In this work, we introduce the magnetic Schrödinger operator corresponding to the generalized Dirichlet problem. We prove its self-adjointness and discreteness of the spectrum in bounded domains in the multidimensional case. We also prove the basis property of its eigenfunctions in the Lebesgue space and in the magnetic Sobolev space. We give a new characteristic of the definition domain of the magnetic Schrödinger operator. We investigate the existence and uniqueness of a solution of the magnetic Schrödinger equation with a spectral parameter. It is proved that if the spectral parameter is different from the eigenvalues, then the first generalized Dirichlet problem has a unique solution. We then find the solvability condition for the generalized Dirichlet problem when the spectral parameter coincides with the eigenvalue of the Schrödinger magnetic operator.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.