Related papers: Asymptotic approximation of the eigenvalues and th…
The WKB approximation plays an essential role in the development of quantum mechanics and various important results have been obtained from it. In this paper, we introduce another method, {\it the so-called uniform asymptotic…
The model for Orr--Sommerfeld equation with quadratic profile on the finite interval is considered. The behavior of the spectrum of this problem is completely investigated for large Reynolds numbers. The limit curves are found to which the…
The Orr-Sommerfeld equation with linear profile on the finite interval is considered. The behavior of the spectrum of this problem is completely investigated for large Reynolds numbers. The limit curves are found to which the eigenvalues…
Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the WKB semiclassical expansion for the eigenvalues. However, all examples studied to date have…
We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies…
A method for solving the half-space Sommerfeld problem is proposed, which allows us to obtain exact solutions in the form of Sommerfeld integrals, as well as their short-wave asymptotics. The first carried out by reducing the Sommerfeld…
We present a method to calculate the asymptotic behavior of eigenfunctions of Schr\"odinger operators that also works at the threshold of the essential spectrum. It can be viewed as a higher order correction to the well-known WKB method…
The auxiliary field method is a powerful technique to obtain approximate closed-form energy formulas for eigenequations in quantum mechanics. Very good results can be obtained for Schr\"odinger and semirelativistic Hamiltonians with various…
A new approximation scheme to the centrifugal term is proposed to obtain the $l\neq 0$ bound-state solutions of the Schr\"{o}dinger equation for an exponential-type potential in the framework of the hypergeometric method. The corresponding…
Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by…
The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward…
A spectral problem is considered in a thin $3D$ graph-like junction that consists of three thin curvilinear cylinders that are joined through a domain (node) of the diameter $\mathcal{O}(\varepsilon),$ where $\varepsilon$ is a small…
The WKB approximation for deformed space with minimal length is considered. The Bohr-Sommerfeld quantization rule is obtained. A new interesting feature in presence of deformation is that the WKB approximation is valid for intermediate…
A classical problem in acoustic (and electromagnetic) scattering concerns the evaluation of the Green's function for the Helmholtz equation subject to impedance boundary conditions on a half-space. The two principal approaches used for…
In this paper, we revisit the eigenvalue problem of the one-dimensional Schr{\"o}dinger equation for smooth single well potentials. In particular, we provide a new interpretation of the Bohr-Sommerfeld quantization formula. A novel aspect…
An approach is developed for constructing simple analytical formulae accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar approximation theory. In order to derive…
Coulomb wave functions are difficult to compute numerically for extremely low energies, even with direct numerical integration. Hence, it is more convenient to use asymptotic formulas in this region. It is the object of this paper to derive…
Uniform asymptotic approximations are obtained for the prolate spheroidal wave functions, in the high-frequency case. The results are obtained by an application of certain existing asymptotic solutions of differential equations, and involve…
We consider the asymptotic behaviour of small-amplitude gravity water waves in a rectangular domain where the water depth is much smaller than the horizontal scale. The control acts on one lateral boundary, by imposing the horizontal…
An appropriate rational approximation to the eigenfunction of the Schr\"{o}dinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of Hankel determinants. The…