Periodic Darboux q-chains
It is well known that discrete spectrum of the harmonic oscillator consists of an arithmetic sequence and is fully defined by the corresponding algebraic relation; its eigenfunctions can be expressed in terms of the Hermite polynomials and therefore they form a complete family in the Hilbert space $\mathcal L_2(\mathbb R)$. In 1990-ies various difference realizations of the so called $q$-oscillator (i.e. an operator that satisfies $q$-deformed Hiesenberg relation) were introduced. In particular, Atakishijev, Frank and Wolf considered $q$-oscillator that is bounded on the lattice $\mathbb Z$.
Veselov and Shabat examined the dressing chain, that is, the periodically closed chain of Schr\"odinger operators, connected by first order Darboux transformations (such chain of length $1$ leads to the harmonic oscillator).
I will present a discrete $q$-analog of the dressing chain and will show that corresponding operator relations can be realized by bounded difference operators in the Hilbert space $\mathcal L_2(\mathbb Z)$ such that their spectrum consists of several $q$-arithmetic sequences and their eigenfunctions form a complete family in $\mathcal L_2(\mathbb Z)$.