New bounds on the Lieb-Thirring constants

Improved estimates on the constants $L_{\gamma,d}$, for $1/2<\gamma<3/2$, $d\in N$ in the inequalities for the eigenvalue moments of Schr\"{o}dinger operators are established

Eigenvalue Bounds for Perturbations of Schrodinger Operators and Jacobi Matrices With Regular Ground States

We prove general comparison theorems for eigenvalues of perturbed Schrodinger operators that allow proof of Lieb--Thirring bounds for suitable non-free Schrodinger operators and Jacobi matrices.Comment: 11 page

Weighted Supermembrane Toy Model

A weighted Hilbert space approach to the study of zero-energy states of supersymmetric matrix models is introduced. Applied to a related but technically simpler model, it is shown that the spectrum of the corresponding weighted Hamiltonian simplifies to become purely discrete for sufficient weights. This follows from a bound for the number of negative eigenvalues of an associated matrix-valued Schr\"odinger operator.Comment: 18 pages, 2 figures; to appear in Lett. Math. Phys

Lieb-Thirring inequalities for geometrically induced bound states

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schr\"odinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider, in particular, a straight strip and a straight circular tube with either mixed boundary conditions or boundary deformations.Comment: LaTeX2e, 14 page

LpL^p-approximation of the integrated density of states for Schr\"odinger operators with finite local complexity

We study spectral properties of Schr\"odinger operators on \RR^d. The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in \ZZ^d, with the property that frequencies of finite patterns are well defined. We prove that the integrated density of states (spectral distribution function) is approximated by its finite volume analogues, i.e.the normalised eigenvalue counting functions. The convergence holds in the space $L^p(I)$ where $I$ is any finite energy interval and $1\leq p< \infty$ is arbitrary.Comment: 15 pages; v2 has minor fixe

Localization criteria for Anderson models on locally finite graphs

We prove spectral and dynamical localization for Anderson models on locally finite graphs using the fractional moment method. Our theorems extend earlier results on localization for the Anderson model on \ZZ^d. We establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps

We prove, via an elementary variational method, 1d and 2d localization within the band gaps of a periodic Schrodinger operator for any mostly negative or mostly positive defect potential, V, whose depth is not too great compared to the size of the gap. In a similar way, we also prove sufficient conditions for 1d and 2d localization below the ground state of such an operator. Furthermore, we extend our results to 1d and 2d localization in d dimensions; for example, a linear or planar defect in a 3d crystal. For the case of D-fold degenerate band edges, we also give sufficient conditions for localization of up to D states.Comment: 9 pages, 3 figure

A new numerical approach to Anderson (de)localization

We develop a new approach for the Anderson localization problem. The implementation of this method yields strong numerical evidence leading to a (surprising to many) conjecture: The two dimensional discrete random Schroedinger operator with small disorder allows states that are dynamically delocalized with positive probability. This approach is based on a recent result by Abakumov-Liaw-Poltoratski which is rooted in the study of spectral behavior under rank-one perturbations, and states that every non-zero vector is almost surely cyclic for the singular part of the operator. The numerical work presented is rather simplistic compared to other numerical approaches in the field. Further, this method eliminates effects due to boundary conditions. While we carried out the numerical experiment almost exclusively in the case of the two dimensional discrete random Schroedinger operator, we include the setup for the general class of Anderson models called Anderson-type Hamiltonians. We track the location of the energy when a wave packet initially located at the origin is evolved according to the discrete random Schroedinger operator. This method does not provide new insight on the energy regimes for which diffusion occurs.Comment: 15 pages, 8 figure

On the negative spectrum of two-dimensional Schr\"odinger operators with radial potentials

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V$ with the radial potential $V(x)=F(|x|), F(r)\ge 0$, we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues, as the coupling parameter $\alpha$ tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth $N_-(H_{\alpha V})=O(\alpha)$ and for the validity of the Weyl asymptotic law.Comment: 13 page