Norm resolvent convergence of singularly scaled Schr\"odinger operators and \delta'-potentials

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as \epsilon goes to 0, of a family S_\epsilon of one-dimensional Schr\"odinger operators on the line of the form S_\epsilon:= -D^2 + \epsilon^{-2} V(x/\epsilon). Under certain conditions the family of potentials converges in the sense of distributions to the first derivative of the Dirac delta-function, and then the limit of S_\epsilon might be considered as a "physically motivated" interpretation of the one-dimensional Schr\"odinger operator with potential \delta'.Comment: 30 pages, 2 figure; submitted to Proceedings of the Royal Society of Edinburg

Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra

We solve the inverse spectral problem of recovering the singular potentials $q\in W^{-1}_{2}(0,1)$ of Sturm-Liouville operators by two spectra. The reconstruction algorithm is presented and necessary and sufficient conditions on two sequences to be spectral data for Sturm-Liouville operators under consideration are given.Comment: 14 pgs, AmS-LaTex2

Inverse spectral problems for Sturm-Liouville operators with singular potentials, IV. Potentials in the Sobolev space scale

We solve the inverse spectral problems for the class of Sturm--Liouville operators with singular real-valued potentials from the Sobolev space W^{s-1}_2(0,1), s\in[0,1]. The potential is recovered from two spectra or from the spectrum and norming constants. Necessary and sufficient conditions on the spectral data to correspond to the potential in W^{s-1}_2(0,1) are established.Comment: 16 page

Fluctuations of the Phase Boundary in the Ising Ferromagnet

We discuss statistical properties of phase boundary in the 2D low-temperature Ising ferromagnet in a box with the two-component boundary conditions. We prove the weak convergence in C [O, 1] of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them. The limiting Gaussian measure coincides with the conditional distribution of certain Gaussian process obtained by the integral transformation of the white noise

Scattering theory for Schrödinger operators with Bessel-type potentials

We show that for the Schrödinger operators on the semi-axis with Bessel-type potentials κ(κ + 1)/x2, , there exists a meaningful direct and inverse scattering theory. Several new phenomena not observed in the "classical case” of Faddeev-Marchenko potentials arise here; in particular, for κ ≠ 0 the scattering function S takes two different values on the positive and negative semi-axes and is thus discontinuous both at the origin and at infinit

Reconstructing Jacobi Matrices from Three Spectra

Cut a Jacobi matrix into two pieces by removing the n-th column and n-th row. We give neccessary and sufficient conditions for the spectra of the original matrix plus the spectra of the two submatrices to uniqely determine the original matrix. Our result contains Hostadt's original result as a special case

Replacement of buffer gas with nitrogen in gas storage formations (models, methods, numerical experiments)

The paper gives description of the object of study - reservoir of the underground gas storage facility. A problem of replacement of buffer gas with nitrogen is raised and the problem formulations for its candidate solution are shown. A mathematical model of replacing buffer gas with nitrogen is proposed, which includes filtering model and convection model - diffusion of gases with concentrated sources. For the cases of unmixing gases the algorithm was developed for finding the propagation path of nitrogen. Numerical experiments were carried out

Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not assume the Markov property. Existing results that we generalize include a circle of ideas related to the Marcinkiewicz-Zygmund strong law of large numbers, as well as more recent work of Kesten and Maller. Our proofs are robust and use martingale methods. We demonstrate the benefit of the generality of our results by applications to some non-classical models, including random walks with heavy-tailed increments on two-dimensional strips, which include, for instance, certain generalized risk processes