相关论文: Solutions of Heun's general equation and elliptic …
We extend the reduction group method to the Lax-Darboux schemes associated with nonlinear Schr\"odinger type equations. We consider all possible finite reduction groups and construct corresponding Lax operators, Darboux transformations,…
Exceptional Laguerre-type differential expressions make up an infinite class of Schr\"odinger operators having rational potentials and one limit-circle endpoint. In this manuscript, the spectrum of all self-adjoint extensions for a general…
We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that…
We investigate expansive solutions of the $N$-body problem in $\mathbb{R}^d$ ($d\ge2$) driven by homogeneous Newtonian potentials of degree $-\alpha$. We establish the existence of half-entire expansive motions with prescribed initial…
By combining stability analysis of scalar field theories with the Darboux transformation technique, we create models featuring kink-like solutions whose quantum perturbations are all bounded. On the one hand, the stability analysis relates…
We derive 35 five-parametric classes of the quantum time-dependent two-state models solvable in terms of the general Heun functions. Each of the classes is defined by a pair of generating functions the first of which is referred to as the…
We study spherically symmetric static solutions of the most general sixth-derivative gravity using series expansions. Specifically, we prove that the only solutions of the complete theory (i.e., with generic coupling constants) that possess…
We consider a generalized Mathieu series where the summands of the classical Mathieu series are multiplied by powers of a complex number. The Mellin transform of this series can be expressed by the polylogarithm or the Hurwitz zeta…
On even-dimensional Euclidean space for integer powers of the Laplace operator greater than or equal to half the dimension, a fundamental solution of the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer polynomial…
We investigate the backward Darboux transformations (addition of a lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are…
We generalize a recently proposed small-energy expansion for one-dimensional quantum-mechanical models. The original approach was devised to treat symmetric potentials and here we show how to extend it to non-symmetric ones. Present…
We find two series expansions for Legendre's second incomplete elliptic integral $E(\lambda, k)$ in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the $(\lambda, k)$ plane.…
The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invarint solution for it are obtained by means of this technique. Polynomial, trigonometric and elliptic function solutions can be…
Firstly, we construct kernels of integral relations among solutions of the confluent Heun equation (CHE) and its limit, the reduced CHE (RCHE). In both cases we generate additional kernels by systematically applying substitutions of…
We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of…
The Fokker-Planck equation associated with the two - dimensional stationary Schr\"odinger equation has the conservation low form that yields a pair of potential equations. The special form of Darboux transformation of the potential…
By exploiting a recently developed connection between Heun's differential equation and the generalized associated Lam\'e equation, we not only recover the well known periodic solutions, but also obtain a large class of new, quasi-periodic…
The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques…
We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the…
Using the disconjugacy properties of the Schr\"odinger equation, it is possible to develop a new type of generalized SUSY QM partnership which allows to generate new solvable rational extensions for translationally shape invariant…