Intertwining operators associated with a Dunkl type operator on the real line and applications (Q2910614)

The authors consider a generalization of the one-dimensional Dunkl operator NEWLINE\[NEWLINED_{\gamma}f(x)=f'(x)+\left(\gamma+\frac{1}{2}\right)\frac{f(x)-f(-x)}{x}.NEWLINE\]NEWLINE In particular, they consider the operator \(\Lambda\) defined by NEWLINE\[NEWLINE\Lambda f(x):=f'(x)+\left(\gamma+\frac{1}{2}\right)\frac{f(x)-f(-x)}{x}+q(x)f(x),NEWLINE\]NEWLINE where \(\gamma>-\frac{1}{2}\) and \(q\) is an odd \(\mathcal{C}^{\infty}\)-function on \(\mathbb{R}\). The authors provide solutions of the eigenvalue problem NEWLINE\[NEWLINE\Lambda f(x)=i\lambda f(x),\quad f(0)=1.NEWLINE\]NEWLINE In addition, they study certain of the properties of the dual operator \(\widetilde{\Lambda}\) as well as provide a discussion of the generalized Fourier transform associated to the operator \(\Lambda\).