On the basis property of the Riesz system of root functions of a differential operator with integral conditions
The author studies the differential operator \[ L(y)= -y''(x)+ q(x) y(x), \] \[ \int^1_0 y(x)\overline{\varphi_i}(x) dx= (y,\varphi_i)= 0,\quad i= 1,2, \] \[ D(L)= \{y\in W^2(0,1): (y,\varphi_i)= 0,\;i=1,2\}, \] with \[ \varphi^{(k)}_i(0)= \varphi^{(k)}_i(1)= 0,\quad k= 0,\dots, s_i- 1,\quad i= 1,2, \] \[ \left|\begin{matrix} \varphi^{(s_1)}_1(0) & (-1)^{s_1} \varphi^{(s_1)}_1(1)\\ \varphi_2^{(s_2)}(0) & (-1)\varphi^{(s_2)}_2(1)\end{matrix}\right|\neq 0. \] He proves the following theorem: Let \(g\in C^m(0, 1)\) for \(m= \max(0,2[s_1/2]- 2\), \(2[s_2/2]- 2)\), and \(\varphi_i\in C^{s_i+2}(0, 1)\), \(i= 1,2\). Then the spectrum of \(L\) is discrete and the system of root functions forms a Riesz basis with brackets in the space \(H= \{y\in L_2(0, 1): (y,\psi_{k,i})= 0\), \(k= 0,1,\dots\) \(\left[{s_i\over 2}\right]\), \(i= 1,2\}\), \(\psi_{k,i}= \overline L^k(\varphi_i)\).
scientific article; zbMATH DE number 1439619