Asymptotics for Solutions of Linear Differential Equations Having Turning Points With Applications

Asymptotics for Solutions of Linear Differential Equations Having Turning Points With Applications Synopsis

Asymptotics are built for the solutions $y_j(x,\lambda)$, $y_j^{(k)}(0,\lambda)=\delta_{j\,n-k}$, $0\le j,k+1\le n$ of the equation $L(y)=\lambda p(x)y,\quad x\in [0,1],$ where $L(y)$ is a linear differential operator of whatever order $n\ge 2$ and $p(x)$ is assumed to possess a finite number of turning points. The established asymptotics are afterwards applied to the study of: the existence of infinite eigenvalue sequences for various multipoint boundary problems posed on $L(y)=\lambda p(x)y,\quad x\in [0,1],$, especially as $n=2$ and $n=3$ (let us be aware that the same method can be successfully applied on many occasions in case $n>3$ too), and asymptotical distribution of the corresponding eigenvalue sequences on the complex plane.