A. D. Baute, I. L. Egusquiza, J. G. Muga
The Moller operators and the asociated Lippman-Schwinger equations obtained from different partitionings of the Hamiltonian for a step-like potential barrier are worked out, compared and related.
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It is pointed out that for a general short-ranged potential the Lippmann-Schwinger-Low scattering state $|ψ^L_k>$ does not strictly satisfy the Schrodinger eigen equation, and the pair $|ψ^L_n>$, $|ψ^L_k>$ is mutually nonorthogonal if $E_n=E_k$. For this purpose, we carefully use an infinitesimal adiabatic parameter $ε$, a nonlinear relation among transition amplitudes, and a separable interaction as illustration.
We review the way to analytically continue the Lippmann-Schwinger bras and kets into the complex plane. We will see that a naive analytic continuation leads to nonsensical results in resonance theory, and we will explain how the non-obvious but correct analytical continuation is done. We will see that the physical basis for the non-obvious but correct analytic continuation lies in the invariance of the Hamiltonian under anti-unitary symmetries such as time reversal or PT.
The analytic continuation of the Lippmann-Schwinger bras and kets is obtained and characterized. It is shown that the natural mathematical setting for the analytic continuation of the solutions of the Lippmann-Schwinger equation is the rigged Hilbert space rather than just the Hilbert space. It is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution, resulting into a semigroup time evolution. Physically, the semigroup time evolution is simply a (retarded or advanced) propagator.
We calculate eigenvalues of one-dimensional quantum-systems by the exact numerical solution of the Lippmann-Schwinger equation, analogous to the scattering problem. To illustrate our method, we treat elementary problems: the harmonic and quartic oscillator, a symmetric and a skew double-well potential, and potentials with finite and infinite depth. Furthermore, we show how our method can be used for eigenvalue-engineering.
We exemplify the way the rigged Hilbert space deals with the Lippmann-Schwinger equation by way of the spherical shell potential. We explicitly construct the Lippmann-Schwinger bras and kets along with their energy representation, their time evolution and the rigged Hilbert spaces to which they belong. It will be concluded that the natural setting for the solutions of the Lippmann-Schwinger equation--and therefore for scattering theory--is the rigged Hilbert space rather than just the Hilbert space.
The classical Lippmann-Schwinger equation plays an important role in the scattering theory (non-relativistic case, Schrödinger equation). In the present paper we consider the relativistic analogue of the Lippmann-Schwinger equation. We represent the corresponding equation in the integral form. Using this integral equation we investigate the stationary scattering problems (relativistic case, Dirac equation). We consider the dynamical scattering problems (relativistic case, Dirac equation) as well.
We first present two possible analytic continuations of the Lippmann-Schwinger eigenfunctions to the second sheet of the Riemann surface, and then we compare the different Gamow vectors that are obtained through each analytic continuation.
Solutions of the Lippmann-Schwinger equation for confocal parabolic billiards
本文解析和数值求解了共焦抛物线台球的Lippmann-Schwinger方程，研究了共振、隧穿及特定能量下的透明现象。
Many useful concepts for a quantum theory of scattering and decay (like Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially decaying Gamow vectors, causality) are not well defined in the mathematical frame set by the conventional (Hilbert space) axioms of quantum mechanics. Using the Lippmann-Schwinger equations as the takeoff point and aiming for a theory that unites resonances and decay, we conjecture a new axiom for quantum mechanics that distinguishes mathematically between prepared states and detected observables. Suggested by the two signs $\pm iε$ of the Lippmann…
The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation-registration arrow …
The Schrodinger equation with the trigonometric Rosen-Morse potential in flat three dimensional Euclidean space, E3, and its exact solutions are shown to be also exactly transformable to momentum space, though the resulting equation is purely algebraic and can not be cast into the canonical form of an integral Lippmann-Schwinger equation. This is because the cotangent function does not allow for an exact Fourier transform in E3. In addition we recall, that the above potential can be also viewed as an angular function of the second polar angle parametrizing the three dimensional spherical surfa…
A self-contained discussion of integral equations of scattering is presented in the case of centrally-symmetric potentials in one dimension, which will facilitate the understanding of more complex scattering integral equations in two and three dimensions. The present discussion illustrates in a simple fashion the concept of partial-wave decomposition, Green's function, Lippmann-Schwinger integral equations of scattering for wave function and transition operator, optical theorem and unitarity relation. We illustrate the present approach with a Dirac delta potential.
The classical Lippmann-Schwinger equation (LS equation) plays an important role in the scattering theory for the non-relativistic case (Schrödinger equation). In our previous paper arXiv:1801.05370, we consider the relativistic analogue of the Lippmann-Schwinger equation (RLS equation). We represent the corresponding equation in the integral form. In the present paper, we use the corresponding integral equation and investigate the scattering problems for both stationary and dynamical cases. Our approach allows us to develop an RLS equation theory which is comparable in its completeness with th…
The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger type equation, where the Green's function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the $(e^-,\, e^-,\, {{He}^{++}})$ system is solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of…
We first deduce the analytical continuation in the complex planes of the initial and final three-momenta of the Lippmann-Schwinger equation in coupled or uncoupled partial-wave amplitudes. This result allows us to deduce a master equation whose solution is the exact discontinuity of the on-shell partial-wave amplitudes along the left-hand cut. This equation is always a linear non-singular integral equation whose solution is fixed exclusively by the knowledge of the potential, applicable to either regular or singular potentials. The capability of calculating exactly this discontinuity allows on…
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