Exterior complex scaling and the AC-Stark effect in a Coulomb field

By means of the exterior complex scaling of B. Simon an existence proof of resonances is given for the time-dependent Schrödinger equation $$i\frac{{\partial \psi }}{{\partial t}} = - \left( { - \Delta + V + \mu x_1 \cos \omega t} \right)\psi ,$$ whereV belongs to a class of potentials which includes the Coulomb one. The resonance width is given by the Fermi Golden Rule to second order perturbation theory and is nonzero for μ small and almost every ε.     

Resonances in the Stark effect of atomic systems

Generalizing earlier results on the Hydrogen case it is proved, through a dilation analyticity technique different from the canonical one, that the action of a weak electric field shifts the isolated eigenvalues of any atomic system into resonances of the Stark effect, uniquely determined by the perturbation series through the Borel summation method.     

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 Simon's results on anharmonic oscillators.Partially supported by G.N.F.M., C.N.R.Partially supported by I.N.F.N., Sezione di Bologna     

Adiabatic Theorems for Quantum Resonances

We study the adiabatic time evolution of quantum resonances over time scales which are small compared to the lifetime of the resonances. We consider three typical examples of resonances: The first one is that of shape resonances corresponding, for example, to the state of a quantum-mechanical particle in a potential well whose shape changes over time scales small compared to the escape time of the particle from the well. Our approach to studying the adiabatic evolution of shape resonances is based on a precise form of the time-energy uncertainty relation and the usual adiabatic theorem in quantum mechanics. The second example concerns resonances that appear as isolated complex eigenvalues of spectrally deformed Hamiltonians, such as those encountered in the N-body Stark effect. Our approach to study such resonances is based on the Balslev-Combes theory of dilatation-analytic Hamiltonians and an adiabatic theorem for nonnormal generators of time evolution. Our third example concerns resonances arising from eigenvalues embedded in the continuous spectrum when a perturbation is turned on, such as those encountered when a small system is coupled to an infinitely extended, dispersive medium. Our approach to this class of examples is based on an extension of adiabatic theorems without a spectral gap condition. We finally comment on resonance crossings, which can be studied using the last approach.     

Integral Equations for Three-Body Coulomb Resonances

 We propose a novel method for calculating resonances in three-body systems with repulsive Coulomb interactions. The method is based on the solution of a set of Faddeev and Lippmann-Schwinger integral equations. The resonances of the three-body system are defined as the complex-energy solutions of the homogeneous Faddeev integral equations. We show how the kernels of the integral equations should be continued analytically in order to get the resonances. As a numerical illustration a model for the three-α system is solved. Received October 1, 1999; revised February 25, 2000; accepted for publication June 30, 2000     

Resonances in the ωω system

Results of a partial-wave analysis for the reaction π ^? → ωωn studied at the VES spectrometer (Institute for High Emergy Physics, Protvino) are presented. The behavior of the J ^PC = 2⁺⁺ amplitudes in the ωω system is described by the f ₂(1565) and f ₂(1910) resonances, whose parameters were fixed at M = 1.590 ± 0.010 GeV and Γ = 0.140 ± 0.011 GeV for the former and at M = 1.890 ± 0.010 GeV and Γ = 0.165 ± 0.019 GeV for the latter. The decay f ₄(2050) → ωω was observed at parameters values of M = 1.960 ± 0.015 GeV, Γ = 0.290 ± 0.020 GeV.     

An Inverse Problem for Trapping Point Resonances

We consider semi-classical Schrödinger operator P(h) = ? h ²Δ + V(x) in ${{\mathbb R}^n}$ such that the analytic potential V has a non-degenerate critical point x ₀ = 0 with critical value E ₀ and we can define resonances in some fixed neighborhood of E ₀ when h > 0 is small enough. If the eigenvalues of the Hessian are ${\mathbb Z}$ -independent the resonances in h ^δ-neighborhood of E ₀ (δ > 0) can be calculated explicitly as the eigenvalues of the semi-classical Birkhoff normal form. Assuming that potential is symmetric with respect to reflections about the coordinate axes we show that the classical Birkhoff normal form determines the Taylor series of the potential at x ₀. As a consequence, the resonances in a h ^δ-neighborhood of E ₀ determine the first N terms in the Taylor series of V at x ₀. The proof uses the recent inverse spectral results of V. Guillemin and A. Uribe.     

