In physical problems, this differential equation must be solved with the input of an additional set of initial and/or boundary conditions for the specific physical system studied.
is a mathematical technicality necessary for the calculation of the integrals needed to solve the equation.
The Lippmann–Schwinger equation is useful in a very large number of situations involving two-body scattering.
For three or more colliding bodies it does not work well because of mathematical limitations; Faddeev equations may be used instead.
For example, in a collision between electrons and molecules, there may be tens or hundreds of particles involved.
But the phenomenon may be reduced to a two-body problem by describing all the molecule constituent particle potentials together with a pseudopotential.
Of course, the main motivations of these approaches are also the possibility of doing the calculations with much lower computational efforts.
into the mix, the Schrödinger equation reads[clarification needed]
As is described below, this singularity is eliminated in two distinct ways by making the denominator slightly complex:
The "in" (+) and "out" (−) states are assumed to form bases too, in the distant past and distant future respectively having the appearance of free particle states, but being eigenfunctions of the complete Hamiltonian.
For high energies and/or weak potential it can also be solved perturbatively by means of Born series.
Very important class of methods is based on variational principles, for example the Schwinger-Lanczos method combining the variational principle of Schwinger with Lanczos algorithm.
Intuitively, it consists of elementary particles or bound states that are sufficiently well separated that their interactions with each other are ignored.
The S-matrix is more symmetric under relativity than the Hamiltonian, because it does not require a choice of time slices to define.
This paradigm allows one to calculate the probabilities of all of the processes that we have observed in 70 years of particle collider experiments with remarkable accuracy.
For example, if one wishes to consider the dynamics inside of a neutron star sometimes one wants to know more than what it will finally decay into.
In the 1960s, the S-matrix paradigm was elevated by many physicists to a fundamental law of nature.
This idea was inspired by the physical interpretation that S-matrix techniques could give to Feynman diagrams restricted to the mass-shell, and led to the construction of dual resonance models.
The uncertainty principle now allows the interactions of the asymptotic states to occur over a timescale
wavepackets is given by an integral over the energy E. This integral may be evaluated by defining the wave function over the complex E plane and closing the E contour using a semicircle on which the wavefunctions vanish.
factor in a Schrödinger picture state forces one to close the contour on the lower half-plane.
Both of these varieties of poles occur at finite imaginary energies and so are suppressed at very large times.
The pole in the energy difference in the denominator is on the upper half-plane in the case of
In this case the contour needs to be closed over the upper half-plane, which therefore misses the energy pole of
One may obtain a formula relating the S-matrix to the potential V using the above contour integral strategy, but this time switching the roles of
's on both sides of the equation one finds the desired formula relating S to the potential
In the Born approximation, corresponding to first order perturbation theory, one replaces this last
which expresses the S-matrix entirely in terms of V and free Hamiltonian eigenfunctions.
These formulas may in turn be used to calculate the reaction rate of the process
With the use of Green's function, the Lippmann–Schwinger equation has counterparts in homogenization theory (e.g. mechanics, conductivity, permittivity).