zbMATH — the first resource for mathematics

Asymptotics of the eigenvalues for a boundary value problem with \(\delta\)-like coefficients. (English. Russian original) Zbl 0907.34065
Differ. Equations 33, No. 4, 469-477 (1997); translation from Differ. Uravn. 33, No. 4, 470-477 (1997).
The paper is centred on the spectral problem \[ (D^2- k^2)w= -k^2\theta e^{UH},\quad (D^2- k^2)\theta+ UH_z'D\theta= RUH_z'w, \] \[ D= d/dz,\quad -\textstyle{{1\over 2}}\leq z\leq\textstyle{{1\over 2}},\quad D\theta(\pm\textstyle{{1\over 2}})= 0,\quad w(\mp\textstyle{{1\over 2}})= Dw(\mp\textstyle{{1\over 2}})= 0, \] with \(\delta\)-shaped coefficient, where \(w(z)\), \(\theta(z)\) are the unknown functions, \(H(z,z_0)\) is a given function depending on the parameter \(z_0\), \(k\) and \(U\) are parameters, and \(R\) is the spectral parameter. The authors reduce this problem to an integral equation and studies the asymptotics of Green functions for differential operators from the above equations as parameter \(U\to+\infty\). On this basis, the authors construct the asymptotics of maximal- and minimal-in-modulus eigenvalues and eigenfunctions associated with them for the considered spectral problem. Additionally, the obtained asymptotics are justified and their remainder terms are estimated for large values of parameter \(U\). As an example, a specific spectral problem related to linear theory of hydrodynamical stability is analyzed.
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
34A30 Linear ordinary differential equations and systems, general