Theory of the resonances of the LoSurdo-Stark effect

Abstract

When an atom or a molecule is exposed to an electric field, the discrete electronic spectrum turns into a spectrum of resonances with complex energies due to the mixing of the bound wave function of each unperturbed stationary state with the asymptotic form of the Airy function. The complex energies are eigenvalues of a non-Hermitian Schrödinger equation whose solutions have a special outgoing-wave asymptotic form, which is derived rigorously. At least three coordinate transformations (rotation, translation, and a combination of these) regularize the dc-field-induced resonance function, the most efficient one being the long-honored rotation, f(x)=${\mathit{xe}}^{\mathit{i}\mathrm{\theta }}$, which causes the function to dissipate asymptotically as ${\mathit{e}}^{\mathrm{-}\gamma \mathit{x}3/2}$. This finding, obtained from first principles, explains previous results that have been obtained either via trial computations or via elaborate mathematical analyses of the spectrum. The theory is further developed toward the efficient computation of such resonances with ${\mathit{scrL}}^2$ functions. Starting from the resonance-wave-function form ${\mathrm{\Psi }}^{\mathrm{res}}$=a${\mathrm{\Psi }}_0$+${\mathit{bX}}_{\mathrm{as}}$, the computation of the localized ${\mathrm{\Psi }}_0$ is carried out on the real axis, and only the free-electron function belonging to ${\mathit{X}}_{\mathrm{as}}$ is rotated and optimized in the complex plane. Two applications are presented. The first involves a numerically solvable one-dimensional model of a shape resonance in a dc field for a large range of field strengths.