Spectral properties of the Schrödinger operator on lattice

We consider the total Hamiltonian <i><u>Ĥ</u>_&mu;</i>, <i>&mu;</i> &gt; 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_&mu;</i> is transformed into the momentum representation of the corresponding Schrödinger operator <i>H_&mu;</i>. For various values of the parameter <i>&mu</i>;, the spectral characteristics of the operator <i>H_&mu;</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_&mu;</i> are obtained.