Related papers: On Psi-function for finite-gap potentials
Motivated by the interest in non-relativistic quantum mechanics for determining exact solutions to the Schrodinger equation we give two potentials that are conditionally exactly solvable. The two potentials are partner potentials and we…
We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive…
Let $T\colon X\to X$ be a bounded operator on Banach space, whose spectrum $\sigma(T)$ is included in the closed unit disc $\overline{\mathbb D}$. Assume that the peripheral spectrum $\sigma(T)\cap{\mathbb T}$ is finite and that $T$…
A set of exactly solvable one-dimensional quantum mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and P\"oschl-Teller potentials.…
A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. We extend the addition formulae for sigma functions…
This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function…
We present a method for constructing global analytical expressions that approximate a function over its entire range. These approximations not only mirror the original function as accurately as desired, but are purposefully created to…
In this paper we define the polygamma functions $\psi^{(n)}(x)$ for negative integers by using neutrix calculus.
We present a conditionally exactly solvable singular potential for the one-dimensional Schr\"odinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general…
On compact Riemannian manifolds, we prove a decomposition theorem for arbitrarily bounded energy sequence of solutions of a singular elliptic equation.
In this article we first establish the maximum principle of the antisymmetric functions for parabolic fractional $p$-equations. Then we use it and the parabolic inequalities to provide a different proof of symmetry and monotonicity for…
We show that electronic wave functions Psi of atoms and molecules have a representation Psi=F*phi, where F is an explicit universal factor, locally Lipschitz, and independent of the eigenvalue and the solution Psi itself, and phi has…
Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the…
A general procedure is presented to construct conditionally solvable (CES) potentials using the techniques of supersymmetric quantum mechanics.The method is illustrated with potentials related to the harmonic oscillator problem.Besides…
The cubic H\'enon-Heiles system contains parameters, for most values of which, the system is not integrable. In such parameter regimes, the general solution is expressible in formal expansions about arbitrary movable branch points, the…
We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…
This work presents problems of constructing finite-difference formulas in the Hilbert space, i.e., setting problems of constructing finite-difference formulas using functional methods. The work presents a functional statement of the problem…
In this paper we study some properties of $\psi$-Hilfer fractional integrodifferential equations. We obtain the existence and uniqueness and other properties such as continuous dependence of solution. The tools used for obtaining our result…
We consider an overdetermined problem arising in potential theory for the capacitary potential and we prove a radial symmetry result.
In this paper, we summarize the technique of using Green functions to solve electrostatic problems. We start by deriving the electric potential in terms of a Green function and a charge distribution. We then provide a variety of example…