Abstract
This paper is concerned with the construction of a quasimode for the Laplace operator in a bounded domain Ω in R n , n ≥ 2, with a Dirichlet (Neumann) boundary condition. The quasimode is associated either with a closed gliding ray on the boundary or with a closed broken ray in T*Ω. The frequency set of the quasimode consists of the conic hull of the union of the bicharacteristics of the cosphere bundle S*Ω issuing from a family of invariant tori of the billiard ball map. To construct a quasimode near a gliding ray we find a global symplectic normal form for a pair of glancing hypersurfaces.
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References
V. Babich & V. Lasutkin, Eigenfunctions concentrated near a closed geodesic, Topics in Math. Phys., 2 (M. Birman, ed.), Consultant’s Bureau, New York (1968), pp. 9–18.
Y. Colin de Verdiere, Quasimodes sur les variétés Riemanniennes, Inventiones math., 43 (1977), pp. 15–52.
J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure and Appl. Math., 27 (1974), pp. 207–281.
W. Klingenberg, Lectures on closed geodesics, Springer, Berlin-Heidelberg-New York (1978).
V. Kovachev & G. Popov, Invariant tori for the billiard ball map, (to appear in Transactions AMS).
V. Lasutkin, The existence of caustics for a billiard problem in a convex domain, Math. USSR Izvestija, 7 (1973), pp. 185–214.
V. Lasutkin, Asymptotics of the eigenvalues of the Laplacian and quasi-modes…, Math. USSR Izvestija, 7 (1973), pp. 439–466.
R. Melrose, Equivalence of glancing hypersurfaces, Inventiones Math., 37 (1976), pp. 165–191.
R. Melrose, Transformation of boundary problems, Acta Mathematica, 147 (1981), pp. 149–236.
R. Melrose & M. Taylor, Boundary problems for wave equations with grazing and gliding rays, (preprint).
G. Popov, Glancing hypersurfaces near a closed gliding ray, (submitted).
G. Popov, Quasimodes for the Laplace operator, (submitted).
J. Ralston, Approximate eigenfunctions for the Laplacian, J. Differential Geometry, 12 (1977), pp. 87–100.
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© 1991 Springer-Verlag New York, Inc.
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Popov, G.S. (1991). Quasimodes for the Laplace Operator and Glancing Hypersurfaces. In: Beals, M., Melrose, R.B., Rauch, J. (eds) Microlocal Analysis and Nonlinear Waves. The IMA Volumes in Mathematics and its Applications, vol 30. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9136-4_12
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DOI: https://doi.org/10.1007/978-1-4613-9136-4_12
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