Spectrum of the Scattering Integral Operator in the Physical Energy Sheet

The kernel of the integral equation for the nonrelativistic scattering of a spinless particle by a potential —λV can easily be symmetrized, if V has a definite sign. Under suitable conditions for the potential the symmetrized kernel is square integrable and generates a completely continuous transformation of the space L 2 that has a pure point spectrum. The scattering kernel is a two‐valued function of the energy s. For negative real values of s in the first Riemann sheet it is symmetric (Hermitian) and can be spectral decomposed in the usual way. Eigenvalues and eigenelements can be analytically continued, at least into the first s sheet and even into the finite second s sheet, if the potential decreases faster than any exponential function for r → ∞. Eigenelements at two complex conjugate points s, s * form a complete biorthogonal system of L 2. The original and the resolvent kernel can be expressed in terms of that system and the eigenvalues, for s in the first and eventually finite second sheet. If the potential is indefinite, the scattering kernel is of polar type for negative real s and can be represented in terms of polar eigenvalues and eigenelements. These may be continued in the same way as for a definite potential. The distribution of eigenvalues in the λ plane is studied for arbitrary complex s and related to the occurrence of bound states and resonances. It is shown that Born expansions for resolvent quantities do converge, if and only if neither λV nor —λV create bound states.