Operator extension theory models for periodic array of quantum dots and double quantum layer in a magnetic field

Solvable models for periodic array of quantum dots and for double quantum electron layer in a magnetic field based on the theory of self-adjoint extensions of symmetric operators is suggested. The spectral problem reduces to finite-dimensional one by using harmonic analysis on magnetic translations group. The dispersion equations are obtained. Properties of a gap between symmetric and antisymmetric states are discussed.