The analytic continuation of the scattering kernel associated with the Schrödinger operator with a penetrable wall interaction (Q1332788)

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.