Differential equations whose solution of the Cauchy problem displays nonclassical behaviour with respect to the parameter~$\lambda$

V. Ya. Yakubov

Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1565-1574

Résumé

The behaviour of solutions of the equation $y''+\lambda\rho(x,\lambda)y=0$ with respect to the spectral parameter $\lambda$ is investigated under the assumption that the function $\rho(x,\lambda)$ does not satisfy the classical conditions. Two types of equations are considered: the Sturm-Liouville equation $y''+\lambda\rho(x)y=0$, whose solutions grow like $c(\rho)\lambda^m$ in the norm of $C[0,l]$ (where $m>0$ is arbitrary), and equations of the form $y''+\lambda\rho(x,\lambda)y=0$, $\lim_{\lambda\to+\infty}\rho(x,\lambda)=1$, whose solutions can grow like $c\lambda^m$ in the norm of $C[0,l]$ (where $m>0$ is arbitrary) and even like $\exp\{m\lambda^{1-\gamma}\}$ where $0\gamma1$. Bibliography: 3 titles.

Détail

Keywords: eigenfunctions, nonclassical estimates for eigenfunctions, Cauchy problem.
Mots-clés : Sturm-Liouville problem