We show that for a Jacobi operator with coefficients whose (j + 1)’th moments are summable the j’th derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve the known dispersive estimates with integrable time decay for the time dependent Jacobi equation in the resonant case.

We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the recently added condition (by P. Bálint, N. Chernov, D. Szász, and I. P. Tóth, in order to save this fundamental result) on the algebraic character of the smooth boundary components of the configuration space is unnecessary. Having saved the theorem in its or...

We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the recently added condition on the algebraic character of the smooth boundary components of the configuration space (by P. Bálint, N. Chernov, D. Szász, and I. P. Tóth) is unnecessary. Having saved the theorem in its original form by using additional ideas in t...

We obtain lower bounds on the ground state energy, in one and three dimensions, for the spinless Salpeter equation (Schrödinger equation with a relativistic kinetic energy operator) applicable to potentials for which the attractive parts are in L(R) for some p > n (n = 1 or 3). An extension to confining potentials, which are not in L(R), is also presented.

We study a Fermi Hamilton operator K̂ which does not commute with the number operator N̂ . The eigenvalue problem and the Schrödinger equation is solved. Entanglement is also discussed. Furthermore the Lie algebra generated by the two terms of the Hamilton operator is derived and the Lie algebra generated by the Hamilton operator and the number operator is also classified.

In our earlier paper ([AKT1]), by interpreting the formal transformation to the Airy equation near a simple turning point as the symbol of a microdifferential operator, we derived the Voros connection formula or, equivalently, the discontinuity function of a Borel transformed WKB solution at its movable singularities. In this paper we extend this approach to the two turning points problem; by c...

We shall study a weak solution in the Sobolev space of the transmission problem for the Laplace equation using the integral equation method. First we use the indirect integral equation method. We look for a solution in the form of the sum of the double layer potential corresponding to the skip of traces on the interface and a single layer potential with an unknown density. We get an integral eq...

We consider the problem of fitting a linear operator induced equation to point sampled data. In order to do so we systematically exploit the duality between minimizing a regularization functional derived from an operator and kernel regression methods. Standard machine learning model selection algorithms can then be interpreted as a search of the equation best fitting given data points. For many...

We analyze the extension of the well known relation between Brownian motion and Schrödinger equation to the family of Lévy processes. We propose a Lévy– Schrödinger equation where the usual kinetic energy operator – the Laplacian – is generalized by means of a pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy–Khintchin formula sh...

In this article we study the properties of the hyperinterpolation operator on the unit disk D in R. We show how the hyperinterpolation can be used in connection with the Kumar-Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disk (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove...