On localization of pseudo-relativistic energy

We present a Kato-type inequality for bounded domain Omega \subset R^n, n>1.Comment: 17 page

Sharp two-sided heat kernel estimates for critical Schr\"odinger operators on bounded domains

On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odinger operators -\Delta -V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Scr\"odinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a serier of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincar\'e. As a byproduct of our technique we are able to answer positively to a conjecture of E.B.Davies.Comment: 40 page

Semi-classical analysis of non self-adjoint transfer matrices in statistical mechanics. I

We propose a way to study one-dimensional statistical mechanics models with complex-valued action using transfer operators. The argument consists of two steps. First, the contour of integration is deformed so that the associated transfer operator is a perturbation of a normal one. Then the transfer operator is studied using methods of semi-classical analysis. In this paper we concentrate on the second step, the main technical result being a semi-classical estimate for powers of an integral operator which is approximately normal.Comment: 28 pp, improved the presentatio

Heat Kernel Bounds for the Laplacian on Metric Graphs of Polygonal Tilings

We obtain an upper heat kernel bound for the Laplacian on metric graphs arising as one skeletons of certain polygonal tilings of the plane, which reflects the one dimensional as well as the two dimensional nature of these graphs.Comment: 8 page

Quantum Mechanics as a Simple Generalization of Classical Mechanics

A motivation is given for expressing classical mechanics in terms of diagonal projection matrices and diagonal density matrices. Then quantum mechanics is seen to be a simple generalization in which one replaces the diagonal real matrices with suitable Hermitian matrices.Comment: 9 pages, LaTe

Heat kernel bounds on manifolds with cusps

AbstractWe describe a method of obtaining pointwise upper bounds for the heat kernel of a Riemannian manifold with cusps. We apply our results to a class of approximately hyperbolic manifolds, by which we mean manifolds which have bounded geometry with respect to a hyperbolic structure with cusps (these manifolds include all asymptotically hyperbolic manifolds). For such manifolds we obtain upper bounds on the heat kernels which we believe to be nearly optimal

Wave Packet Pseudomodes of Variable Coefficient Differential Operators

The pseudospectra of nonselfadjoint linear ordinary differential operators with variable coefficients are considered. It is shown that when a certain winding number or twist condition is satisfied, closely related to Hörmander's commutator condition for partial differential equations, \varepsilon-pseudoeigenfunctions of such operators for exponentially small values of \varepsilon exist in the form of localized wave packets. In contrast to related results of Davies and of Dencker, Sjöstrand, and Zworski, the symbol need not be smooth

Robustness of adiabatic quantum computation

We study the fault tolerance of quantum computation by adiabatic evolution, a quantum algorithm for solving various combinatorial search problems. We describe an inherent robustness of adiabatic computation against two kinds of errors, unitary control errors and decoherence, and we study this robustness using numerical simulations of the algorithm.Comment: 11 pages, 5 figures, REVTe

Spectral gap of segments of periodic waveguides

We consider a periodic strip in the plane and the associated quantum waveguide with Dirichlet boundary conditions. We analyse finite segments of the waveguide consisting of $L$ periodicity cells, equipped with periodic boundary conditions at the ``new'' boundaries. Our main result is that the distance between the first and second eigenvalue of such a finite segment behaves like $L^{-2}$.Comment: 3 page