# zbMATH — the first resource for mathematics

Localization of surface spectra. (English) Zbl 0952.60059
Summary: We study spectral properties of the discrete Laplacian $$H$$ on the half-space $$\mathbb{Z}_+^{d+1}=\mathbb{Z}^d\times\mathbb{Z}_+$$ with random boundary condition $$\psi(n,-1)=\lambda V(n)\psi(n,0)$$; the $$V(n)$$ are independent random variables on a probability space $$(\Omega,{\mathcal F},P)$$ and $$\lambda$$ is the coupling constant. It is known that if the $$V(n)$$ have densities, then on the interval $$[-2(d+1)$$, $$2(d+1)] (=\sigma(H_0)$$, the spectrum of the Dirichlet Laplacian) the spectrum of $$H$$ is $$P$$-a.s. absolutely continuous for all $$\lambda$$ [see the first author and Y. Last, “Corrugated surfaces and a.c. spectrum” (subm. for publ.)]. Here we show that if the random potential $$V$$ satisfies the assumption of M. Aizenman and S. Molchanov [Commun. Math. Phys. 157, No. 2, 245-278 (1993; Zbl 0782.60044)], then there are constants $$\lambda_d$$ and $$\Lambda_d$$ such that for $$|\lambda|<\lambda_d$$ and $$|\lambda |>\Lambda_d$$ the spectrum of $$H$$ outside $$\sigma (H_0)$$ is $$P$$-a.s. pure point with exponentially decaying eigenfunctions.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34L05 General spectral theory of ordinary differential operators
Full Text: