On the spectrum of a discrete Laplacian on ℤ with finitely supported potential

Higuchi, Yusuke; Matsumoto, Tomonori; Ogurisu, Osamu

doi:10.1080/03081087.2010.536981

Cited by 8 publications

(6 citation statements)

References 7 publications

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“…When n < ∞, H is a finite rank perturbation of g. Then the absolutely continuous spectrum and the essential spectrum of H are [−1, 1]. In this case the discrete spectrum is studied in e.g., [HMO11] for d = 1. See also [DKS05].…”

Section: Discussionmentioning

confidence: 99%

Note on the spectrum of discrete Schrödinger operators

Hiroshima,

Sasaki,

Shirai

et al. 2012

Preprint

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Section: Discussionmentioning

confidence: 99%

Note on the spectrum of discrete Schrödinger operators

Hiroshima,

Sasaki,

Shirai

et al. 2012

Preprint

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“…Two of the authors gave explicit expressions for N ± (L) on Z 1 in [5]. H. Isozaki and H. Morioka proved that L has no eigenvalue in (−1, 1) on Z d in [7].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Number of Discrete Eigenvalues of a Discrete Schrödinger Operator with a Finitely Supported Potential

Hayashi

Higuchi

Nomura³

et al. 2016

Lett Math Phys

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On the d-dimensional lattice Z d and the r-regular tree T r , an exact expression for the number of discrete eigenvalues of a discrete Laplacian with a finitely supported potential is described in terms of the support and the intensities of the potential on each case. In particular, the number of eigenvalues less than the infimum of the essential spectrum is bounded by the number of negative intensities.Mathematics Subject Classifications. 39A12, 47A75, 34L40.

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“…(1. 8) In [6] we showed that λ 0 (0) = E(ξ(0, 0)) and λ 0 (κ) > E(ξ(0, 0)) for κ ∈ (0, ∞) as soon as the limit in (1.8) exists. In [3] we proved the following:…”

Section: Parabolic Anderson Modelmentioning

confidence: 97%

The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent

Erhard

Hollander

Maillard

2014

Probab. Theory Relat. Fields

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35 pages, 4 figures.International audienceWe continue our study of the parabolic Anderson equation $\partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t)$, $x\in\Z^d$, $t\geq 0$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, and $\xi$ plays the role of a \emph{dynamic random environment} that drives the equation. The initial condition $u(x,0)=u_0(x)$, $x\in\Z^d$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2d\kappa$, split into two at rate $\xi \vee 0$, and die at rate $(-\xi) \vee 0$. We assume that $\xi$ is stationary and ergodic under translations in space and time, is not constant and satisfies $\E(|\xi(0,0)|)<\infty$, where $\E$ denotes expectation w.r.t.\ $\xi$. Our main object of interest is the quenched Lyapunov exponent $\lambda_0 (\kappa) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t)$. In earlier work we showed that under certain mild space-time mixing assumptions the limit exists $\xi$-a.s., is finite and continuous on $[0,\infty)$, is globally Lipschitz on $(0,\infty)$, is not Lipschitz at 0, and satisfies $\lambda_0(0) = \E(\xi(0,0))$ and $\lambda_0(\kappa) > \E(\xi(0,0))$ for $\kappa \in (0,\infty)$.In the present paper we show that $\lim_{\kappa\to\infty} \lambda_0(\kappa) =\E(\xi(0,0))$ under an additional space-time mixing condition on $\xi$. This result shows that the parabolic Anderson model exhibits space-time ergodicity in the limit of large diffusivity. This fact is interesting because there are choices of $\xi$ that fulfill our assumption for which the annealed Lyapunov exponent $\lambda_1(\kappa) = \lim_{t\to\infty} \frac{1}{t}\log \E(u(0,t))$ is infinite on $[0,\infty)$, a situation that is referred to as strongly catalytic behavior

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On the spectrum of a discrete Laplacian on ℤ with finitely supported potential

Cited by 8 publications

References 7 publications

Note on the spectrum of discrete Schrödinger operators

Note on the spectrum of discrete Schrödinger operators

On the Number of Discrete Eigenvalues of a Discrete Schrödinger Operator with a Finitely Supported Potential

The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent

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