Explicit solutions of the Dirichlet–Schrödinger problem via the new cylindrical Poisson–Schrödinger kernel method

Abstract

Abstract In this paper, we study the cylindrical Schrödinger operator and construct a modified cylindrical Poisson–Schrödinger kernel, in analogy to the classical theory of the Laplacian in a strip. A new method expressing explicitly the modified cylindrical Poisson–Schrödinger kernels of any degree is obtained, and a formula expressing the modified cylindrical Poisson–Schrödinger integrals in terms of Nevanlinna–Schrödinger norms is deduced. Applications in the solutions of Dirichlet–Schrödinger problem are described. More specifically, the existence and uniqueness of explicit solutions to the problem mentioned above for the cylindrical Schrödinger equation are obtained. It is shown that our method yields more explicit results than do other methods. Much more challenging geometric configurations, such as a cylinder with a rough base bear new and interesting challenges arising from the lateral boundary conditions. These have-at least to our knowledge-not been addressed before.