Spectral properties of the Schrödinger operator on lattice
Abstract
We consider the total Hamiltonian <i><u>Ĥ</u><sub>μ</sub></i>, <i>μ</i> > 0, which describes the dynamics of one quantum particle moving in a one-dimensional lattice under the influence of an external field. By applying the Fourier transform, the Hamiltonian <i>Ĥ<sub>μ</sub></i> is transformed into the momentum representation of the corresponding Schrödinger operator <i>H<sub>μ</sub></i>. For various values of the parameter <i>&mu</i>;, the spectral characteristics of the operator <i>H<sub>μ</sub></i> are analyzed. Specifically, the necessary conditions are derived under which the operator exhibits: two discrete eigenvalues, one discrete eigenvalue along with one resonance, or a simple isolated eigenvalue. Furthermore, explicit analytical expressions for the eigenfunctions corresponding to the eigenvalues of the operator <i>H<sub>μ</sub></i> are obtained.