Distribution of eigenvalues in the presence of higher order turning points

Leung, Anthony

doi:10.1090/S0002-9947-1977-0447699-3

Distribution of eigenvalues in the presence of higher order turning points
HTML articles powered by AMS MathViewer

by Anthony Leung PDF

Trans. Amer. Math. Soc. 229 (1977), 111-135 Request permission

Abstract:

This article is concerned with the eigenvalue problem $u''(x) - {\lambda ^2}p(x)u(x) = 0,u(x) \in {L_2}( - \infty ,\infty )$, where $p(x)$ is real, analytic and possesses zeroes of arbitrary orders. Under certain conditions on $p(x)$, approximate formulas for the eigenvalues are found. The problem considered is of interest in the study of particle scattering and wave mechanics. The formula is analogous to the quantum rule of Bohr-Sommerfeld.

References

M. A. Evgrafov and M. V. Fedorjuk, Asymptotic behavior of solutions of the equation $w^{\prime \prime }(z)-p(z,\,\lambda )w(z)=0$ as $\lambda \rightarrow \infty$ in the complex $z$-plane, Uspehi Mat. Nauk 21 (1966), no. 1 (127), 3–50 (Russian). MR 0209562
Anthony Wing Kwok Leung, Connection formulas for asymptotic solutions of second order turning points in unbounded domains, SIAM J. Math. Anal. 4 (1973), 89–103. MR 333382, DOI 10.1137/0504010
Anthony Leung, Lateral connections for asymptotic solutions around higher order turning points, J. Math. Anal. Appl. 50 (1975), 560–578. MR 372356, DOI 10.1016/0022-247X(75)90010-4
Yasutaka Sibuya, Subdominant solutions of the differential eqauation $y^{\prime \prime }-\lambda ^{2}(x-a_{1})(x-a_{2}) \cdots (x-a_{m})y=0$, Acta Math. 119 (1967), 235–272. MR 224930, DOI 10.1007/BF02392084
Wolfgang Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0203188
Laurence Weinberg, The asymptotic distribution of eigenvalues for the boundary value problem $y^{\prime \prime }(x)-\lambda ^{2}p(x)y(x)=0,\,\ y\in L_{2}(-\infty ,\,+\infty )$, SIAM J. Math. Anal. 2 (1971), 546–566. MR 306656, DOI 10.1137/0502056
Anthony Wing Kwok Leung, Connection formulas for asymptotic solutions of second order turning points in unbounded domains, SIAM J. Math. Anal. 4 (1973), 89–103. MR 333382, DOI 10.1137/0504010