Related papers: On the eigenvalues of the spheroidal wave equation
The coordinate asymptotics of the wave function for the problem of scattering of three particles with Coulomb interaction is constructed. Representation of hyperspherical functions is used to reduce the Schr\"odinger equation to a system of…
In this work, we follow the idea of the De Broglie's matter waves and the analogous method that Schr\"{o}dinger founded wave equation, but we apply the more essential Hamilton principle instead of the minimum action principle of Jacobi…
New travelling wave solutions to the Fornberg-Whitham equation are investigated. They are characterized by two parameters. The expresssions for the periodic and solitary wave solutions are obtained.
We study the wave equation on one-dimensional self-similar fractal structures that can be analyzed by the spectral decimation method. We develop efficient numerical approximation techniques and also provide uniform estimates obtained by…
This paper investigates oscillation-free stability conditions of numerical methods for linear parabolic partial differential equations with some example extrapolations to nonlinear equations. Not clearly understood, numerical oscillations…
We derive a spectral representation for the oblate spheroidal wave operator which is holomorphic in the aspherical parameter $\Omega$ in a neighborhood of the real line. For real $\Omega$, estimates are derived for all eigenvalue gaps…
We propose a new approach for modeling the quantum ring single particle energy spectrum. The approach is based on separation of variables in the Schr\"odinger equation in oblate spheroidal coordinates. We consider a model of a spheroidal…
We show that in quantum dots the physical quantities probed by local tunneling spectroscopies, namely the quasi-particle wavefunctions of interacting electrons, can considerably deviate from their single-particle counterparts as an effect…
The objective of this paper is to construct geometrically Riemann $k$-wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations. To this end, we adapt and combine elements of two…
We consider a planar waveguide with combined Dirichlet and Neumann conditions imposed in a "twisted" way. We study the discrete spectrum and describe it dependence on the configuration of the boundary conditions. In particular, we show that…
We study random Hamiltonians on finite-size cubes and waveguide segments of increasing diameter. The number of random parameters determining the operator is proportional to the volume of the cube. In the asymptotic regime where the cube…
We derive out a complete series expression of Hamiltonian eigenvalues without any approximation and cut in the general quantum systems based on Wang's formal framework \cite{wang1}. In particular, we then propose a calculating approach of…
We present a systematic account of the separation of the angular degrees of freedom from the nonrelativistic Schr\"{o}dinger equation for a three-body quantum system with arbitrary masses, charges, total angular momentum, and parity. The…
In this paper, we consider the resonance problem for the cubic nonlinear Helmholtz equation in the subwavelength regime. We derive a discrete model for approximating the subwavelength resonances of finite systems of high-contrast resonators…
In this paper we aim to give various explicit and local estimates of ball prolate spheroidal wave functions defined in [25] as eigenfunctions of both finite Fourier transform and some differential operator. In particular, we give further…
The present paper is a numerical study of the dynamics of solitary wave solutions of the fractional nonlinear Schr\"{o}dinger equation, whose existence was analyzed by the authors in the first part of the project. The computational study…
In this paper new criteria are established for the existence of positive radial solutions of a semilinear elliptic system depending on the gradient. These criteria are determined by some relationships between the upper and lower bounds on…
Using matrix identities, we construct explicit pseudo-exponential-type solutions of linear Dirac, Loewner and Schr\"odinger equations depending on two variables and of nonlinear wave equations depending on three variables.
We investigate the existence and the singular structure of delta wave solutions to a semilinear strictly hyperbolic equation with strongly singular initial and boundary conditions. The boundary conditions are given in nonlocal form with a…
We seek to introduce a mathematical method to derive the relativistic wave equations for two-particle system. According to this method, if we define stationary wave functions as special solutions like…