# zbMATH — the first resource for mathematics

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

##### MSC:
 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