On two different kinds of resonances in one-dimensional quantum-mechanical models

Abstract

We apply the Riccati-Padé method and the Rayleigh-Ritz method with complex rotation to the study of the resonances of a one-dimensional well with two barriers. The model exhibits two different kinds of resonances and we calculate them by means of both approaches. While the Rayleigh-Ritz method reveals each set at a particular interval of rotation angles the Riccati Padé method yields both of them as roots of the same Hankel

On the complex solution of the Schrödinger equation with exponential potentials

“…This pole can be moved through any ray in the complex plane, and for this reason the solutions of equation ( 17) yield approximations to the eigenvalues of equation ( 14) whose corresponding eigenfunctions decay asymptotically through different rays in the complex plane. Concordantly, the roots of equation ( 17) persist if a complex rotation of the variable is performed [36]. Another important characteristic of the RPM is that H D d typically exhibits a great number of roots in the neighborhood of each eigenvalue; this makes it easier to differentiate the roots that approximate an eigenvalue from those that are spurious, but it also complicates the task to find the optimal sequence of roots that converges to each eigenvalue by means of iterative methods.…”

“…It was originally proposed for the computation of bound states, but later it was found that it is also able to yield resonances without resorting explicitly to a complex rotation [27,28]. Its ensuing applications showed that it is able to compute them very accurately and with comparatively little cost [31][32][33][34][35][36][37]. For example, the resonances for the Stark effect in the hydrogen atom computed by us in [37] are to our knowledge the most accurate available in literature.…”

“…, introduced by Moiseiev to model pre-dissociation resonances of diatomic molecules [40], but it was later explained in [36] that the incorrect eigenvalues can also be obtained by complex rotation, provided that the rotation angle is set to be greater than the critical value θ crit = π/4. In fact, such eigenvalues had already been obtained and discussed by other authors [7,8,41,42].…”

Highly accurate potential energy curves for the hydrogen molecule ion

“…The RPM has been thoroughly described in previous works [33][34][35][36][37][38][39][41][42][43][44], but for the sake of completeness we briefly recall its main features here. We also revisit the main generalities of the Schrödinger equation for H + 2 , and detail the application of the RPM to solve it.…”