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Limiting set of second order spectra

Lyonell Boulton

TL;DR

The paper develops and rigorously analyzes a second order spectrum method for detecting isolated eigenvalues of self-adjoint operators in gaps of the essential spectrum. By defining the second order spectrum $\mathrm{Spec}_2(M|\mathcal{L}_n)$ via zeros of $\det[P_n(M-z)^2|\mathcal{L}_n]$ and its Lipschitz scalar surrogate $\sigma_n(z)$, it constructs the limiting set $\Lambda$ and proves that, under the mild hypothesis (H), every isolated eigenvalue of finite multiplicity lies in $\Lambda$, while the spectra avoid spurious pollution away from the real axis. The paper also provides a concrete model with $M$ as a piecewise-constant multiplier where $\Lambda(M)=\mathbb{T}$, illustrating maximal pollution in linear Ritz methods and the non-polluting behavior of the second order approach. Finally, it offers practical criteria to verify (H) for unbounded operators, notably via a dominating operator with compact resolvent (e.g., the harmonic oscillator) and an explicit analysis for $M=-\partial_x^2+V$. This combination of theory and explicit examples demonstrates that second order spectra can achieve simultaneous non-pollution and accurate eigenvalue approximation in broad settings.

Abstract

M.Levitin and E.Shargorodsky purposed in a recent article, [math.SP/0212087], the use of the so called ``second order relative spectrum'', to find eigenvalues of self-adjoint operators in gaps of the essential spectrum. Let $M$ be a self-adjoint operator acting on the Hilbert space $H$. A complex number $z$ is in the second order spectrum of $M$ relative to a finite dimensional subspace $Λ\subset dom M^2$ if and only if the truncation to $Λ$ of $(M-z)^2$ is not invertible. It is remarkable that these sets seem to provide a general method to estimating eigenvalues free from the problems of spectral pollution present in most linear methods. In this notes we investigate rigorously various aspects of the use of second order spectrum to finding eigenvalues. Our main result shows that, under certain fairly mild hypothesis on $M$, the uniform limit of the second order spectra, as $Λ$ increases toward $H$, contains the isolated eigenvalues of $M$ of finite multiplicity. In applications the essential spectrum can be computed analytically, while precisely these eigenvalues are the ones that should be approximated numerically. Hence this method seems to combine non-pollution and approximation at a very high level of generality.

Limiting set of second order spectra

TL;DR

The paper develops and rigorously analyzes a second order spectrum method for detecting isolated eigenvalues of self-adjoint operators in gaps of the essential spectrum. By defining the second order spectrum via zeros of and its Lipschitz scalar surrogate , it constructs the limiting set and proves that, under the mild hypothesis (H), every isolated eigenvalue of finite multiplicity lies in , while the spectra avoid spurious pollution away from the real axis. The paper also provides a concrete model with as a piecewise-constant multiplier where , illustrating maximal pollution in linear Ritz methods and the non-polluting behavior of the second order approach. Finally, it offers practical criteria to verify (H) for unbounded operators, notably via a dominating operator with compact resolvent (e.g., the harmonic oscillator) and an explicit analysis for . This combination of theory and explicit examples demonstrates that second order spectra can achieve simultaneous non-pollution and accurate eigenvalue approximation in broad settings.

Abstract

M.Levitin and E.Shargorodsky purposed in a recent article, [math.SP/0212087], the use of the so called ``second order relative spectrum'', to find eigenvalues of self-adjoint operators in gaps of the essential spectrum. Let be a self-adjoint operator acting on the Hilbert space . A complex number is in the second order spectrum of relative to a finite dimensional subspace if and only if the truncation to of is not invertible. It is remarkable that these sets seem to provide a general method to estimating eigenvalues free from the problems of spectral pollution present in most linear methods. In this notes we investigate rigorously various aspects of the use of second order spectrum to finding eigenvalues. Our main result shows that, under certain fairly mild hypothesis on , the uniform limit of the second order spectra, as increases toward , contains the isolated eigenvalues of of finite multiplicity. In applications the essential spectrum can be computed analytically, while precisely these eigenvalues are the ones that should be approximated numerically. Hence this method seems to combine non-pollution and approximation at a very high level of generality.

Paper Structure

This paper contains 5 sections, 8 theorems, 81 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Second order spectrum and norm of inverses
  3. Proof of theorem 1
  4. An example of limiting set
  5. The condition (H)

Key Result

Theorem 1

If $\lambda$ is an isolated eigenvalue of finite multiplicity of $M$ and (H) holds, then $\lambda \in \Lambda$.

Figures (3)

  • Figure 1: $\mathrm{Spec}_2\,(M+K|\mathcal{L}_n)$ for three values of $n$. The operator corresponds to $a=1$, $\psi(x)=1$ and $E=(0,\pi]$. See table 1.
  • Figure 2: $\mathrm{Spec}_2\,(M+K|\mathcal{L}_n)$ for three values of $n$. The operator corresponds to $a=1$, $\psi(x)=1$ and $E=(-15\pi/16, \pi]$. See table 2.
  • Figure 3: Two zoom pictures of figure 2. Clustering near $\pm 1$. The star shows the location of the theoretical eigenvalue.

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 2
  • Proposition 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 8