RO  EN
IMI/Publicaţii/BASM/Ediţii/BASM n.3 (28), 1998/

On the spectrum of some integro-differential operators. (Romanian)

Authors: Stanescu M. M.

Abstract

Some conditions are established, which quarante the absence of eigenvalues for the integro-differential operator $$ H=\sum\limits_{j=0}^{n}H_{j}D^{n-j}\quad, $$ where $(H_{j}u)(t)=a_{j}(t)u(t)+b_{j}(t) u(t)+\int\limits_{\Bbb{R}_{+}} k_{j}(t,s)u(s)ds;$ $a_j(t+T)=a_j(t)$ \linebreak $(j=0,\ldots, n; a_0(t)=1; t\in \Bbb{R}_{+})$ are functions of period $T$, $b_j(t)$, $k_j(t, s)$ $(j=0, \ldots, n; t, s \in (\Bbb{R}_{+})$; $b_0(t)=k_0(t, s)=0)$ are measurable functions (in general complex). Operator is acting in Hilbert space $L_2(\Bbb{R}_{+})$ (the choice of boundary is not important), but similar rezults can be obtained in spaces $L_p(\Bbb{R}_{+})$, $L_p(\Bbb{R}) (1\leq p< \infty)$. The rezults are true for integro-differential operator with matrix coefficients.

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