Spectral properties of the biconfluent Heun differential equation

Biconfluent Heun equation in quantum chemistry: Harmonium and related systems

“…On one hand, the existence of simple polynomial solutions of this equation, discovered in connection with studies on electron correlation, motivated numerous works in the community of quantum chemists [18][19][20][21][22][23][24][25][26][27][28]. However, this equation is known in mathematics since more than a century as the biconfluent Heun equation and its properties were studied from both purely mathematical perspective [29,30,[35][36][37][38] and in the context of its applications in different areas of physics [39][40][41]. Very recently a brief review on its physical applications has been published by Hortaçsu [42].…”

“…As it was already mentioned, the family of the Heun equations, including the BHE, was studied by the mathematicians since more than a century [29,30,[35][36][37][38]. A very rich bibliography of the texts published on the Heun functions throughout the years has been collected in the framework of The Heun Project: Heun functions, their generalizations and applications created at the University of Sofia [49].…”

Untitled

“…In this section we will consider some particular cases when (x) is some of the classical weight functions, i.e., the Jacobi, Laguerre, Hermite or Gegenbauer weight functions, respectively. Moreover, since in expressions (8) and (9) the kernel polynomials Ker n 1 (x; x i ) appear we will consider the case when we add some delta Dirac masses at the origin x = 0 or at the ends of the interval of orthogonality of the classical polynomials. The last consideration allows us to obtain explicit formulas for the kernel polynomials in terms of the classical polynomials and their derivatives 5], 6].…”

The energy levels and eigen wave functions of electrons in quantum rings in the magnetic field

“…Для того чтобы волновая функ-ция обращалась в ноль на бесконечности, ограниченное в нуле частное ре-шение уравнения Хойна (7) должно представлять собой полином. Решения такого вида существуют, если параметры функции HB(α, β, γ, δ, x) удовлетво-ряют двум соотношениям [6]:…”