Spectral asymptotics for elliptic second order differential operators
Georgi S. Popov · Kyoto journal of mathematics · 1985
O . Introduction and statement of the results Let X be an open set in R , n> 2, with a smooth boundary Y and R n\X c B R = Ix ; x 0. Suppose that Rn is provided with a smooth Riemannian metric ds 2 =g 1 i(x)dx 1dx1 which is Euclidean outside th e b a ll B R . S et g(x)= det (g i i (x)), g o f ' =6t (Kronecker's index), i, k=1,..., n where th e summation convention is u sed . Let A9, g u (x )= g(x) - 1 -/ 2 o a x i (g(x)i/ 2 gti(x) e a x i ( x ) ) , u e (R n), be the corresponding Laplace-Beltrami operator and H = —Gig + V (X ) with some real-valued function Ve C,T(BR ). The operator H will be considered as a self-adjoint operator in L 2 (X ) (the scalar product in L 2 (X ) is given by (u, v)=1 x u(x)v(x)g 1