Related papers: On Psi-function for finite-gap potentials
An expression of the multivariate sigma function associated with a (n,s)-curve is given in terms of algebraic integrals. As a corollary the first term of the series expansion around the origin of the sigma function is directly proved to be…
We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local…
Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the…
Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields.
We construct finite-gap solutions to the modified Novikov-Veselov equations, describe their spectral properties and the reduction to the modified Korteweg--de Vries equation and explain its relation to soliton deformations of tori and the…
We extend the notion of some energy-type expressions based on two sets, developed in the abstract potential theory. We also give the discretized version of the quantities defined, similar to Chebyshev constant. This extension allows to…
In the framework of differential Galois theory we treat the classical spectral problem $\Psi"-u(x)\Psi=\lambda\Psi$ and its finite-gap potentials as exactly solvable in quadratures by Picard--Vessiot without involving special functions; the…
Reciprocal space methods for solving Poisson's equation for finite charge distributions are investigated. Improvements to previous proposals are presented, and their performance is compared in the context of a real-space density functional…
In this paper we find explicit conditions on the periodic PT-symmetric complex-valued potential q for which the number of gaps in the real part of the spectrum of the one-dimensional Schrodinger operator L(q) is finite.
A systematic procedure to derive exact solutions of the associated Lame equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are…
Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…
We apply a method recently devised by one of the authors to obtain an approximate analytical formula for the spectrum of a quantum anharmonic potential. Due to its general features the method can be applied with minimal effort to general…
We propose in this paper to study the solutions of some nonlinear elliptic equations with singular potential.
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are P-functions. Some applications to special means of real…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
In this paper we show that the one-dimensional Schr\"odinger equation can be viewed as the geodesic equation of a certain metric on a 2-surface. In case of the Schr\"odinger equation with a finite-gap potential, the metric and geodesics are…
We obtained a new class of exactly-solvable potentials by means of the hypergeometric equation for Schrodinger equation, which different from the exactly-solvable potentials introduced by Bose and Natanzon. Using the new class of solvable…
We consider both finite and infinite power chi expansions of $f$-divergences derived from Taylor's expansions of smooth generators, and elaborate on cases where these expansions yield closed-form formula, bounded approximations, or analytic…
A new expansion for integral powers of the hypergeometric function corresponding to a special case of the incomplete beta function is summarized, and consequences, including two new sums involving digamma (psi) functions are presented.
We obtain pointwise estimates for solutions of semilinear parabolic equations with a potential on connected domains both of $\mathbb R^n$ and of general Riemannian manifolds.