Eigenvalue bounds for polynomial central potentials in d dimensions

TL;DR

This paper derives bounds and approximation formulas for eigenvalues of quantum particles in polynomial central potentials in multiple dimensions, improving understanding of spectral properties and providing comparisons with recent results.

Contribution

It introduces new bounds and approximation formulas for eigenvalues in polynomial potentials, extending semi-classical methods to higher dimensions and specific anharmonic oscillators.

Findings

Derived lower and upper bounds for eigenvalues in polynomial potentials.

Provided algebraic expressions for eigenvalues of anharmonic oscillators.

Compared new bounds with recent literature results.

Abstract

If a single particle obeys non-relativistic QM in R^d and has the Hamiltonian H = - Delta + f(r), where f(r)=sum_{i = 1}^{k}a_ir^{q_i}, 2\leq q_i < q_{i+1}, a_i \geq 0$, then the eigenvalues E = E_{n\ell}^{(d)}(\lambda) are given approximately by the semi-classical expression E = \min_{r > 0}[\frac{1}{r^2} + \sum_{i = 1}^{k}a_i(P_ir)^{q_i}]. It is proved that this formula yields a lower bound if P_i = P_{n\ell}^{(d)}(q_1), an upper bound if $P_i = P_{n\ell}^{(d)}(q_k) and a general approximation formula if P_i = P_{n\ell}^{(d)}(q_i). For the quantum anharmonic oscillator f(r)=r^2+\lambda r^{2m},m=2,3,... in d dimension, for example, E = E_{n\ell}^{(d)}(\lambda) is determined by the algebraic expression \lambda={1\over \beta}({2\alpha(m-1)\over mE-\delta})^m({4\alpha \over (mE-\delta)}-{E\over (m-1)}) where \delta={\sqrt{E^2m^2-4\alpha(m^2-1)}} and \alpha, \beta are constants. An improved lower bound to the lowest eigenvalue in each angular-momentum subspace is also provided. A comparison with the recent results of Bhattacharya et al (Phys. Lett. A, 244 (1998) 9) and Dasgupta et al (J. Phys. A: Math. Theor., 40 (2007) 773) is discussed.