε*). Let *g*(*w*)=*w*−*μ* and *h*(*z*)=*g*(*z*)^{−1}. The operator *h*(*A*) is bounded and *g*(*w*) satisfies the hypothesis of lemma 2.6. Let . Then . Indeed, any can be expressed as *u*=*g*(*A*)*v* for *v*∈D(*A*), and since , we can find such that *g*(*A*)*v*_{n}→*g*(*A*)*v*. By virtue of theorem 3.4,The fact that *g*(*w*) is a conformal mapping, lemma 2.6 and the spectral mapping theorem ensure the desired conclusion. ■

### Example 3.7

Let *A* be the operator of examples 1.3 and 2.5. By virtue of (2.5) and corollary 3.6, for any .

### Corollary 3.8

*Let A be an operator with a compact resolvent. For all* , .

When *A* is not semi-bounded, this statement is in stark contrast to Example 1.2.

## 4. Numerical applications

The identity (2.6) leads to guaranteed intervals of enclosures for *λ*∈Spec_{disc}(*A*). Theorem 3.4 yields *a priori* upper bounds for the length of these enclosure in terms of bounds on the distance from the test space, , to an orthonormal basis of eigenfunctions . In practice, these upper bounds are computed from interpolation estimates for bases of . We now examine various aspects of the applicability of these results.

### (a) On multiplicity and approximation

Suppose that *λ* is of multiplicity *m*, so that (3.4) ensures an upper bound on the estimation of *λ* by *m* conjugate pairs of . If captures well a basis of only *l*<*m* eigenfunctions and rather poorly the remaining *m*−*l* elements of , then it would be expected that only *l* conjugate pairs of will be close to *λ* while the remaining *m*−*l* will lie far from this eigenvalue. We can illustrate this locking effect on the multiplicity in a simple numerical experiment.

Let subject to Dirichlet boundary conditions on . Then and a family of eigenfunctions is given by . Some eigenvalues of *A* are simple and some are multiple. In particular, the ground eigenvalue 2=1+1 is simple and 1+*k*^{2} for *k*=2,3,4,5 are double. Contrast between the two eigenfunctions *u*_{k1} and *u*_{1k} will increase as *k* increases. The former will be highly oscillatory in the *x*-direction while the latter will be so in the *y*-direction. If captures well oscillations in only one direction, we should expect one conjugate pair of to be close to 1+*k*^{2} and the other to be not so close to this eigenvalue.

In order to implement a finite element scheme for the computation of the second-order spectra of the Dirichlet Laplacian, the condition prescribes the corresponding basis to be at least *C*^{1}-conforming. We let be generated by a basis of Argyris elements on given triangulations of *Ω*. Contrast between the residues in the interpolation of *u*_{1k} and *u*_{k1} is achieved by considering triangulation that are stretched either in the *x* or in the *y* direction (figure 1*b*).

Below, we generate triangulations with a fixed total number of 240 elements, resulting from diagonally bisecting a decomposition of *Ω* as the union of 120 rectangles of equal ratio. We consider mesh with *s*=3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30 and 40 elements of equal size on the lower edge . We then compute the pairs , which are closer to the eigenvalues 1+*k*^{2} than to any other point in Spec(*A*). We know that Re *z*_{1k}(*s*)±|Im *z*_{1k}(*s*)| and Re *z*_{k1}(*s*)±|Im *z*_{k1}(*s*)| are guaranteed bounds for this eigenvalue.

Even though we have the same number of elements in each of the above mesh, in general they do not have the same amount of elements intersecting ∂*Ω*. Then typically for *s*≠*t*, although these numbers do not differ substantially. The precise dimension of the test spaces are: , , , , and .

In figure 1 we have depicted the residues 2|Im *z*_{1k}(*s*)| and 2|Im *z*_{k1}(*s*)| for each one of the eigenvalues 1+*k*^{2} in the vertical axis, versus *s* in the horizontal axis on a semi-log scale. The graph suggests that the order of approximation for all the eigenvalues changes by at least two orders of magnitude as *s* varies. The minimal residue in the approximation of the ground eigenvalue is achieved when *s*=10 and *s*=12. This corresponds to low contrast in the basis of . When the eigenvalue is multiple, however, the minimal residue is achieved by increasing the contrast in the basis. As this contrast increases, one conjugate pair will get closer to the real axis while the other will move away from it. The greater the *k* is, the greater contrast is needed to achieve a minimal residue and the further away the conjugate pairs travel from each other.

This experiment suggests a natural extension for theorem 3.4. If only *l*<*m* members of are close to , then only *l* conjugate pairs on will be close to the corresponding eigenvalue. At present it is not clear whether a more precise statement in this respect can be formulated along the lines of theorem 3.4.

### (b) Optimality of convergence to eigenvalues

We saw in example 3.5 that the upper bound established in theorem 3.4 is sub-optimal. We now examine this assertion from a computational perspective.

