Non-Selfadjoint Sturm-Liouville Operators with Multiple Spectra

Abstract

We consider Sturm-Liouville operators

$$H = - \frac{{d^2 }}{{dx^2 }} + q\left( x \right),q\left( x \right) \in \mathfrak{L}^2 \left[ {0,\pi } \right],$$

((1))

with one of the following boundary conditions

$$\begin{gathered} \left. D \right)y\left( 0 \right) = 0,\left. N \right)y\left( 0 \right) = 0,\left. P \right)y\left( 0 \right) = y\left( \pi \right),\left. {AP} \right)y\left( 0 \right) = - y\left( \pi \right), \hfill \\ y\left( \pi \right) = 0;y'\left( \pi \right) = 0;y'\left( 0 \right) = y'\left( \pi \right);y'\left( 0 \right) = - y'\left( \pi \right). \hfill \\ \end{gathered} $$

Spectrum of each problem is discrete and behaves asymptotically as the spectrum of the corresponding operator with q(x)≡0. Namely,

$$\begin{gathered} \lambda _n \left( D \right) = \left( {n + \frac{Q}{n} + \frac{{f_n \left( D \right)}}{n}} \right)^2 ,Q \in C; \hfill \\ \lambda _n \left( N \right) = \left( {n - \frac{1}{2} + \frac{Q}{n} + \frac{{f_n \left( N \right)}}{n}} \right)^2 ,\left\{ {f_n \left( \cdot \right)} \right\}_{n = 1}^\infty \in \ell ^2 ; \hfill \\ \lambda _n^ \pm \left( {AP} \right) = \left( {2n + 1 + \frac{Q}{{2n + 1}} \pm \frac{{f_n \left( {AP} \right)}}{n} + \frac{{g_n \left( {AP} \right)}}{{n^2 }}} \right)^2 . \hfill \\ \end{gathered} $$

((2))

Our aim is to find additional conditions, if any, which guarantee that a given sequence satisfying one of equations (2) is the spectrum of the corresponding boundary problem. In particular, we would like to know whether some points of the spectra may be multiple and whether there are some restrictions on their multiplicities, i.e., on dimensions of corresponding root subspaces.