Journal of Mathematical Sciences, Vol. 138, No. 2, 2006 BEHAVIOR OF GAUSSIAN BEAMS IN AN ANISOTROPIC MEDIUM WITH INTERFACE A. S. Kirpichnikova ∗ UDC 517.9 A region that consists of two parts with anisotropic Riemannian metrics is considered. The metric has a jump on the interface. Asymptotic solutions of the wave equation, reflected and transmitted from the interface, i.e., Gaussian beams (“quasiphotons”), are constructed. Bibliography: 7 titles. 1. Results 1.1. Main definitions. Consider two anisotropic media Ω − and Ω + with a common part, which we call a common interface γ. Assume that the dimension of Ω ± is n, and the interface γ is a hypersurface. Suppose the parts Ω ± are Riemannian manifolds C ∞ -smooth up to the boundary ∂Ω ± . Denote the Riemannian metric tensor on Ω ± by g ± , i.e., g ± = g(Ω ± ), assuming that g − | γ = g + | γ . (1) 1.1.1. Coordinates. We use boundary normal coordinates (also called semigeodesic) (see [1, 2, 4]), i.e., coordinates, associated with the interface γ, such that (q,σ ± )=(q 1 ,...,q n−1 ,σ ± )= {q α ,σ ± },α =1,...,n − 1, where q denotes some smooth coordinates on γ and σ ± is the distance to γ in the metric g ± , i.e., σ =      σ + > 0 in Ω + , σ =0 on γ, −σ − < 0 in Ω − . We use Greek letters α, β, δ, . . . for enumeration of (n − 1) coordinates of the interface, {q α } = {q 1 ,...,q n−1 }. Also, we need some notation for any regular (inner) coordinates x =(x 1 ,...,x n )= {x i },i =1, . . . , n, i.e., coordinates, smooth in some vicinity V ⊂ Ω −  Ω + of a point on γ. Latin letters enumerate inner coordinates of media {x i } = {x 1 ,...,x n }. We choose an origin such that it belongs to γ; for instance, we take the point M 1 with coordinates σ =0,q α = 0 to be the origin of semigeodesic coordinates. Since σ is orthogonal to all q α , the metric tensor matrix in semigeodesic coordinates takes the form {g ± ij } =  g ± αβ 0 0 1  , where g ± αβ =( dx dq α , dx dq β ) g ± is the (n − 1) × (n − 1) smooth matrix of tangential components of the metric. 1.1.2. The Laplace–Beltrami operator and the Dirichlet Problem. Consider the Laplace-Beltrami operator ∆ ± on both sides from γ; it has the following form in local semigeodesic coordinates: ∆ g = 1 √ g (∂ α g αβ √ g∂ β + ∂ n √ g∂ n ), g = det g ij , i,j =1, . . . , n, (2) where ∂ α := ∂ ∂q α ,∂ n := ∂ σ := ∂ ∂σ , and g =  g + , σ> 0 g − , σ< 0 or g =  g + , (q,σ + ) ∈ Ω + g − , (q,σ − ) ∈ Ω − . Recall metric jump condition (1) on the interface. The worst case (in the sense of the smoothness class of the operator) is the case where the metric tensor g has a jump on a set γ of zero measure, i.e., g ∈ L ∞ . The Dirichlet form is defined for such an operator. The latter means that the operator is well defined in the weak sense, i.e., * St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 77–109. Original article submitted June 3, 2005. 5524 1072-3374/06/1382-5524 c 2006 Springer Science+Business Media, Inc.