Let acting on , where *V* is a smooth real-valued bounded potential. Let D(*A*)={*u*∈*H*^{2}[0,*π*]:*u*(0)=*u*(*π*)=0}, so that *A* defines a self-adjoint semi-bounded operator. Note that *λ*∈Spec(*A*)=Spec_{disc}(*A*) if and only if *λ* solves the Sturm–Liouville eigenvalue problem *Au*=*λu* subject to homogeneous Dirichlet boundary conditions at 0 and *π*.

Let *Ξ* be an equidistant partition of [0,*π*] into *n* sub-intervals *I*_{l}=[*x*_{l−1},*x*_{l}] of length *h*=*π*/*n*=*x*_{l}−*x*_{l−1}. Letbe the finite element space generated by *C*^{k}-conforming elements of order *r* subject to Dirichlet boundary conditions at 0 and *π*. An implementation of standard interpolation error estimates for finite elements combined with theorem 3.4 ensures the following.

### Lemma 4.1

*Let A,* *and* *be as above. Let Spec(A)={λ*_{1}*<λ*_{2}*< … }. Let a*_{j}*= (1/4)λ*_{j}*+(3/4)λ*_{j−1} *and b*_{j}*=(1/4)λ*_{j}*+(3/4)λ*_{j+1}*, where* *. For all r>k≥3, there exist a constant c>0, dependent on j, k and r, but independent of h, such that* *for all h>0 sufficiently small.*

### Proof.

Use the well-known estimate ∥*v*−*v*_{h}∥_{Hp[0,π]}≤*ch*^{r+1−p}, where is the finite element interpolant of and note that all eigenvalues of *A* are simple while their associated eigenfunctions are . ■

Therefore, each individual eigenvalue *λ*_{j} is approximated by second-order spectral points at a rate *O*(*h*^{(r−3)/2}) for test subspaces generated by a basis of *C*^{3}-conforming finite elements of order *r*>4. Because of the high regularity required on the approximating basis, this result is only of limited practical use. In fact, only *k*≥1 (and *r*≥3) is required for . The following simple numerical experiment confirms that the exponent predicted by lemma 4.1 can be improved.

### Example 4.2

Suppose that *V* =0 so that . In figure 2 we have fixed an equidistant partition *Ξ* and let be the space of Hermite elements of order *r*=3,4,5 satisfying Dirichlet boundary conditions in 0 and *π*. We then find the conjugate pairs , which are close to *λ*_{j}=*j*^{2} in Spec(*A*) for *j*=1,2,3,4,5.

### Example 4.3

The previous example suggests an order of approximation of the eigenvalue *λ* from its closest of Indeed, interpolation by Hermite elements of order *r* has an *H*^{2}-error proportional to *h*^{r−1}. The same convergence rate is confirmed by example 3.5. Let be the Mathieu potential. Let be as in example 4.2. The exponents *p* reported in table 1 further confirm this conjecture.

It is well established that the error in the estimation of the eigenvalues of a one-dimensional elliptic problem of order 2*p* by the Galerkin method using Hermite elements of order *r* is proportional to *h*^{2(r+1−p)}. If *A* is a second-order differential operator, the quadratic eigenvalue problem (2.2) gives rise to a non-self-adjoint fourth-order problem, which is to be solved by a projection-type method. Thus, example 4.3 is consistent with this estimate if we consider the improved enclosure (2.7).

### (c) Improved accuracy

Guaranteed enclosures for *λ*∈Spec_{disc}(*A*) can be improved by combining (2.7) with (2.6). For this we should find an upper bound , a lower bound and such that Here *μ* and *ν* can be elements of the discrete or the essential spectrum of *A*. The preliminary bounds *a* and *b* can be found from or by analytical means. If *b*−*a* is sufficiently large and |Im *z*| is sufficiently small, then (2.7) improves upon (2.6). This technique can be implemented as follows.

### Example 4.4

Let , and for *r*=3,4,5 be as in examples 4.2 and 4.3. In table 1 we have computed inclusions for the first five eigenvalues of *A* by directly employing (2.6) and by the technique described in the previous paragraph. In the latter case, we have found *a*_{j}=*a* from the computed upper bound for *λ*_{j−1}<*λ*_{j} () using (2.6). Similarly for *b*_{j}=*b*. Compare these calculations with those of Arbenz (1983).

We can also consider an example from solid-state physics to illustrate the case where *μ* and *ν* are not in the discrete spectrum.

### Example 4.5

Let on , where . Then *A* is a semi-bounded operator but now Spec_{ess}(*A*) consists of an infinite number of non-intersecting bands, separated by gaps, determined by the periodic part of the potential. The endpoints of these bands can be found analytically. They correspond to the so called Mathieu characteristic values. The addition of a fast decaying perturbation gives rise to a non-trivial discrete spectrum. Non-degenerate isolated eigenvalues can appear below the bottom of the essential spectrum or in the gaps between bands.

In table 2 we report on computation of inclusions for the first three eigenvalues in Spec_{disc}(*A*): , *λ*_{2} in the first gap of the essential spectrum and *λ*_{3} in the second gap. No other eigenvalue is to be found in any of these gaps. Here is the space of Hermite elements of order 3 on a mesh of 20*l* segments of equal size *h*=0.1 in [−*l*,*l*] subject to Dirichlet boundary conditions at ±*l*. For the improved enclosure, *a* and *b* are approximations of the endpoints of the gaps where the *λ*_{j} lie.

