Eigenvalue asymptotics for almost flat compact hypersurfaces (Q416915)

Let \(h_\pm\) be a pair of smooth functions on a bounded domain \(\omega\subset\mathbb{R}^n\) which vanish on the boundary \(\partial\omega\). Under further assumptions on \(h_\pm\), and for small \(\varepsilon>0\), the authors consider the Laplace--Beltrami operator, \(\mathcal{H}_\varepsilon\), on the union, \(\mathcal{S}_\varepsilon\), of the graphs of the functions \(\varepsilon h_\pm\). In the limit \(\varepsilon\to 0\), \(\mathcal{S}_\varepsilon\) contracts to a flat two-sided surface. There is a limiting self-adjoint operator, the Laplacian \(\mathcal{H}_0\) defined by certain boundary conditions. For \(m\)-fold eigenvalues \(\lambda\) in the discrete spectrum of \(\mathcal{H}_0\), it is shown that \(\mathcal{H}_\varepsilon\) has \(m\) eigenvalues which satisfy an asymptotic expansion \[ \lambda_k(\varepsilon) = \lambda + \varepsilon^2 \ln\varepsilon \mu_k(\varepsilon) +O(\varepsilon^{2+\rho}), \] \(0<\rho<1/2\), \(k=1,2,\dots,m\). The parameters \(\mu_k\) arise as eigenvalues of a matrix which is constructed from traces at \(\partial\omega\) of \(\mathcal{H}_0\)-eigenfunctions with eigenvalue \(\lambda\).