Vincenzo Grecchi

Resonances in Stark effect and perturbation theory

It is proved that the action of a weak electric field shifts the eigenvalues of the Hydrogen atom into resonances of the Stark effect, uniquely determined by the perturbation series through the Borel method.This is obtained by combining the Balslev-Combes technique of analytic dilatations with Simons results on anharmonic oscillators.

Stark Wannier ladders

We study the Schrödinger equation for an electron in a one dimensional crystal submitted to a constant electric field. We prove the existence of ladders of resonances, the imaginary part of which is exponentially small with the field.

Double wells: Nevanlinna analyticity, distributional Borel sum and asymptotics

AbstractWe consider the energy levels of a Stark family, in the parameterj, of quartic double wells with perturbation parameterg:

Padé summability of the cubic oscillator

H(g,j) = p^2 + x^2 (1 - gx)^2 - j\left( {gx - \frac{1}{2}} \right).

Stark resonances: asymptotics and distributional Borel sum

For non-evenj (and for the symmetric casej=0) we prove analyticity in the full Nevanlinna disk ℜg−2 >R−1 of theg2-plane, as predicted by Crutchfield. By means of an approximation we give a heuristic estimate of the asymptotic smallg behaviour, showing the relation between the avoided crossings and the failure of the usual perturbation series.

Tunnelling instability via perturbation theory

Abstract It is proved that the 1 R expansion for H2+ is divergent and Borel summable to a complex eigenvalue of a non-self-adjoint operator, which has the same 1 R expansion. The Borel sum is related to the H2+ system as follows: its real part agrees with the eigenvalue doublet asymptotically to all orders, and its imaginary part determines the asymptotics of the 1 R expansion coefficients via a dispersion relation. A rigorous estimate of the leading behavior of the imaginary part is obtained, and as a consequence the approximate formula of Brezin and Zinn-Justin relating the square of the eigenvalue gap to the asymptotics of the 1 R expansion is put on a rigorous basis.

Non-linear Stark effect and molecular localization

We prove the Pade (Stieltjes) summability of the perturbation series of any energy level En,1(β), , of the cubic anharmonic oscillator, , as suggested by the numerical studies of Bender and Weniger. At the same time, we give a simple proof of the positivity of every level of the -symmetric Hamiltonian H1(β) for positive β (Bessis–Zinn Justin conjecture). The n zeros, of a state ψn,1(β), stable at β = 0, are confined for β on the cut complex plane, and are related to the level En,1(β) by the Bohr–Sommerfeld quantization rule (semiclassical phase-integral condition). We also prove the absence of non-perturbative eigenvalues and the simplicity of the spectrum of our Hamiltonians.

Resonances in one‐dimensional Stark effect and continued fractions

We complete and correct some proofs of an earlier paper on distributional Borel summability and we add an application which can be useful in the discussion of semiclassical problems.