Complex eigenvalues of the Schrödinger equation with a Gaussian barrier

The Schrödinger equation with a unidimensional Gaussian potential barrier admits a finite number of complex eigenvalues associated with purely outgoing wavefunctions, called resonances.

We compute the resonances for this problem with varying barrier strength parameter by means of the well-known Rayleigh–Ritz method (RR) with complex rotation, using a harmonic oscillator basis set, and the Riccati–Padé method (RPM), which is based on the application of a Padé approximant to the Riccati equation for the regularized logarithmic derivative of the wavefunction. The latter method does not need to perform a complex rotation explicitly. We also show that a second set of complex eigenvalues exists, which is obtainable by both methods; in the case of the RR method, the computation of either of them requires different choices of the rotation parameter, setting it to be either below or above a certain threshold, respectively, whereas the RPM yields both sets of resonances. The lowest-lying eigenvalues of the second set are close to the resonances of the first set.

We perform a simple asymptotic analysis of the eigenfunctions, which allows us to determine the threshold value of the complex rotation parameter, as well as analyze the characteristics of both sets of eigenfunctions.
Finally, we draw parallels with similar results obtained in previous examples that present similar behavior, such as the radial exponential potential barrier and a well-known barrier potential that presents pre-dissociating resonances akin to those found in diatomic molecules.