Spectral Analysis of q-Sturm-Liouville Problem with the Spectral Parameter in the Boundary Condition

This paper is concerned with q-Sturm-Liouville boundary value problem in the Hilbert space with a spectral parameter in the boundary condition. We construct a self-adjoint dilation of the maximal dissipative q-difference operator and its incoming and outcoming spectral representations, which make it possible to determine the scattering matrix of the dilation. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of operator generated by boundary value problem

The different tongues of q-calculus

In this review paper we summarize the various dialects of q-calculus: quantum calculus, time scales, and partitions. The close connection between Γq(x) functions on the one hand, and elliptic functions and theta functions on the other hand will be shown. The advantages of the Heine notation will be illustrated by the (q-)Euler reflection formula, q-Appell functions, Carlitz– AlSalam polynomials, and the so-called q-addition. We conclude with some short biographies about famous scientists in q-calculus

Spectral Analysis of Q-difference Equations with Spectral Singularities

In this paper we investigate the eigenvalues and the spectral singularities of non-selfadjoint q-difference equations of second order with spectral singularities