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Exponential asymptotics for an eigenvalue of a problem involving parabolic cylinder functions. (English) Zbl 0759.34054
The authors consider the problem of finding the imaginary part of the “eigenvalue” $\lambda$ of the problem ${y}^{\text{'}\text{'}}\left(x\right)+\left(\lambda +\epsilon {x}^{2}\right)y\left(x\right)=0$ on $\left(0,\infty \right)$, ${y}^{\text{'}}\left(0\right)+hy\left(0\right)=0$, ${lim}_{x\to \infty }W\left[y\left(x\right),exp\left(\frac{i}{2}{\epsilon }^{1/2}{x}^{2}\right)\right]=0$, where $\epsilon \ge 0$, $h$ is a positive constant, $W$ is the Wronskian. They obtain the asymptotic formula (1) $\text{Im}\lambda \sim -2{h}^{2}exp\left[-\frac{{h}^{2}\pi }{2\sqrt{\epsilon }}\right]$ as $\epsilon \to 0+$. This result is obtained by two different methods, both of which use results which the authors establish on asymptotics of the parabolic cylinder function and on exponentially improved asymptotics of a quotient of gamma functions. The first method consists of construction of a linear combination of parabolic cylinder functions that satisfy the boundary condition at infinity. Insertion into the boundary condition at the origin yields (1). The second method consists of construction of the Titchmarsh-Weyl function $m\left(\lambda \right)$ and determination of its pole approximated by (1). A problem similar to the one in this paper with potential $\epsilon x$ instead of $\epsilon {x}^{2}$ has been studied by R. B. Paris and A. D. Wood [IMA J. Appl. Math. 43, No. 3, 273-384 (1989; Zbl 0695.34057)]. Problems of the type studied in this article have arisen in optical tunnelling and in the theory of surface waves.
##### MSC:
 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 34B20 Weyl theory and its generalizations 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE) 34B24 Sturm-Liouville theory 33B15 Gamma, beta and polygamma functions 33E99 Other special functions