Wave functions and spectrum of electrons in a bounded crystal

Problem of determining the spectrum and the wave function of an electron in a semi-bounded crystal with periodical potential of the muffin-tin type is considered. Solution of the Schrodinger equation is reduced to the solution of the system of the Hilbert boundary value problems, the condition of solvability of which determines the electron spectrum. It is shown that the wave functions and the spectrum of an electron can be expressed in terms of the parameters relating to one layer parallel to the boundary of the crystal.