On discreteness of spectrum of a second order differential operator

A new form of a necessary and sufficient conditions for the discreteness of the spectrum of singular operator $-\frac{1}{\rho(x)}(p(x)u')'$, $-\infty\le a\le x\le b\le+\infty$ is obtained. A simpler proof of the necessity is obtained

On discreteness of spectrum of a functional differential operator

summary:We study conditions of discreteness of spectrum of the functional-differential operator $$\mathcal {L} u=-u''+p(x)u(x)+\int _{-\infty }^\infty (u(x)-u(s)) {\rm d}_s r(x,s)$$ on $(-\infty ,\infty )$. In the absence of the integral term this operator is a one-dimensional Schrödinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum

On discreteness of spectrum of a second order differential operator

A new form of a necessary and sufficient conditions for the discreteness of the spectrum of singular operator − 1 ρ(x) (p(x)u 0 0 , −∞ ≤ a ≤ x ≤ b ≤ +∞ is obtained. A simpler proof of the necessity is obtained

On non-oscillation on semi-axis of solutions of second order deviating differential equations

summary:We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations \begin {equation*} u''(x)+\sum _i p_i(x) u'(h_i(x))+\sum _i q_i(x) u(g_i(x)) = 0 \end {equation*} without the delay conditions $h_i(x),g_i(x)\le x$, $i=1,2,\ldots$, and $$ u''(x)+\int _0^{\infty }u'(s){\rm d}_sr_1(x,s)+\int _0^{\infty } u(s){\rm d}_sr_0(x,s) = 0. $

On monotone solutions and a self-adjoint spectral problem for a functional-differential equation of even order

For a self-adjoint boundary value problem for a functional-differential equation of even order, the basis property of the system of eigenfunctions and the equivalence of such statements as the positivity of the corresponding quadratic functional, the Jacobi condition and the positivity of the Green function are established