On a boundary value problem with a spectral parameter in the boundary conditions
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 \(\lambda^2\).
boundary value problem for an ordinary differential equation
spectral parameter in the boundary conditions
asymptotic formulas for eigenvalues and eigenfunctions
scientific article; zbMATH DE number 1432313