Dirk Hundertmark

Finite-Volume Fractional-Moment Criteria¶for Anderson Localization

Abstract: A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient “Lifshitz tail estimates” on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the “multiscale analysis”. In the converse direction, the analysis rules out fast power-law decay of the Green functions at mobility edges.

On sharp Strichartz inequalities in low dimensions

Recently Foschi gave a proof of a sharp Strichartz inequality in one and two dimensions. In this note, a new representation in terms of an orthogonal projection operator is obtained for the space time norm of solutions of the free Schrödinger equation in dimension one and two. As a consequence, the sharp Strichartz inequality follows from the elementary property that orthogonal projections do not increase the norm.

Lieb-Thirring Inequalities for Jacobi Matrices

For a Jacobi matrix J on ?2(Z+) with Ju(n)=an?1u(n?1)+bnu(n)+anu(n+1), we prove that??E?>2(E2?4)1/2??n?bn?+4?n?an?1?. We also prove bounds on higher moments and some related results in higher dimension.

A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator

We give a proof of the Lieb-Thirring inequality in the critical case d=1, γ = 1/2, which yields the best possible constant.

CONTINUITY PROPERTIES OF SCHRÖDINGER SEMIGROUPS WITH MAGNETIC FIELDS

The objects of the present study are one-parameter semigroups generated by Schrodinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schrodinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-continuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownian-bridge expectation. Altogether, the article is meant to extend some of the results in B. Simons landmark paper [Bull. Amer. Math. Soc.7 (1982) 447] to non-zero vector potentials and more general configuration spaces.

Bounds on the Spectral Shift Function and the Density of States

We study spectra of Schrödinger operators on ℝd. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values μn of the difference of the semigroups as n→∞ and deduce bounds on the spectral shift function of the pair of operators.Thereafter we consider alloy type random Schrödinger operators. The single site potential u is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be Hölder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Hölder continuity of the integrated density of states.

Variational estimates for discrete Schrödinger operators with potentials of indefinite sign

AbstractLet H be a one-dimensional discrete Schrödinger operator. We prove that if Σess(H)⊂[−2,2], then H−H0 is compact and Σess(H)=[−2,2]. We also prove that if

A unified theoretical framework for fluctuating-charge models in atom-space and in bond-space.

{{H_0 + \frac 14 V^2}}

Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators

has at least one bound state, then the same is true for H0+V. Further, if

An optimalL p -bound on the Krein spectral shift function

{{H_0 + \frac 14 V^2}}