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Register a new userEigenfunctions localised on a defect in high-contrast random media
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We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrum of a high-contrast random operator. We consider a family of elliptic operators $mathcal{A}^varepsilon$ in divergence form whose coefficients possess double porosity type scaling and are perturbed on a fixed-size compact domain. The coefficients of $mathcal{A}^varepsilon$ are random variables generated, in an appropriate sense, by an ergodic dynamical system. Working in the gaps of the limiting spectrum of the unperturbed operator $widehat{mathcal{A}}^varepsilon$, we show that the point spectrum of $mathcal{A}^epsilon$ converges in the sense of Hausdorff to the point spectrum of the homogenised operator $mathcal{A}^mathrm{hom}$ as $varepsilon to 0$. Furthermore, we prove that the eigenfunctions of $mathcal{A}^varepsilon$ decay exponentially at infinity uniformly for sufficiently small $varepsilon$. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of $mathcal{A}^mathrm{hom}$.
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We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we present a mathematically precise formulation of Berrys conjecture for a compact negatively curved manifold and formulate a Berry-type conjecture for sequences of locally symmetric spaces. We prove some we
We develop a Hilbert-space approach to the diffusion process of the Brownian motion in a bounded domain with random jumps from the boundary introduced by Ben-Ari and Pinsky in 2007. The generator of the process is introduced by a diffusion elliptic differential operator in the space of square-integrable functions, subject to non-self-adjoint and non-local boundary conditions expressed through a probability measure on the domain. We obtain an expression for the difference between the resolvent of the operator and that of its Dirichlet realization. We prove that the numerical range is the whole complex plane, despite the fact that the spectrum is purely discrete and is contained in a half-plane. Furthermore, for the class of absolutely continuous probability measures with square-integrable densities we characterise the adjoint operator and prove that the system of root vectors is complete. Finally, under certain assumptions on the densities, we obtain enclosures for the non-real spectrum and find a sufficient condition for the non-zero eigenvalue with the smallest real part to be real. The latter supports the conjecture of Ben-Ari and Pinsky that this eigenvalue is always real.
Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices are established. Two cases are considered separately: (i) the case in which the spectral parameter lies in a general gap of the spectrum of the Jacobi matrix and (ii) the case of a lower semi-bounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp. We apply these results to obtain a many barriers-type criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobi matrix at almost every value of the spectral parameter in suitable open sets. As an application, we provide examples of unbounded Jacobi matrices with a spectral mobility edge.
We consider the self-adjoint Schrodinger operator in $L^2(mathbb{R}^d)$, $dgeq 2$, with a $delta$-potential supported on a hyperplane $Sigmasubseteqmathbb{R}^d$ of strength $alpha=alpha_0+alpha_1$, where $alpha_0inmathbb{R}$ is a constant and $alpha_1in L^p(Sigma)$ is a nonnegative function. As the main result, we prove that the lowest spectral point of this operator is not smaller than that of the same operator with potential strength $alpha_0+alpha_1^*$, where $alpha_1^*$ is the symmetric decreasing rearrangement of $alpha_1$. The proof relies on the Birman-Schwinger principle and the reduction to an analogue of the P{o}lya-SzegH{o} inequality for the relativistic kinetic energy in $mathbb{R}^{d-1}$.
In this note the three dimensional Dirac operator $A_m$ with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that $A_m$ is self-adjoint in $L^2(Omega;mathbb{C}^4)$ for any open set $Omega subset mathbb{R}^3$ and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in $Omega$. In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of $A_m$ consists of discrete eigenvalues that accumulate at $pm infty$ and one additional eigenvalue of infinite multiplicity.
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