New series of asymptotics of eigenvalues of the Sturm-Liouville problem with rapidly oscillating coefficients (Q1324908)

This article deals with asymptotics of eigenvalues \(\lambda^{(n)}(\varepsilon)\) for the Sturm-Liouville problem \(\partial_ x a(\varepsilon^{-1} x)\partial_ x u(\varepsilon,x)+ \lambda(\varepsilon)\rho(\varepsilon^{-1} x)u(\varepsilon,x)= 0\) \((x\in [-1,1])\), \(u(\varepsilon,\pm 1)= 0\). These asymptotics are written in the form \(\lambda^{(n)}_ m(\varepsilon)= \varepsilon^{-2} \mu_ m+ \varepsilon^{-1}\Lambda_ m^{(n)}+ \lambda^{(n)}_ m+\cdots\), where \(m\) is number of series of asymptotics, \(\mu_ m\) is the eigenvalue with number \(m\) of the degenerated problem \(\partial_ x a(y)\partial_ x \varphi(y)+ \mu\rho(y) \varphi(y)= 0\) \((y\in [0,1])\), \(\varphi(1)= \varphi(0)\), \(\partial_ y \varphi(1)= \partial_ y \varphi(0)\).