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

Authors: BİLENDER PAŞAOĞLU, HÜSEYİN TUNA

Abstract: We consider the singular Hahn-Dirac system defined by $-\frac{1}{q}D_{-\omega q^{-1},q^{-1}}y_{2}+p\left( x\right) y_{1} =\lambda y_{1},$ $D_{\omega,q}y_{1}+r\left( x\right) y_{2} & =\lambda y_{2},$ where $\lambda$ is a complex spectral parameter and $p$ and $r$ are real-valued functions defined on $(-\infty,\infty)$ and continuous at $\omega_{0}$. We prove the existence of a spectral function for such a system. We also prove the Parseval equality and the spectral expansion formula in terms of the spectral function for this system on the whole line.

Keywords: Hahn-Dirac system, singular point, Parseval equality, spectral function, spectral expansion

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