Romaniuk N. Functional-discrete method for the solution of spectral problems with multiple eigenvalues.

We developed and justified a new scheme of the functional-discrete (FD-) method for the spectral problems of Sturm-Liouville type on a finite interval. The scheme is capable of solving problems with multiple eigenvalues as well as the problems where such multiplicity occur during the solution process. For the Schrodinger equation in the cases when the potential is piecewise constant and when it belongs to the negative Sobolev space, we derived the analytical estimates on corrections to the eigenvalues which are unimprovable with respect to the eigenvalue's index. For such case of the negative Sobolev space we present a sufficient condition for the FD-method's convergence rate to be exponential. We present new algorithmic implementation of FD-method for the Sturm-Liouville problems for the Schrodinger equations with polynomial potential. The advantage of this implementation come from the fact that it relies on the simple algebraic operations only and does not involve the solution of any supplementary boundary problems or integral evaluations unlike the previously known implementations. We also derive new algorithmic implementation of FD-method for the Schrodinger equation with the potential being a derivative of the function with bounded variation and containing a finite linear combination of Dirac delta functions. For such case we established the sufficient conditions of the method's exponential convergence rate. A new scheme of the FD-method's algorithm for the eigenvalue problem in the abstract setting is proposed. We study the cases when the involving linear operators act in the Hilbert and in the Banach spaces and have the discrete spectrum. The basic problem is allowed to have eigenvalues of arbitrary multiplicity. We received the sufficient conditions of super-exponential convergence rate of the proposed approach.