Discrete Quantum Scattering Theory

We formulate quantum scattering theory in terms of a discrete $L_2$-basis of eigen differentials. Using projection operators in the Hilbert space, we develop a universal method for constructing finite-dimensional analogues of the basic operators of the scattering theory: $S$- and $T$-matrices, resolvent operators, and Möller wave operators as well as the analogues of resolvent identities and the Lippmann–Schwinger equations for the $T$-matrix. The developed general formalism of the discrete scattering theory results in a very simple calculation scheme for a broad class of interaction operators.