an:01227230
Zbl 0907.34065
Zhukov, M. Yu.; Sazonov, L. I.
Asymptotics of the eigenvalues for a boundary value problem with \(\delta\)-like coefficients
EN
Differ. Equations 33, No. 4, 469-477 (1997); translation from Differ. Uravn. 33, No. 4, 470-477 (1997).
00050064
1997
j
34L20 47E05 34A30
spectral problem; integral equation; asymptotics; Green functions; differential operators; eigenvalues; eigenfunctions; hydrodynamical stability
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.
V.Chernyatin (Szczecin)