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Asymptotics of the spectrum of a boundary value problem. (Russian) Zbl 0623.58024
Let G be a compact n-dimensional \(C^{\infty}\)-manifold with the smooth boundary, A a selfadjoint elliptic 2mth order scalar partial differential operator on G with elliptic selfadjoint boundary conditions, semibounded from below. Then there is an asymptotic formula for the eigenvalue counting function \[ N(\lambda)=a\lambda^{n/2m}+O(\lambda^{(n- 1)/2m})\quad as\quad \lambda \to +\infty. \] Moreover, if mes \(\Lambda\) \(=mes \Pi =0\) where \(\Lambda\) is the set of all the deadlock points of the Hamiltonian flow with reflection and (perhaps) branching at the boundary and \(\Pi\) is the set of all the points absolutely periodic with respect to this flow then \[ N(\lambda)=a\lambda^{n/2m}+(b+o(1))\lambda^{(n-1)/2m}\quad as\quad \lambda \to +\infty. \] Remark. The first asymptotic formula holds for arbitrary elliptic selfadjoint matrix operators (and its semiboundedness is not necessary) and under a condition of global nature (more restrictive than the author’s) the second holds too [the reviewer, Sib. Mat. Zh. 28, No.1, 107-114 (1987)].
Reviewer: V.Ya.Ivrij

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J45 Hyperbolic equations on manifolds
35P25 Scattering theory for PDEs
58J32 Boundary value problems on manifolds