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On a boundary value problem with a spectral parameter in the boundary conditions. (English. Russian original) Zbl 0942.34067
The authors study the boundary value problem with a spectral parameter in the equation and the boundary conditions: $$-u'' + q(x)u = \lambda^2u,\quad 0<x<1,$$ $$(\alpha_0 + \alpha_1\lambda + \alpha_2\lambda^2)u(0) + u'(0) = 0,\quad (\beta_0 + \beta_1\lambda + \beta_2\lambda^2)u(1) + u'(1) = 0.$$ Here, $\lambda$ is a spectral parameter, $q(x)$ is a nonnegative continuous function on the interval $[0,1]$, and $\alpha_i$ and $\beta_i$ are real constants. The authors study properties of the eigenvalues and eigenfunctions of the boundary value problem. The main result is a theorem on the number of zeros of eigenfunctions under their natural enumeration and a theorem on asymptotic formulas for eigenvalues and eigenfunctions. It was Shkalikov who drew the authors’ attention to the nontriviality of this problem in the case when the boundary conditions involve $\lambda$ and $\lambda^2$.

34L05General spectral theory for OD operators
34B24Sturm-Liouville theory
34L20Asymptotic distribution of eigenvalues for OD operators
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory