Investigation of the asymptotics of the eigenvalues of a second order quasidifferential

M. Yu. Vatolkin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2024), pp. 15-37

Résumé

We construct the asymptotics of the eigenvalues for a quasidifferential Sturm–Liouville boundary value problem on eigenvalues and eigenfunctions considered on a segment $J=[a,b]$, with the boundary conditions of type I on the left – type I on the right, i.e., for a problem of the form (in the explicit form of record) \begin{gather*} p_{22}(t)\Big(p_{11}(t)\big(p_{00}(t)x(t)\big)^{\prime} +p_{10}(t)\big(p_{00}(t)x(t)\big)\Big)^{\prime}+ p_{21}(t)\Big(p_{11}(t)\big(p_{00}(t)x(t)\big)^{\prime} +p_{10}(t)\big(p_{00}(t)x(t)\big)\Big)+ \\ +p_{20}(t)\big(p_{00}(t)x(t)\big)= -\lambda \big(p_{00}(t)x(t)\big) \ (t\in J=[a,b]),\\ p_{00}(a)x(a)=p_{00}(b)x(b)=0, \end{gather*} The requirements for smoothness of the coefficients (i.e., functions $p_{ik}(\cdot):J\to {\mathbb R}, k\in 0:i, i\in0:2)$ in the equation are minimal, namely, these are: functions $p_{ik}(\cdot):J\to {\mathbb R}$ are such that functions $p_{00}(\cdot) $ and $ p_{22}(\cdot) $ are measurable, nonnegative, almost everywhere finite and almost everywhere nonzero, functions $p_{11}(\cdot)$ and $p_{21}(\cdot)$ are also nonnegative on segment $J$, and in addition, functions $p_{11}(\cdot) $ and $ p_{22}(\cdot) $ are essentially bounded on $J,$ functions $ \dfrac{1}{p_{11}(\cdot)}, \dfrac{p_{10}(\cdot)}{p_{11}(\cdot)}, $ $ \dfrac{p_{20}(\cdot)}{p_{22}(\cdot)}, \dfrac{p_{21}(\cdot)}{p_{22}(\cdot)}, \dfrac{1}{\min \{ p_{11}(t) p_{22}(t), 1 \}} $ are summable on segment $J.$ Function $p_{20}(\cdot)$ acts as a potential. It is proved that under the condition of nonoscillation of a homogeneous quasidifferential equation of the second order on $J,$ the asymptotics of the eigenvalues of the boundary value problem under consideration has the form $$ \lambda_k=\big(\pi k\big)^2 \Big(D+O\big({1}\big{/}{k^2}\big)\Big) $$ as $k \rightarrow \infty,$ where $D$ is a real positive constant defined in some way.

Détail

Keywords: eigenfunction, eigenvalue, power series, estimate for coefficients, quasidifferential equation, boundary value problem, sum of series, representation of eigenfunctions as sums of power series.

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