scholarly journals Exact solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation

2020 ◽  
Vol 5 (1) ◽  
Author(s):  
Farrokh Atai ◽  
Edwin Langmann
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Schrodinger Equation ◽  
Kernel Functions ◽  
Integral Representations ◽  
Explicit Solutions ◽  
Jack Polynomials ◽  
Stationary Schrödinger Equation ◽  
Sutherland Model ◽  
Generalized Kernel

Abstract We use generalized kernel functions to construct explicit solutions by integrals of the non-stationary Schrödinger equation for the Hamiltonian of the elliptic Calogero–Sutherland model (also known as elliptic Knizhnik–Zamolodchikov–Bernard equation). Our solutions provide integral representations of elliptic generalizations of the Jack polynomials.

A METHOD OF GENERATING EXACT SOLUTIONS OF THE TIME-DEPENDENT SCHRÖDINGER EQUATION WITH TIME-DEPENDENT MASS

2002 ◽  
Vol 17 (24) ◽  
pp. 1567-1573
Author(s):  
AXEL SCHULZE-HALBERG
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Large Class ◽  
Power Law ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Stationary Schrödinger Equation ◽  
Dependent Mass

Extending the method presented in our previous paper,12 we map the time-dependent Schrödinger equation (TDSE) with time-dependent mass on a stationary Schrödinger equation for a nonconstant potential. On solving the latter, we can thus generate a large class of exact solutions of the original TDSE. Several examples are given, including potentials of power-law and modified Pöschl–Teller type.

scholarly journals Solving Integral Representations Problems for the Stationary Schrödinger Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yudong Ren
Keyword(s):  
Schrödinger Equation ◽  
Half Space ◽  
Harmonic Functions ◽  
Schrodinger Equation ◽  
Integral Representations ◽  
Representation Theorems ◽  
Stationary Schrödinger Equation

When solutions of the stationary Schrödinger equation in a half-space belong to the weighted Lebesgue classes, we give integral representations of them, which imply known representation theorems of classical harmonic functions in a half-space.

scholarly journals ON p – q DUALITY AND EXPLICIT SOLUTIONS IN c ≤ 1 2D GRAVITY MODELS

1995 ◽  
Vol 10 (08) ◽  
pp. 1219-1236 ◽  
Author(s):  
S. KHARCHEV ◽  
A. MARSHAKOV
Keyword(s):  
String Theory ◽  
Exact Solutions ◽  
2D Gravity ◽  
Integral Formula ◽  
Integral Representations ◽  
Gravity Models ◽  
Explicit Solutions ◽  
String Equation ◽  
Duality Transformation

We study the role of integral representations in the description of nonperturbative solutions to c ≤ 1 string theory. A generic solution is determined by two functions, W(x) and Q(x), which behave at infinity like xp and xq respectively. The integral formula for arbitrary (p, q) models is derived, which explicitly realizes a duality transformation between (p, q) and (q, p) 2D gravity solutions. We also discuss the exact solutions to the string equation and reduction condition and present several explicit examples.

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