THE HEUN–SCHRÖDINGER RADIAL EQUATION WITH TWO AUXILIARY PARAMETERS FOR H-LIKE ATOMS

2002 ◽  
Vol 16 (25) ◽  
pp. 937-941 ◽  
Author(s):  
V. F. TARASOV
Keyword(s):  
Fine Structure ◽  
Coulomb Potential ◽  
Radial Function ◽  
Standard Form ◽  
Nuclear Motion ◽  
Heun Equation ◽  
Radial Equation ◽  
Schrödinger’S Equation ◽  
Relativistic Corrections ◽  
Schrödinger's Equation

This article deals with the connection between the confluent Heun equation (with two singularities 0 and ∞) and Schrödinger's equation for H-like atoms with two "auxiliary" parameters μ and ν which "influence" the spectrum of eigenvalues, the Coulomb potential and also the radial function. The case of μ = ν = 1 corresponds to the "standard" form of Schrödinger's equation. In terms of the parameter ν, for example, "relativistic corrections" of the type: Dirac's formula of the fine structure, the nuclear motion, etc., may be considered.

THE HEUN–SCHRÖDINGER RADIAL EQUATION FOR DH-ATOMS

2005 ◽  
Vol 19 (19n20) ◽  
pp. 981-989 ◽  
Author(s):  
V. F. TARASOV
Keyword(s):  
Coulomb Potential ◽  
Radial Function ◽  
Standard Form ◽  
Electron Cloud ◽  
Quantum Corrections ◽  
Heun Equation ◽  
Radial Equation ◽  
Radial Functions ◽  
Kummer’S Function ◽  
Multidimensional Equation

This article deals with the connection between Schrödinger's multidimensional equation for DH-atoms (D≥1) and the confluent Heun equation with two auxiliary parameters ν and τ, where |1-ν| = o(1) and τ∈ℚ+, which influence the spectrum of eigenvalues, the Coulomb potential and the radial function. The case τ = ν = 1 and D = 3 corresponds to the "standard" form of Schrödinger's equation for a 3H-atom. With the help of parameter ν, e.g., some "quantum corrections" may be considered. The cases 0<τ<1 and τ>1, but â = (n-l-1)τ≥0 is an integer, change the "geometry" of the electron cloud in the atom, i.e. the so-called "exotic" 3H-like atoms arise, where Kummer's function 1F1(-â; c; z) has â zeros and the discrete spectrum depends only on Z/(νn) but not on l and τ. Diagrams of the radial functions [Formula: see text] as n≤3 are given.

scholarly journals Orthogonal structure on a quadratic curve

2020 ◽  
Author(s):  
Sheehan Olver ◽  
Yuan Xu
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Extension Problem ◽  
Orthogonal Expansions ◽  
Schrödinger’S Equation ◽  
Fourier Extension ◽  
Quadratic Curve ◽  
Interpolation Of Functions ◽  
Fourier Orthogonal Expansions ◽  
Schrödinger's Equation

Abstract Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.

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