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We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with δ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As the main result, we find a connection between the arrangement of the gaps and the coefficients in a continued fraction associated with the ratio of edge lengths of the lattice. This knowledge enables a straightforward construction of a periodic quantum graph with any required number of spectral gaps and—to some degree—to control their positions; i.e. to partially solve the inverse spectral problem. [ABSTRACT FROM AUTHOR]

The Güting algorithm for constructing multidimensional continued fractions is considered. It is proved that, in the case of dimension 2, this algorithm can be used to find the coefficients of the linear dependence of numbers; a criterion is given for verifying that the partial quotients furnished by the algorithmare, indeed, elements of the continued fraction for the expanded (generally irrational) numbers. [ABSTRACT FROM AUTHOR]