Sergey Labovskiy

On Discreteness Criterion for the Spectrum of a Differential Operator of Even Order

abstract:
Necessary and sufficient conditions are obtained for the discreteness of the spectrum of the operator
\begin{equation*}
\mathcal{L} u (x):= \frac{1}{\rho(x)}\,(-1)^m (p(x)u^{(m)}(x))^{(m)}, \;\; x\in I=[0,\infty),\;\; m\ge 1.
\end{equation*}
In the case of $p(x)=x^\nu$, $\nu\in[0,1)$, they are the same:
\begin{equation*}
\lim_{s\to\infty}s^{2m-1-\nu}\int\limits_s^\infty\rho(x)\,dx=0.
\end{equation*}

Mathematics Subject Classification: 34L05

Key words and phrases: Differential operator, discreteness of the spectrum