Converging bounds for the eigenvalues of multiminima potentials in two-dimensional space

Abstract

The eigenvalue problem in L 2 (R 2) of Schrödinger operators with a polynomial perturbation has been replaced by one corresponding to the system confined in a box with impenetrable walls. It is shown that the Dirichlet and the von Neumann problems in L 2 () generate upper and lower bounds, respectively, to the eigenvalues of the unbounded system. To illustrate the method, rapidly converging two-sided bounds for the energy levels of two-and three-well oscillators are presented, using simple trigonometric basis functions.

Exact solutions for vibrational levels of the Morse potential

“…Furthermore, in section 3 we formulate the Dirichlet and von Neumann boundary value problems over a truncated interval of r, r ∈ [0, ], for the Schrödinger equation (1.6). In this way, we generate converging upper and lower bounds to the eigenvalues of the nonconfined system [10,12]. The last section is devoted to the discussion of the specimen numerical applications and the concluding remarks.…”

“…again leads to a standard matrix eigenvalue problem of form (3.2) yielding lower bounds to E [12]. Now the matrix elements are given by…”

“…As a matter of fact, regarding as a nonlinear optimization parameter [10,12], we have an occasion of determining two-sided bounds to the eigenvalues of the nonconfined system and, hence, numerically exact results for the Morse potential.…”

“…It is interesting to note that such a bounded system has been used for some time as a powerful method to determine the spectrum of the usual Schrödinger equation with asymptotic boundary conditions. Actually, in the preceding articles of the present author and his co-workers, the truncated interval approach has been applied very effectively in solving both symmetrical [8,9] and asymmetrical [9,10] quantum problems in one dimension as well as nonseparable [11,12] and isotropic [13,14] potentials in two and N dimensions, respectively. More recently, Aquino [15] used the same approximation to compute the energy levels of the three-dimensional isotropic harmonic oscillator and the hydrogen atom for which the exact eigenvalues are known analytically in the unbounded interval.…”

Article

“…In contrast, double-well potentials in multidimensional space have not received so much attention although some progress has been made in calculating their energy eigenvalues [16][17][18][19].…”