The spectral expansion for the Hahn–Dirac system on the whole line

Abstract

We consider the singular Hahn–Dirac system defined by$-frac1qD_{-wq^{-1},q^{-1}}y_2+p(x)y_1=lambda y_1$$D_{w,q}y_1+r(x)y_2=lambda y_2$where $lambda$ is a complex spectral parameter and p and r are real-valued functions defined on $(-infty,infty)$ and continuousat $omega_0$ . We prove the existence of a spectral function for such a system. We also prove the Parseval equality and thespectral expansion formula in terms of the spectral function for this system on the whole line.