Relativistic Coupled-Channel Quark Model for Meson Resonances

We present results from a study of meson ground and resonant states within a relativistic coupled-channel constituent-quark model. Along such an approach in particular the resonance character of hadron excitations can be fully taken into account. Thus it becomes possible to describe the decay of a hadron resonance into a lower lying state more realistically than in usual single-channel approaches. In a simplified model we demonstrate the viability and the advantages of our method by producing relativistic invariant results for the finite decay width of a meson resonance.     

Distribution of Resonances for Open Quantum Maps

We analyze a simple model of quantum chaotic scattering system, namely the quantized open baker’s map. This model provides a numerical confirmation of the fractal Weyl law for the semiclassical density of quantum resonances. The fractal exponent is related to the dimension of the classical repeller. We also consider a variant of this model, for which the full resonance spectrum can be rigorously computed, and satisfies the fractal Weyl law. For that model, we also compute the shot noise of the conductance through the system, and obtain a value close to the prediction of random matrix theory.     

Resonances within chaos

A chaotic system under periodic forcing can develop a periodically visited strange attractor. We discuss simple models in which the phenomenon, quite easy to see in numerical simulations, can be completely studied analytically.     

Degenerate Elliptic Resonances

Quasi-periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations. The parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations. Here we prove that, for a class of elliptic tori, a resummation algorithm can be devised and proved to be convergent, thus extending to such lower-dimensional invariant tori the methods employed to prove convergence of the Lindstedt series either for the maximal (i.e. KAM) tori or for the hyperbolic lower-dimensional invariant tori.     

High-Order Resonances in the Water Molecule

The water-vapor spectra in the near-infrared and visible region were reanalyzed with the purpose of finding experimental evidences of unusual high-order resonance between "dark" high-bending and "bright" stretch vibration states. About 70 transitions to the (050), (060), (070), (080), (160), (061), (170), (071), and, even (0 10 0) bending states, and their resonating partners were assigned in the spectra that gives the experimental energy levels lying near or above the potential energy barrier to linearity. The assignments were confirmed by combination differences and simultaneous observation of both perturbed and perturbing levels. It was found that the high-order resonances with large changing of vibration quantum numbers are typical for the water molecule and they are caused by the strong centrifugal distortion near the linear configuration. These resonances destroy the usual polyad scheme originating from well-known Coriolis, Darling-Dennison, and Fermi resonances in H(2)O molecule. Copyright 2001 Academic Press.     

Resonances in chaotic dynamics

We present a discussion and some numerical results on the actual possibility of making accessible, by numerical techniques, the complex singularities of the power spectrum (resonances) for a chaotic signal. Hénon's transformation is investigated in detail, showing that the position of the leading resonance in the complex frequency plane determines the kind of mixing rate in the time evolution.     

Resonances in neutral Bremsstrahlung

A strong resonance in the Bremsstrahlung emitted by electrons scattered by atomic nitrogen has been predicted by John and Williams.⁽¹⁾ This radiation resonance corresponds to the predicted elastic scattering resonance between electrons and atomic nitrogen. At 3360°K, the effective oscillator strength (f-number) for the free p to free s wave has a value of about 0.5, indicating that this resonance is a large fraction of the low temperature Bremsstrahlung in this system.     

Resonances in Positron-He Scattering

We present the results for resonances in positron-He scattering at low impact energy （19.3-24.0eV） by using the momentum space coupled-channel optical （CCO） method. The S-partial wave resonance at 20.16 eV is found for the first time.     

Clusters as Many-Body Resonances

A formalism to evaluate the resonant states produced by two particles moving outside a closed shell core is presented. The two-particle states are calculated by using a single-particle representation consisting of bound states, Gamow resonances and scattering states in the complex energy plane (Berggren representation). Two representative cases are analysed corresponding to whether the Fermi level is below or above the continuum threshold. It is found that long-lived resonances are mostly determined by either bound states or by narrow Gamow resonances. However, they are significantly affected by wide resonances and the continuum background itself.