Determination of Resonances by the Optimized Spectral Approach

TL;DR

This paper introduces an optimized spectral approach using complex parameters in the Rayleigh-Ritz method to efficiently determine energies and lifetimes of resonance states in quantum systems.

Contribution

It extends the Rayleigh-Ritz method with complex, adjustable parameters to accurately compute resonance energies and lifetimes without iterative procedures.

Findings

Effective for one-dimensional and spherical potentials

Computationally inexpensive and fast convergence

Outperforms some existing methods in accuracy and efficiency

Abstract

The Rayleigh-Ritz procedure for determining bound-states of the Schr\"{o}dinger equation relies on spectral representation of the solution as a linear combination of the basis functions. Several possible extensions of the method to resonance states have been considered in the literature. Here we propose the application of the optimized Rayleigh-Ritz method to this end. The method uses a basis of the functions containing adjustable nonlinear parameters, the values of which are fixed so as to make the trace of the variational matrix stationary. Generalization to resonances proceeds by allowing the parameters to be complex numbers. Using various basis sets, we demonstrate that the optimized Rayleigh-Ritz scheme with complex parameters provides an effective algorithm for the determination of both the energy and lifetime of the resonant states for various one-dimensional and spherically symmetric potentials. The method is computationally inexpensive since it does not require iterations or predetermined initial values. The convergence rate compares favorably to other approaches.