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Asymptotics of fundamental solutions to Sturm–Liouville problem with respect to spectral parameter
V. E. Vladykina Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
We consider the Sturm–Liouville equation
$$-(r^2y')'+py'+qy=\lambda^2\rho^2 y,\qquad x\in[a,b]\subset\mathbb{R},$$
where
$\lambda^2$ is a spectral parameter,
$r$ and
$\rho$ are positive functions while
$p$ and
$q$ are complex-valued ones. An asymptotic representation for the fundamental system of solutions with respect to the spectral parameter
$\lambda\to\infty$ is obtained in the half-planes $\operatorname{Im}\lambda\geqslant\operatorname{const}$ and $\operatorname{Im}\lambda\leqslant\operatorname{const}$ under the following conditions on the coefficients:
$$p\in L_1[a,b],\quad q\in W_2^{-1}[a,b],\quad\rho,r\in W_1^1[a,b],\quad\rho'u,r'u,pu\in L_1[a,b], \quad\text{where}\quad u=\int q~dx,$$
and the antiderivative is understood in the sense of distributions.
Key words:
Sturm–Liouville equation, asymptotics of solutions with large parameter.
UDC:
517.928 +
517.984 Received: 22.06.2018
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