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We consider the problem of how to compute eigenvalues of a self-adjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic methods for addressing this problem. Recently a new perturbation approach has emerged, the idea being to perturb eigenvalues off the real line and, consequently, away from regions where the Galerkin method fails. We propose a simplified perturbation method which requires no a priori information and for which we provide a rigorous convergence analysis. The latter shows that, in general, our approach will significantly outperform the quadratic methods. We also present a new spectral enclosure for operators of the form $A+iB$ where $A$ is self-adjoint, $B$ is self-adjoint and bounded. This enables us to control, very precisely, how eigenvalues are perturbed from the real line. The main results are demonstrated with examples including magnetohydrodynamics, Schrodinger and Dirac operators.
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Let $Gamma$ be an arbitrary $mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $mathcal{H}_varepsilon$ on $Gamma$ with the action $-varepsilon^{-1}{mathrm{d}^2/mathrm{d} x^2}$ on its edges; here $varepsilon>0$ is a small parameter. Let $minmathbb{N}$. We show that under a proper choice of vertex conditions the spectrum $sigma(mathcal{H}^varepsilon)$ of $mathcal{H}^varepsilon$ has at least $m$ gaps as $varepsilon$ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of $sigma(mathcal{H}^varepsilon)$ as $varepsilonto 0$ can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) $varepsilon$ the precise coincidence of the left endpoints of the first $m$ spectral gaps with predefined numbers.
The spectral gap of the graph Laplacian with Dirichlet boundary conditions is computed for the graphs of several communication networks at the IP-layer, which are subgraphs of the much larger global IP-layer network. We show that the Dirichlet spectral gap of these networks is substantially larger than the standard spectral gap and is likely to remain non-zero in the infinite graph limit. We first prove this result for finite regular trees, and show that the Dirichlet spectral gap in the infinite tree limit converges to the spectral gap of the infinite tree. We also perform Dirichlet spectral clustering on the IP-layer networks and show that it often yields cuts near the network core that create genuine single-component clusters. This is much better than traditional spectral clustering where several disjoint fragments near the periphery are liable to be misleadingly classified as a single cluster. Spectral clustering is often used to identify bottlenecks or congestion; since congestion in these networks is known to peak at the core, our results suggest that Dirichlet spectral clustering may be better at finding bona-fide bottlenecks.
We consider the spectrum of the almost Mathieu operator $H_alpha$ with frequency $alpha$ and in the case of the critical coupling. Let an irrational $alpha$ be such that $|alpha-p_n/q_n|<c q_n^{-varkappa}$, where $p_n/q_n$, $n=1,2,dots$ are the convergents to $alpha$, and $c$, $varkappa$ are positive absolute constants, $varkappa<56$. Assuming certain conditions on the parity of the coefficients of the continued fraction of $alpha$, we show that the central gaps of $H_{p_n/q_n}$, $n=1,2,dots$, are inherited as spectral gaps of $H_alpha$ of length at least $cq_n^{-varkappa/2}$, $c>0$.
This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with ``Dirichlet boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szego-Weinberger, Payne-Polya-Weinberger, Sperner, Hile-Protter, and H. C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, S^n or H^n.
We prove sharp lower bounds on the spectral gap of 1-dimensional Schrodinger operators with Robin boundary conditions for each value of the Robin parameter. In particular, our lower bounds apply to single-well potentials with a centered transition point. This result extends work of Cheng et al. and Horvath in the Neumann and Dirichlet endpoint cases to the interpolating regime. We also build on recent work by Andrews, Clutterbuck, and Hauer in the case of convex and symmetric single-well potentials. In particular, we show the spectral gap is an increasing function of the Robin parameter for symmetric potentials.
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