Related papers: Root asymptotics of spectral polynomials for the L…
We study asymptotic distribution of eigen-values $\omega$ of a quadratic operator polynomial of the following form $(\omega^2-L(\omega))\phi_\omega=0$, where $L(\omega)$ is a second order differential positive elliptic operator with…
It is shown that the regular-at-infinity solution of the 1D Schrodinger equation with the hyperbolic Poschl-Teller (h-PT) potential with integer parameters is expressible in terms of a n-order Heun polynomial in y=thr at an arbitrary…
I consider the power series expansion of Lame function in the Weierstrass's form and its integral forms applying three term recurrence formula[1]. I investigate asymptotic expansions of Lame function for the cases of infinite series and…
We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems…
We study the spectral polynomial of the Treibich-Verdier potential. Such spectral polynomial, which is a generalization of the classical Lame polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
We study an indefinite spectral problem for a second-order self-adjoint elliptic operator in an asymptotically thin cylinder. The operator coefficients and the spectral density function are assumed to be locally periodic in the axial…
In this paper, we derive sharp asymptotics for the spectral data (eigenvalues and weight numbers) of the fourth-order linear differential equation with a distribution coefficient and three types of separated boundary conditions. Our methods…
For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problem $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the…
Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the…
For integers $m\geq 3$, we study the non-self-adjoint eigenvalue problems $-u^{\prime\prime}(x)+(x^m+P(x))u(x)=E u(x)$, $0\leq x<+\infty$, with the boundary conditions $u(+\infty)=0$ and $\alpha u(0)+\beta u^{\prime}(0)=0$ for some $\alpha,…
In this paper the asymptotic distributions are exactly solved for linearly independent solutions considering problems of the second order and for the coefficients of asymptotic destribution the recurent formulas are obtained. Further, using…
We present asymptotically exact solutions of an incommensurate Harper equation---one-dimensional Schroedinger equation of one particle on a lattice in a cosine potential. The wave functions can be written as an infinite product of string…
The band structure of the Lam\'e equation, viewed as a one-dimensional Schr\"odinger equation with a periodic potential, is studied. At integer values of the degree parameter l, the dispersion relation is reduced to the l=1 dispersion…
We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly…
Small perturbations of the Jacobi matrix with weights \sqrt n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical…
Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\to\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and…
We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue…
We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends…
We first show the existence and nature of convergence to a limiting set of roots for polynomials in a three-term recurrence of the form $p_{n+1}(z) = Q_k(z)p_{n}(z)+ \gamma p_{n-1}(z)$ as $n$ $\rightarrow$ $\infty$, where the coefficient…