The asymptotic distribution of eigenvalues for \(\partial ^ 2/\partial x^ 2+Q(x)(\partial ^ 2/\partial y^ 2)\) in a strip domain (Q1064475)

Consider the eigenvalue problem: (*) \(-(\partial^ 2u/\partial x^ 2+Q(x)(\partial^ 2u/\partial y^ 2))=\lambda u\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(\Omega =]0,+\infty [\times]0,\pi [\) in \({\mathbb{R}}^ 2\). Q is smooth and satisfies: \(\inf_{x>0}Q(x)>0\) and \(\lim_{x\to +\infty}Q(x)=+\infty\). The author states the following Weyl formula for the distribution function N(\(\lambda)\) for (*): \[ N(\lambda)=(1/\pi)\sum^{\infty}_{j=1}\int_{\lambda \geq j^ 2Q(x)}(\lambda -j^ 2 Q(x))^{1/2} dx+O(\lambda^{1/2}). \] He gives two methods: one is using the zeta function of the eigenvalues of (*) the other is splitting (*) into a family of Sturm-Liouville problems.