Investigation of the dimension of the spectral projection of a self-adjoint second order quasidifferential operator

M. Yu. Vatolkin

Kalashnikov Izhevsk State Technical University, 7 Studencheskaya str., Izhevsk, 426069 Russia

Abstract: Let $\lambda_1$ and $\lambda_2$ be real, $\lambda_1<\lambda_2,$ functions $\psi_{-}(\lambda_i,t)$ be solutions to the second order quasidifferential equations $L\psi_-={\lambda_i }_P^0\psi_-$, $i=1,2$, satisfying a homogeneous boundary condition at point $a.$ We express the number of eigenvalues of operator $L,$ belonging to the interval $(\lambda_1,\lambda_2)$ (or the dimension of its spectral projection relative to the interval $(\lambda_1,\lambda_2)$), in terms of the number of zeros of the Vronskian composed for the functions $\psi_{-}(\lambda_1,t)$ and $\psi_{-}(\lambda_2,t).$

Keywords: quasidifferential operator, spectral projection, self-adjoint quasidifferential expression.

UDC: 517.925

Received: 01.06.2023
Revised: 06.12.2023
Accepted: 26.12.2023

DOI: 10.26907/0021-3446-2024-7-47-62

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2024, 68:7, 34–48