Lipmann-Schwinger Equations for a System of Multiband Energy-Eigenvalue Spectrum

Abstract

Abstract. For the first time, we establish the Lippmann-Schwinger equations for a system of multiband energy-eigenvalue spectrum so that the involved time-independent Schrödinger wavefunctions become matrix elements. As a matter of fact, a supremum-seminorm of the total wavefunction matrix is estimated in terms of the same seminorm for the partial wavefunction matrix and for two key matrices of operators. In addition, an associated matrix-tensor formalism is presented. Keywords:Lippmann-Schwinger equations; multiband eigenvalue spectrum; supremum-seminorm; matrices of operators; matrix formalism. IntroductionTreating by using elegant mathematical-physics methods, quantum systems of multiband energyeigenvalue spectra is certainly interesting because these systems play an important role in several areas of Physics. Let us regard, for instance, quantum systems of two-band energy eigenvalue spectra [1,2]. A typical example of these systems can be found in magnetism of amorphous solids [1][2][3]. In particular, we refer to determine the magnetic susceptibility of amorphous solids [1][2][3]. In this context, a Wannier-type representation was employed in ref.[1] while a generalization of this representation, by introducing a matrix formalism, was done in ref. [2]. Really, the physics of condensed matter exhibits a number of examples of quantum systems with multiband eigenvalue spectra. Consider, for instance, a significant two-band eigenvalue spectrum consisting of spin-up and spin-down electronic bands (as, for example, in metamagnetic systems [4,5] and nanophysics [6][7][8][9][10]) and a typical system with three-band spectrum as, for instance, a semiconductor with the conduction band, the valence band, and the forbidden band (band gap). Certainly, systems with two or three eigenvalue bands are relatively frequent in Physics. Therefore, it is natural to think on generalizing for many bands so that elaborating analytical models to characterize multiband systems is desirable. On the other hand, the so-called Lippmann-Schwinger model is suitable to discuss relevant issues in Nuclear Physics as, for example, neutron-nucleus scattering [11] and in Solid State Physics as, for instance, problems related to optical potential [12]. It is well-known (see, for instance, ref. [11]) that the Lippmann-Schwinger approach consists of a system of equations (one equation for each quantum state) relative to the time-independent non-relativistic or relativistic Schrödinger equation with a potential viewed as a perturbation. Here, we will consider the static non-relativistic Schrödinger equation with a time-independent external potential. Within this picture, electron-atom scattering or electron-molecule scattering may be considered by employing the Lippmann-Schwinger equations in relation to a multiband energy-eigenvalue spectrum. In fact, among other things, we will establish a matrix formalism to describe the aforementioned scattering. TheoryWe consider the ( )