Abstract
We study analytically the radial Schrödinger equation with long-range attractive potentials whose asymptotic behaviors are dominated by inverse power-law tails of the form V(r)=-beta _n r^{-n} with n>2. In particular, assuming that the effective radial potential is characterized by a short-range infinitely repulsive core of radius R, we derive a compact analytical formula for the threshold energy E^{text {max}}_l=E^{text {max}}_l(n,beta _n,R), which characterizes the most weakly bound-state resonance (the most excited energy level) of the quantum system.
Highlights
The main goal of the present paper is to present a simple and elegant mathematical technique for the calculation of the most excited energy levels Emax(n, βn),1 which characterize the family (1) of attractive inverse power-law potentials
The Schrödinger differential equation with inverse powerlaw attractive potentials has attracted the attention of physicists and mathematicians since the early days of quantum mechanics
We have studied analytically the Schrödinger differential equation with attractive radial potentials whose asymptotic behaviors are dominated by inverse power-law tails of the form V (r ) = −βnr −n with n > 2
Summary
Where the effective radial potential V (r ) in (2) is characterized by a long-range inverse power-law attractive part and a short-range infinitely repulsive core. The bound-state (E < 0) resonances of the Schrödinger differential equation (2) that we shall analyze in the present paper are characterized by exponentially decaying radial eigenfunctions at spatial infinity: ψl (r → ∞) ∼ e−κr ,. The repulsive core of the effective radial potential (3) dictates the inner boundary condition ψl (r = R) = 0. The Schrödinger equation (2), supplemented by the radial boundary conditions (4) and (6), determine the discrete spectrum of bound-state eigen-wavenumbers {κ(n, βn, R)} [or equivalently, the discrete spectrum of binding energies E(n, βn, R)] which characterize the effective radial potential (3). As we shall explicitly show, the most weakly bound-state resonance (that is, the most excited energy level), which characterizes the quantum system (3), can be determined analytically in the regime5,6 [12]. Of small binding energies, where the characteristic length scale rn is defined by the relation
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