Schwinger variational principle

Schwinger variational principle is a variational principle which expresses the scattering T-matrix as a functional depending on two unknown wave functions. The functional attains stationary value equal to actual scattering T-matrix. The functional is stationary if and only if the two functions satisfy the Lippmann-Schwinger equation. The development of the variational formulation of the scattering theory can be traced to works of L. Hultén and J. Schwinger in 1940s.

The T-matrix expressed in the form of stationary value of the functional reads

where ${\displaystyle \phi }$ and ${\displaystyle \phi '}$ are the initial and the final states respectively, ${\displaystyle V}$ is the interaction potential and ${\displaystyle G_{0}^{(+)}(E)}$ is the retarded Green's operator for collision energy ${\displaystyle E}$. The condition for the stationary value of the functional is that the functions ${\displaystyle \psi }$ and ${\displaystyle \psi '}$ satisfy the Lippmann-Schwinger equation

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