Since the eigenvectors of *A* decay exponentially fast as , the members of Spec_{disc}(*A*) are close to eigenvalues of the regular Sturm–Liouville problem *Au*=*λu*, subject to *u*(−*l*)=*u*(*l*)=0, for sufficiently large . The numerical method considered in this example does not distinguish between the and the large eigenvalue problem. For instance, the inclusion found for *λ*_{1} is also an inclusion for the Dirichlet ground eigenvalue of . In the case of *λ*_{2} and *λ*_{3}, which lie in gaps of Spec_{ess}(*A*), they should be close to high-energy eigenvalues of the finite interval problem. Indeed, for the parameters considered in table 2, the inclusion for *λ*_{2} is also an inclusion for the 17*th* eigenvalue of . Similarly that for *λ*_{3} is an inclusion for the 65*th* eigenvalue of .

## Acknowledgements

We are grateful to M. Langer for fruitful discussions. L.B. wishes to acknowledge the hospitality of Ceremade where part of this research was carried out. M.S. gratefully acknowledges the support of EPSRC grant no. EP/E037844/1.

- Received May 5, 2010.
- Accepted July 9, 2010.

- This journal is © 2010 The Royal Society

## Appendix A. Accumulation points outside the real line

Since is second countable and the essential spectrum of *A* is closed, there always exists a family of open intervals such that . Throughout this section we will be repeatedly referring to the following two *A*-dependent regions of the complex plane: , where . For and *ε*>0 we will denote the open *ε*-*neighbourhood* of *Ω* by .

### Lemma A.1

*Let* *be compact and such that* *. Let* *. There exists* *and N>0 such that* *for all* *and n≥N.*

### Proof.

It follows from lemmas 2.3 and 3.1 applied in (*a*_{j}+*ε*,*b*_{j}−*ε*), where is a finite covering of , by analogous arguments as in the proof of theorem 3.4. ■

Combining this lemma with an argument as in the proof of corollary 3.6, we immediately achieve a generalization of corollary 3.8:We now examine accumulation of in compact subsets of .

### Lemma A.2

*Let c∈Spec*_{ess}*(A). For any given* *and δ>0 there exists* *such that*

*(i) ∥x*_{l}*∥=1 for all l=1, … ,m*

*(ii)* *for all* *and p,q=0,1*

*(iii)* *where* *for any l=1, … ,m.*

### Proof.

If *c* is isolated from the other points in the spectrum, the proof is elementary. Otherwise, there exists *c*_{l}→*c*, such that *c*_{l}∈Spec(*A*)\{*c*}. By substituting *c*_{l} by a sub-sequence if necessary, we can assume that |*c*_{l}−*c*|=*δ*(*l*)<*δ*/2 for 0<*δ*(*l*+1)<*δ*(*l*), such that If we pick *x*_{l}∈1_{(cl−(δ(l)/2),cl+(δ(l)/2))}(*A*) for *l*=1, … ,*m* with ∥*x*_{l}∥=1, the desired conclusion is guaranteed. ■

For *c*_{1}≤*c*_{2}≤*c*_{3} denote .

### Lemma A.3

*Let* *. For δ>0 let x*_{1}*,x*_{2}*,x*_{3}*∈D(A) be such that (i)–(iii) of lemma A.2 hold for m=3. There exist* *independent of δ such that if* *, the polynomial 〈Ay,Ay〉−2λ〈Ay,y〉+λ*^{2}*〈y,y〉 has roots λ*_{+}*=z+O(δ) and λ*_{−}*=z+O(δ).*

### Proof.

The cases where for are easy, so we assume that *c*_{1}<*c*_{2}<*c*_{3}. Moreover, without loss of generality we can fix *c*_{1}=−1, *c*_{2}=*c*, where |*c*|<1 and *c*_{3}=1. Then *z* and are roots of the polynomial for *z*=*ae*^{iθ}. If we getThe solution of the systemis , and , where . The proof is completed by an elementary trigonometric argument, showing that *β*_{±}≥0 for any . ■

### Theorem A.4

Let be compact. For all *ε*>0 there exists such that .

### Proof.

Let and *ε*>0 be fixed. Since is compact, there exists , such that . The proof will be completed if is such that . Below we will choose *δ*>0 small enough.

Let be such that . By virtue of lemma A2 there is a family of vectors , such that: ∥*x*_{kj}∥=1 for all *k*=1,2,3 and *j*=1, … ,*n*; for all and *p*,*q*=0,1; , where for any *k*=1,2,3 and *j*=1, … ,*n*. Let *α*_{kj}=*α*_{k} be as in lemma A3 for *z*=*z*_{j}. Choosing *δ*>0 small enough and defining and completes the proof. ■

This result has two straightforward consequences. Given any compact subset , there exists , such that . Evidently, might fall outside *Λ*_{0} in general. On the other hand, we can always find , such that .