The analytic continuation of the scattering kernel associated with the Schrödinger operator with a penetrable wall interaction
The author studies the analytical properties of the scattering matrix with respect to the wave vector \(k\) for the Schrödinger operator \(H = - \Delta + q(x)\) \(\delta (| x | - a)\) in \(L_ 2 (\mathbb{R}^ 3)\) where the function \(q(x)\) is real and smooth and \(\delta\) means the Dirac delta-function. He proves that the scattering kernel can be continued to the whole complex plane as a meromorphic function. If an imaginary part of a variable \(k\) is positive, the scattering kernel may have only simple poles and only on the imaginary axis and at most a finite number. They correspond to the negative eigenvalues of the Schrödinger operator. On the real axis the scattering kernel may have a zero resonance if any. The author examines domains of the complex plane free of poles and deals with the case where \(q(x)\) is constant.
analytical properties of the scattering matrix
scientific article; zbMATH DE number 633570