Abstract

Spectral Properties of a $\lambda$-Rational Sturm-Liouville Problem H. Langer We consider the boundary value problem $$ y''(x)-p(x)y(x)+\lambda y(x)+\frac{q(x)y(x)}{u(x)-\lambda}=0,\,\quad y(0)=y(1)=0, $$ on the interval $[0,1]$. The functions $q\,(\ge0), u,p$ have to satisfy certain assumptions. This equation contains a so--called floating singularity for $\lambda$ in the range $u([0,1])$. Roughly, this range is the essential spectrum of the block matrix operator $$ \left( \begin{array}{cc} -D^2+p&q^{1/2}\\q^{1/2}&u \end{array} \right) $$ where $D^2$ denotes the second derivative including the boundary conditions. In the lecture results about estimations of the eigenvalues, in particular of the embedded eigenvalues, the spectral function and the corresponding Fourier transformation, nonreal resonances on the nonphysical sheet are presented (joint work with V.Adamyan and M.Langer).

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