Eigenoscillations of a string with an additional mass. (English. Russian original) Zbl 0702.34050

Sib. Math. J. 29, No. 5, 744-760 (1988); translation from Sib. Mat. Zh. 29, No. 5(171), 71-91 (1988).
The eigenvalue problem \[ u''(x)+\lambda (\rho (x)+\epsilon^{-m} h(x/\epsilon))u(x)=0,\quad x\in (-a,b),\quad u(-a)=u(b)=0, \] where \(\epsilon >0\) is a small parameter, \(a,b>0\), \(0<\rho_ 0\leq \rho (x)\leq \rho_ 1\) and h(t) is a positive function on [-1,1], \(h(t)=0\) for \(| t| >1\), is investigated. The asymptotic expansions for the eigenvalues \(\lambda_ k(\epsilon)\) and the eigenfunctions \(u_ k(\epsilon,x)\) of the problem with respect to parameter \(\epsilon\) are constructed and the estimates of remainders are given. The character of these eigenoscillations depends on the exponent m and then different cases for \(m<1\), \(m=1\), \(1<m<2\), \(m\geq 2\) must be treated separately.
Reviewer: M.Kopáčková


34E05 Asymptotic expansions of solutions to ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
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