Perturbation determinant and Levinson’s formula for Schrödinger operators with 1-D general point interaction

We consider the one-dimensional Schrödinger operator with properly connecting general point interaction at the origin. We derive a trace formula for trace of difference of resolvents of perturbed and unperturbed Schrödinger operators in terms of a Wronskian which results in an explicit expression for perturbation determinant. Using the estimate for large-time real argument on the trace norm of the resolvent difference of the perturbed and unperturbed Schrödinger operators we express the spectral shift function in terms of perturbation determinant. Under certain integrability conditions on the potential function, we calculate low-energy asymptotics for the perturbation determinant and prove an analog of Levinson’s formula