Geodesic
Differential equations whose solution of the Cauchy problem displays nonclassical behaviour with respect to the parameter $\lambda$
Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1565-1574 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $0<\gamma<1$. Bibliography: 3 titles.
Keywords: eigenfunctions, nonclassical estimates for eigenfunctions, Cauchy problem.
Mots-clés : Sturm-Liouville problem 
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     author = {V. Ya. Yakubov},
     title = {Differential equations whose solution of the {Cauchy} problem displays nonclassical behaviour with respect to the parameter~$\lambda$},
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V. Ya. Yakubov. Differential equations whose solution of the Cauchy problem displays nonclassical behaviour with respect to the parameter $\lambda$. Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1565-1574. http://geodesic.mathdoc.fr/item/SM_2009_200_10_a7/

[1] I. G. Petrovsky, Lectures on partial differential equations, Interscience Publ., New York–London, 1954 | MR | MR | Zbl

[2] V. Ya. Yakubov, “Estimates for eigenfunctions of elliptic operators with respect to the spectral parameter”, Funct. Anal. Appl., 33:2 (1999), 128–136 | DOI | MR | Zbl

[3] V. Ya. Yakubov, “Estimates for solutions to a Cauchy problem involving a spectral parameter”, Differential Equations, 34:1 (1998), 59–64 | MR | Zbl