Terms Of Resonant States Research Articles

Properties of the Expansion of the Decaying Wave Solution in Terms of Resonant States

We consider the formulation of quantum decay in terms of resonant states and complex poles of the outgoing Green’s function of the problem to discuss some properties of the decaying wave solution that exhibit its non-Hermitian character and hence illustrate its departure from the standard formalism of quantum mechanics.

Resonant states for complex potentials and spectral singularities

We consider the analytical properties of the outgoing Green's function for complex potentials of finite range to study the relationship between its residues at the complex poles, the resonant states, and the spectral singularities of continuum wave functions. We obtain that the resonant state corresponding to a spectral singularity is a well-defined function, which we call the spectral singularity resonant function, and find a general expression that yields the position of the spectral singularity on the real wave number axis in terms of the imaginary part of the complex potential and the spectral singularity resonant function. We derive an analytical expression for continuum wave solutions as an expansion in terms of resonant states and the corresponding expression near a spectral singularity. Our findings are illustrated by considering an imaginary $\ensuremath{\delta}$-shell potential for $s$ waves.

Resonant-state expansions and the long-time behavior of quantum decay

It is shown that a representation of the decaying wave function as a resonant sum plus a nonexponential integral term may be written as a purely discrete resonant sum by evaluating at long times the integral term by the steepest descents method, and then expanding the resulting expression in terms of resonant states. This leads to a representation that is valid along the exponential and the inverse power in time regimes. A model calculation using the $\ensuremath{\delta}$ potential allows us to make a comparison of the expansion with numerical integrations in terms of continuum wave functions and, in the long time regime, with an exact analytic expression of the integral term obtained using the steepest descents method. The results demonstrate that resonant states give a correct description of the long-time behavior of decay.

Koopman's approach to dissipation

We analyze spectral properties of Koopman's operator corresponding to classical dissipative dynamics. It turns out that apart from the real continuous spectrum such operator gives rise to the family of discrete complex generalized eigenvalues. The corresponding generalized eigenvectors are interpreted as resonant states. We show that Koopman's operator may be entirely characterized in terms of resonant states and the main message is that resonant states are responsible for dissipation.

García-Calderón, Mateos, and Moshinsky Reply:

In our paper [Phys. Rev. Lett. 74, 337 (1995)], we derived an exact expression for the survival and nonescape probabilities as an expansion in terms of resonant states. It was shown that these quantities exhibit at long times a different behavior. Although both decay as a power law, they have different exponents. In this paper we show that, contrary to the claim in the Comment of R. M. Cavalcanti (quant-ph/9704023), the nonescape probability decay for long times as an inverse power law.

An expansion of continuum wave functions in terms of resonant states: (II). Solvable models

This paper applies the formalism developed in part I which provides a purely discrete expansion for a continuum wave solution of the Schrödinger equation in terms of resonant states along the interior region of a finite range interaction. We consider two exactly solvable models for several values of the distance and momenta on and off resonance. Our results are compared with a recent approach by Bang and collaborators which involves subtraction terms. It is found that along the internal region the subtraction terms are not in general negligible. Our work substantiates an expression for the continuum wave function near resonance. We also obtain an equation for time delay in terms of resonant states.

Study of resonances off the energy shell in terms of resonant states

The half-on-shell or off-shell partial wave amplitude is expanded in terms of resonant states. Each off-shell resonant term can be written as the corresponding on-shell term multiplied by a complex off-shell factor. This factor changes the size and shape of the resonance. The above description provides a unified treatment of elastic scattering and stripping leading to resonant states. This theoretical framework may be used to test in a consistent form the perturbation approximations utilized in evaluating stripping amplitudes.

An expansion of continuum wave functions in terms of resonant states

The poles and residues of the complete outgoing Green function in the complex momentum plane are used to obtain, in the case of finite range potentials, an eigenfunction expansion of the continuum wave solution. It is found that in the region r< a the wave solution may be expressed as an infinite sum of discrete terms involving the bound, antibound and resonant states of the problem. At the boundary radius r = a a different expansion is obtained. In this case, in order to get an infinite discrete sum, one has to introduce two subtraction terms. Otherwise the expansion is given by a finite sum of discrete terms and an

R-matrix theory of reactive atom-diatom collisions

For chemical reactions, R-matrix theory is demonstrated to be well suited both as a computational tool and for investigations of the reaction mechanism. Transition probabilities are calculated for three different potential surfaces and the results discussed in terms of resonance states.

What is resonance?

The concept of duality, discussed in the previous paper by Dr C. Schmid, refers to the notion that the analysis of an S matrix amplitude in terms of s channel resonances and its analysis in terms of t channel (and u channel) reggeon exchanges are not independent, but represent alternative descriptions, the former being the more appropriate in the low energy domain and the latter being the more appropriate in the low t (or low u ) domain for high energy interactions. Hence, at the time when this Discussion meeting was planned, it appeared necessary to attempt to clarify the notion of resonance , since there was then a good deal of discussion about how partial wave analyses of scattering data should be interpreted in terms of resonance states (Bransden, O’Donnell &amp; Moorhouse 1965; Lovelace 1968; Donnachie 1968), and even whether such an interpretation was necessarily the case at all (Schmid 1968 a,b ; Collins, Johnson &amp; Squires 1968; Alessandrini &amp; Squires 1968; Alessandrini, Freund, Oehme &amp; Squires 1968; Collins, Johnson &amp; Ross 1968; Kreps &amp; Logan 1969). However, in the actual use of duality in this meeting and in the model amplitudes which have been discussed thus far for the illustration of duality, there is really no controversy about the nature of resonance, since these models are still confined to the cases of resonances with zero width, characterized by poles on the real s axis, and there is essentially no ambiguity about the relation between resonances and poles in this limiting case. All the same, it still seems worthwhile to take a few moments to draw attention to a number of points about the description of resonance, which deserve to be more widely known.

Resonance interpretation of phase shift analyses and the generalized interference model

The phase-shift analysis of πp, K̄p and Kp scattering are discussed in terms of resonance states. The properties of these states are briefly reviewed, in particular their interrelation via the symmetric quark model (or SU(6) x 0(3)). The generalized interference model is introduced as a natural way in which to describe the results.

On potential scattering of relativistic particles

The S -matrix for the problem of potential scattering of relativistic particles can be expressed in terms of resonance states under restrictions on the potential which are similar to those of the non-relativistic case. The S -matrix is a double-valued function of momentum or energy with branch points as given by the relativistic energy-momentum relationship.