arXiv subjects
Publications and source records attributed to Baruch Schneider.
For a two-dimensional curved waveguide, it is well known that the spectrum of the Dirichlet Laplacian is unstable. Any perturbation of the straight strip produces eigenvalues below the essential spectrum. In this paper, a magnetic field is added. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small but non-local deformations of the waveguide.
In this paper we derive Lieb-Thirring estimates for eigenvalues of Dirichlet Laplacians below the threshold of the essential spectrum on asymptotically Archimedean spiral-shaped regions.
In this paper we introduce the concept of matrix-valued $q$-rational functions. In comparison to the classic case we give different characterizations with principal emphasise on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna Pick kernels in this context and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna Pick kernels.
This short report reviews the current state of the research and methodology on theoretical and practical aspects of Artificial Neural Networks (ANN). It was prepared to gather state-of-the-art knowledge needed to construct complex, hypercomplex and fuzzy neural networks. The report reflects the individual interests of the authors and, by now means, cannot be treated as a comprehensive review of the ANN discipline. Considering the fast development of this field, it is currently impossible to do a detailed review of a considerable number of pages. The report is an outcome of the Project 'The Strategic Research Partnership for the mathematical aspects of complex, hypercomplex and fuzzy neural networks' meeting at the University of Warmia and Mazury in Olsztyn, Poland, organized in September 2022.
Using $q$-calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their $q$-calculus counterparts. Furthermore, these new spaces allow us to study intertwining operators between classic backward-shift operators and the q-Jackson derivative.
We consider the magnetic Schrödinger operator $H=(i \nabla +A)^2- V$ with a non-negative potential $V$ supported over a strip which is a local deformation of a straight one, and the magnetic field $B:=\mathrm{rot}(A)$ is assumed to be nonzero and local. We show that the magnetic field does not change the essential spectrum of this system, and investigate a sufficient condition for the discrete spectrum of $H$ to be empty.
In this paper we study a bounded domain with a small hole removed. Our main result concerns the spectrum of the Laplace operator with the Robin conditions imposed at the hole boundary. Moreover we prove that under some suitable assumptions on the parameter in the boundary condition the spectrum of the Laplacian converges in the Hausdorff distance sense to the spectrum of the Laplacian defined on the unperturbed domain.
This article gives a domain with a small compact set of removed and the magnetic Neumann Laplacian on such set. The main theorem of this article shows the description of the holes which do not change the spectrum drastically. In this article we prove that the spectrum of the magnetic Neumann Laplacian converges in the Hausdorff distance sense to the spectrum of the original operator defined on the unperturbed domain.
We consider the extension of the Jackson calculus into higher dimensions and specifically into Clifford analysis.
We consider a domain with a small compact set of zero Lebesgue measure of removed. Our main result concerns the spectrum of the Neumann Laplacian defined on such domain. We prove that the spectrum of the Laplacian converges in the Hausdorff distance sense to the spectrum of the Laplacian defined on the unperturbed domain.
In this paper we consider magnetic Schroedinger operators on the two-dimensional unit disk with a radially symmetric magnetic field which explodes to infinity at the boundary. We prove a bound for the eigenvalue moments and a bound for the number of negative eigenvalues for such operators.
In this article, we present the asymptotic solution for the matrix system of equations representing the multiple scattering coefficients of an infinite grating of insulating dielectric circular cylinders associated with vertically polarized obliquely incident plane electromagnetic waves.
In this article, we present the derivation of the asymptotic forms of the equations corresponding to the scattering coefficients of the exterior electric and magnetic fields of an infinite grating of insulating dielectric circular cylinders for vertically polarized and obliquely incident plane electromagnetic waves. Exploiting the generalized forms of the Twersky's elementary function representations for Schloemilch series, we have deducted an Ansatz describing the behavior of the scattering coefficients of the electric and magnetic fields for obliquely incident waves when the grating spacing is much smaller than the wavelength of the incident electromagnetic radiation. Introducing the statement of this Ansatz into the equations of the scattering coefficients of the infinite grating at oblique incidence, and expanding the scattering coefficients in the form of an asymptotic series as a function of the ratio of the radius of the cylinders to the grating spacing, we have acquired two new infinite sets of algebraic equations associated with the scattering coefficients of the exterior electric and magnetic fields of the grating for vertically polarized and obliquely incident plane waves.
A computational algorithm for the exact equations representing the scattering coefficients of an infinite grating of insulating dielectric circular cylinders associated with obliquely incident vertically polarized plane electromagnetic waves is generated by matrix methods, and the solution for the scattering coefficients is acquired by a matrix inversion procedure.
A rigorous analytical representation for the multiple scattering coefficients of the fields radiated by an infinite grating of dielectric circular cylinders excited by an obliquely incident plane electromagnetic wave is derived in terms of the "well-known scattering coefficients of an isolated dielectric cylinder at oblique incidence" and "Schlömilch series". In addition, a generalized sum-integral grating equation is acquired for the multiple scattered amplitude of a cylinder at oblique incidence in the grating in terms of the scattering coefficients of the insulating dielectric circular cylinder at oblique incidence.