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Journal of Physics A Mathematical and General

Time delay and resonances in potential scattering

Werner O. Amrein, Kalyan B. Sinha

Published Jul 5, 2006• 15 citations
Publisher

Abstract

We present stationary expressions for the on-shell time delay operator and for its trace in quantum-mechanical potential scattering. The trace of this operator is shown to represent the time delay for a monoenergetic and well-collimated beam of initial states. For exponentially decaying potentials the total scattering cross section and the trace of the on-shell time delay operator have analytic extensions to complex energies, with poles at most at the usual resonance poles (poles of the analytically continued S -matrix) in the lower half plane and at their mirror images with respect to the real axis. At real energies close to such a pole both the scattering cross section and the time delay are approximately of Breit–Wigner type, the sign of the time delay being related to that of the virial. These results corroborate the idea that a resonance should show up through a peak in the scattering cross section and a large (though positive) time delay.

DOI
10.1088/0305-4470/39/29/016

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We present stationary expressions for the on-shell time delay operator and for its trace in quantum-mechanical potential scattering. The trace of this operator is shown to represent the time delay for a monoenergetic and well-collimated beam of initial states. For exponentially decaying potentials the total scattering cross section and the trace of the on-shell time delay operator have analytic extensions to complex energies, with poles at most at the usual resonance poles (poles of the analytically continued S -matrix) in the lower half plane and at their mirror images with respect to the real axis. At real energies close to such a pole both the scattering cross section and the time delay are approximately of Breit–Wigner type, the sign of the time delay being related to that of the virial. These results corroborate the idea that a resonance should show up through a peak in the scattering cross section and a large (though positive) time delay.
A formalism to calculate the time delay experienced by a particle undergoing scattering is presented. The formalism is a natural extension of Wigner’s delay time to wave packets which do not exhibit a well defined angular momentum. The delay depends only on the partial wave phase shifts characterizing the scattering event. The formalism is tested by applying it to the 3D square well. The results are shown to agree with the known properties of resonant scattering as deduced from a study of the poles of the scattering matrix in the complex energy plane.
We consider time delay and symmetrised time delay (defined in terms of sojourn times) for quantum scattering pairs {H0 = h(P), H}, where h(P) a dispersive operator of hypoelliptic-type. For instance h(P) can be one of the usual elliptic operators such as the Schrödinger operator h(P) = P 2 or the square-root Klein-Gordon operator h(P) = √ 1 + P 2. We show under general conditions that the symmetrised time delay exists for all smooth even localization functions. It is equal to the Eisenbud-Wigner time delay plus a contribution due to the non-radial component of the localization function. If the scattering operator S commutes with some function of the velocity operator ∇h(P), then the time delay also exists and is equal to the symmetrised time delay. As an illustration of our results we consider the case of a one-dimensionnal Friedrichs Hamiltonian perturbed by a finite rank potential. Our study put into evidence an integral formula relating the operator of differentiation with respect to the kinetic energy h(P) to the time evolution of localization operators. 1 Introduction and main results One can find a large literature on the identity of Eisenbud-Wigner time delay and time delay in quantum scattering defined in terms of sojourn times (see [3, 7, 8, 12, 19, 23,
Abstract The dynamics of absorption is analyzed by using an exactly solvable model that deals with an analytical solution to Schrödinger’s equation for cutoff initial plane waves incident on a complex absorbing potential. A dynamical absorption coefficient which allows us to explore the dynamical loss of particles from the transient to the stationary regime is derived. We find that the absorption process is characterized by the emission of a series of damped periodic pulses in time domain, associated with damped Rabi-type oscillations with a characteristic frequency, ω = ( E + ε )/ℏ, where E is the energy of the incident waves and − ε is energy of the quasidiscrete state of the system induced by the absorptive part of the Hamiltonian; the width γ of this resonance governs the amplitude of the pulses. The resemblance of the time-dependent absorption coefficient with a real decay process is discussed, in particular the transition from exponential to nonexponential regimes, a well-known feature of quantum decay. We have also analyzed the effect of the absorptive part of the potential on the dynamical delay time, which behaves differently from the one observed in attractive real delta potentials, exhibiting two regimes: time advance and time delay.
The Hermitian properties of the Hamiltonian are related not only to the operator itself but also to the functions on which it acts. Hermitian Hamiltonians operate on functions in the L 2 Hilbert space which correspond to boundary conditions which vanish at infinity. In this chapter, in order to move into the non-Hermitian domain, we will impose on the solutions to the time-independent Schrödinger equation (TISE) different boundary conditions which lead to solutions which can be associated with different types of the complex poles of the scattering matrix. These solutions will contain information which was not available within the scope of functions in L 2 . By imposing outgoing boundary conditions on the eigenfunctions of the timeindependent Hamiltonian complex eigenvalues, E res = ε − (i/2)Г, are obtained. These complex energies are associated with decaying resonance states which were discussed in the previous two chapters. The bound states (if they exist) appear as real eigenvalues since they result from exactly such outgoing boundary conditions which appear under the threshold energy. When incoming boundary conditions are imposed two kind of solution are obtained. One type of solution is the complex conjugates of the decay resonance solutions mentioned above. In scattering theory text books (see Taylor for example) the physical resonance solutions are associated with the poles of the scattering matrix which are embedded in the lower half of the complex energy plane. However, in nuclear physics the complex poles embedded in the upper half of the complex energy plane, so-called virtual states , are denoted as capture resonances .
In the standard Breit-Wigner approach to scattering the phase shift is to have a form tan⁡δBW=Γ1/(E1−E)\tanδ_{\rm BW} =Γ_1/(E_1-E)tanδBW​=Γ1​/(E1​−E) at a real energy resonance. This leads to complex energy poles in the scattering amplitude at EBW=E1−iΓ1E_{\rm BW}=E_1-iΓ_1EBW​=E1​−iΓ1​, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting tan⁡δBW=Γ1/(E1−E)\tanδ_{\rm BW} =Γ_1/(E_1-E)tanδBW​=Γ1​/(E1​−E) is not always a good description of the real energy scattering amplitude, that Γ1Γ_1Γ1​ can be negative, that EBWE_{\rm BW}EBW​ is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear PTPTPT symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs E∓=E2∓iΓ2E_{\mp}=E_2\mp i Γ_2E∓​=E2​∓iΓ2​ with E−≠EBWE_- \neq E_{\rm BW}E−​=EBW​, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two.