Spectral properties of second order differential operator determined by non-local boundary conditions

A. N. Shelkovoy

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 21 (2018) no. 4, pp. 18-33.

Résumé

In this work we study the spectral properties of the operator acting in the Hilbert space $L_2[0,2\pi]$ defined by the differential expression $\mathcal{L}y=-\ddot {y}+y$ and nonlocal boundary conditions $$ y(0)=y(2\pi)+\int\limits_0^{2\pi} a_0(t)y(t)dt,\quad \dot {y}(0)=\dot {y}(2\pi)+\int\limits_0^{2\pi} a_1(t)y(t)dt. $$ Here $a_0$ and $a_1$ are functions from $L_2[0,2\pi]$. To investigate spectrum of the operator, $\mathcal{L}$ is used adjoint of the operator $\mathcal{L}^*$ one defined by the differential expression $(\mathcal{L}^{*}x)(t) = (Ax)(t) - (Bx)(t)$ and boundary conditions $ x(0) = x(2\pi), ~ \dot {x}(0) = \dot {x}(2\pi), $ with $A$ generated by the differential expression $Ax = -\ddot {x} + x$ with the domain $$D(A) = \{x\in L_2[0,2\pi] : x, ~ \dot {x} \in C[0,2\pi], ~ \ddot {x} \in L_2[0,2\pi],$$ $$x(0) = x(2\pi), ~ \dot {x}(0) = \dot {x}(2\pi)\},$$ and $(Bx)(t) = \dot {x}(2\pi)a_0(t)-x(2\pi)a_1(t), ~ t\in [0,2\pi], ~ x\in D(A)$. As a method of studying the spectral properties of the operator $A - B$ the similar operators method serves. One of the main results is the following theorem. Theorem 3. Let functions $a_0$ and $a_1$ of bounded variation on a segment $[0,2\pi]$ and sequences $\gamma_1, \gamma_2\colon \mathbb{N}\to \mathbb{R}_+ = [0,\infty)$ defined by formulas: $$ \gamma_1(n) = \Biggl(\frac{\alpha_0^{2}n^{4} + 1}{n^6}~+~\frac{4}{n^2}\sum\limits_{\substack{m\ge 1 \\ m\ne n}}\frac{n^4 + m^4}{m^2|n^2 -m^2|^2}\Biggr)^{1/2} \infty, $$ $$ \gamma_2(n) = 2\max\Biggl\{\frac{1}{2n - 1}; \frac{|a_0^0|}{4n^2} + \sum\limits_{\substack{m\ge 1 \\ m\ne n}}\frac{1}{|n^2 -m^2|}\Biggr\} \infty $$ and $$ \alpha_0 = \sqrt{\frac{|a_0^0|^2 + |a_1^0|^2}{2}},\quad a_0^0 = \frac{1}{\pi}\int\limits_0^{2\pi}a_0(t)dt, \quad a_1^0 = \frac{1}{\pi}\int\limits_0^{2\pi}a_1(t)dt. $$ Let conditions $\lim\limits_{n\to \infty} \gamma_1(n) = 0, \lim\limits_{n\to \infty} \gamma_2(n) = 0$ hold true. Then the spectrum $\sigma(A - B)$ of operator $A - B$ can be represented as $\sigma(A - B) = \bigcup\limits_{n \ge 1}\widetilde {\sigma}_n$ where $\widetilde {\sigma}_n, ~ n \ge 1$, — no more than set of two points. Provided that the estimates: $$\Biggl|\widetilde{\lambda}_{n} - (n^2 + 1) ~ + ~ \frac{(-1)^n}{2}\Biggl| ~ \le ~ c\cdot\frac{\ln{n}}{n},$$ where $\widetilde{\lambda}_n$ — the weighted mean of eigenvalues in $\widetilde {\sigma}_n$. Equally satisfy estimates: $$\Biggl(\int\limits_0^{2\pi}\Biggl|(\widetilde{P}_{n}x)(t) - \frac{1}{\pi}\Biggl(\int\limits_0^{2\pi}x(t)\cos{nt}dt\Biggr)\cos{nt} ~ - $$ $$ - ~ \frac{1}{\pi}\Biggl(\int\limits_0^{2\pi}x(t)\sin{nt}dt\Biggr)\sin{nt}\Biggr|^{2}dt\Biggr)^{1/2} \le c(n)\gamma_1(n), ~ n \ge 1,$$ for some sequence c>0 where $\lim\limits_{n\to \infty} c(n) = 1$. Here $\widetilde{P}_{n}$ is the Riesz projector constructed by spectral of set $\widetilde{\sigma}_{n}$ of operator $A - B$.

Détail

Keywords: eigenvalues, operator spectrum, differential operator of second order operator, spectrum asymptotic, similar operators method.

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