Abstract:
We study the pion-nucleon P_33 and P_11 scattering, where the Delta and the Roper resonances are seen, respectively. We use the Lippmann-Schwinger equation extended to couple to a one-body state, and investigate nature of these resonances by taking and omitting a one-baryon state into account. We see validity and puzzle of the standard quark model interpretation, for the Delta and the Roper resonances, respectively.

Abstract:
The Lippmann-Schwinger equation is an integral equation formulation for acoustic and electromagnetic scattering from an inhomogeneous media and quantum scattering from a localized potential. We present the sparsifying preconditioner for accelerating the iterative solution of the Lippmann-Schwinger equation. This new preconditioner transforms the discretized Lippmann-Schwinger equation into sparse form and leverages the efficient sparse linear algebra algorithms for computing an approximate inverse. This preconditioner is efficient and easy to implement. When combined with standard iterative methods, it results in almost frequency-independent iteration counts. We provide 2D and 3D numerical results to demonstrate the effectiveness of this new preconditioner.

Abstract:
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.

Abstract:
Nogga, Timmermans and van Kolck recently argued that Weinberg's power counting in the few-nucleon sector is inconsistent and requires modifications. Their argument is based on the observed cutoff dependence of the nucleon-nucleon scattering amplitude calculated by solving the Lippmann-Schwinger equation with the regularized one-pion exchange potential and the cutoff Lambda varied in the range Lambda = 2...20 fm^(-1). In this paper we discuss the role the cutoff plays in the application of chiral effective field theory to the two-nucleon system and study carefully the cutoff-dependence of phase shifts and observables based on the one-pion exchange potential. We show that (i) there is no need to use the momentum-space cutoff larger than Lambda ~ 3 fm^(-1); (ii) the neutron-proton low-energy data show no evidence for an inconsistency of Weinberg's power counting if one uses Lambda ~ 3 fm^{-1}.

Abstract:
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.

Abstract:
When homogenizing elliptic partial differential equations, the so-called corrector problem is pivotal to compute the macroscale effective coefficients from the microscale information. To solve this corrector problem in the periodic setting, Moulinec and Suquet introduced in the mid-nineties a numerical strategy based on the reformulation of that problem as an integral equation (known as the Lippmann--Schwinger equation), which is then suitably discretized. This results in an iterative, matrix-free method, which is of particular interest for complex microstructures. Since the seminal work of Moulinec and Suquet, several variants of their scheme have been proposed. The aim of this contribution is twofold. First, we provide an overview of these methods, recast in the language of the applied mathematics community. These methods are presented as asymptotically consistent Galerkin discretizations of the Lippmann--Schwinger equation. The bilinear form arising in the weak form of this integral equation is indeed the sum of a local and a non-local term. We show that most of the variants proposed in the literature correspond to alternative approximations of this non-local term. Second, we propose a mathematical analysis of the discretized problem. In particular, we prove under mild hypotheses the convergence of these numerical schemes with respect to the grid-size. We also provide a priori error estimates on the solution. The article closes on a three-dimensional numerical application within the framework of linear elasticity.

Abstract:
We outline a formalism and develop a computational procedure to treat the process of multiphoton ionization (MPI) of atomic targets in strong laser fields. We treat the MPI process nonperturbatively as a decay phenomenon by solving a coupled set of the integral Lippmann-Schwinger equations. As basic building blocks of the theory we use a complete set of field-free atomic states, discrete and continuous. This approach should enable us to provide both the total and differential cross-sections of MPI of atoms with one or two electrons. As an illustration, we apply the proposed procedure to a simple model of MPI from a square well potential and to the hydrogen atom.

Abstract:
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.

Abstract:
We consider time domain acoustic scattering from a penetrable medium with a variable sound speed. This problem can be reduced to solving a time domain volume Lippmann-Schwinger integral equation. Using convolution quadrature in time and trigonometric collocation in space we can compute an approximate solution. We prove that the time domain Lippmann-Schwinger equation has a unique solution and prove conditional convergence and error estimates for the fully discrete solution for smooth sound speeds. Preliminary numerical results show that the method behaves well even for discontinuous sound speeds.

Abstract:
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.