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Some examples in one-dimensional “geometric” scattering on manifolds. (English) Zbl 0914.58037

Let \(\Omega\) be a compact Riemannian manifold which is inserted between two half lines; this means that one glues on two copies of \([0,\infty)\) to two points \(x_1\) and \(x_2\) of a compact Riemannian manifold \(M\). The standard Laplacian \(\Delta_M\) is chosen on \(M\); the operator \(-\partial/\partial x^2\) is chosen on the two half lines. Boundary conditions are chosen at the gluing points \(x_i\) so that the Laplacian does not decompose as a direct sum with summands acting on the different geometric components. The spectrum of the resulting operator is studied as are the transition coefficient in the scattering problem. The author applies these results to the particular example in which \(M\) is the unit sphere in \(\mathbb{R}^3\) and in which the two half lines are glued on at antipodal points; spherical harmonics are used to analyze the situation and the large energy setting is studied.
Reviewer: P.Gilkey (Eugene)

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

[1] Aubin, T., Nonlinear Analysis on Manifolds: Monge-Ampere Equations (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0512.53044
[2] Avron, J. E.; Exner, P.; Last, Y., Periodic Schrödinger operators with large gaps and Wannier-Stark ladders, Phys. Rev. Lett., 72, 896-899 (1994) · Zbl 0942.34503
[3] Exner, P., A model of resonance scattering on curved quantum wires, Ann. Physik, 47, 123-138 (1990)
[4] Exner, P., The absence of the absolutely continuous spectrum for δ′ Wannier-Stark ladders, J. Math. Phys., 36, 4561-4570 (1995) · Zbl 0884.47049
[5] Exner, P.; Seba, P., Quantum motion on a half-line connected to a plane, J. Math. Phys., 28, 386-391 (1987) · Zbl 0634.46057
[6] Gerasimenko, N. I.; Pavlov, B. S., Scattering problems for noncompact graphs, Theoret. Math., 74, 230-240 (1988) · Zbl 0659.47006
[7] Jimbo, S., Perturbation formula of eigenvalues in a singularly perturbed domain, J. Math. Soc. Japan, 45, 339-356 (1993) · Zbl 0785.35069
[8] Kiselev, A. A.; Pavlov, B. S., Eigenvalues and eigenfunctions of Neuman Laplacian in the system of two connected resonators, Theoret. Math., 100, 1065-1074 (1994) · Zbl 0853.35084
[9] Pavlov, B. S.; Popov, I. Yu., Running wave in the ring resonator, Vestnik LGU, 4, 99-102 (1985) · Zbl 1268.81172
[10] Pavlov, B. S.; Popov, I. Yu., Acoustic model of zero-width slits and hydrodynamic boundary-layer stability, Theoret. Math., 86, 269-276 (1991) · Zbl 0729.76074
[11] Popov, I. Yu., The theory of Helmholtz resonator and the theory of operator extensions in a space with indefinite metric, Russian Sb. Math., 75, 285-315 (1992) · Zbl 0798.35037
[12] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. Methods of Modern Mathematical Physics, Fourier analysis, self-adjointness (1975), Academic Press: Academic Press New York · Zbl 0308.47002
[13] Taylor, M. E., Pseudodifferential Operators (1981), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0453.47026
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