Abstract
In these lectures we consider inverse boundary spectral problems for elliptic operators on manifolds. This means the reconstruction of an unknown manifold and an elliptic operator on it from the knowledge of the boundary spectral data, i.e. the spectrum of the operator and normal derivatives of the normalized eigenfunctions on the boundary. Before we formulate and solve this problem in exact terms, we explain why the manifolds appear in the study of the inverse problems.
The author was partly supported by Russian grant RFFI 99-01-00107
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References
Avdonin, S., Belishev, M., Rozhkov, Y.: A dynamic inverse problem for the nonselfadjoint Sturm-Liouville operator. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 250 (1998), 7–21.
Babich V., Ulin V.: The copmlex space-time ray method and quasiphotons, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 117 (1981), 5–12.
Belishev M.: On an approach to multidimensional inverse problems for the wave equation. Dokl. Akad. Nauk SSSR, 297 (1987), 524–527.
Belishev, M.: Wave bases in multidimensional inverse problems. Mat. Sb. 180 (1989), 584–602, 720.
Belishev, M., Glasman, A.: A dynamic inverse problem for the Maxwell system: reconstruction of the velocity in the regular zone (the BC-method). Algebra i Analiz 12 (2000), 131–187.
Belishev, M., Gotlib, V.: Dynamical variant of the BC-method: theory and numerical testing. J. Inverse Ill-Posed Probl. 7 (1999), 221–240.
Belishev, M., Isakov, V., Pestov, L., Sharafutdinov, V.: On the reconstruction of a metric from external electromagnetic measurements. Dokl. Akad. Nauk 372 (2000), 298–300.
Belishev M., Katchalov A.: Boundary control and quasiphotons in the problem of a Riemannian manifold reconstruction via its dynamical data. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) (203)1992, 21–50.
Belishev M. I.: Boundary control in reconstruction of manifolds and metrics. (the BC-method), Inverse Problems 13 (1997), R1–R45.
Belishev M., Kurylev Y.: Nonstationary inverse problem for the wave equation (in large), Zap. Nauchn. Sein. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 165 (1987), 21–30.
Belishev M., Kurylev Y.: To the reconstruction of a Riemannian manifold via its spectral data (BC-method). Comm. Partial Differential Equations 17 (1992). 767–804.
Blagovestchenskii A.: On an inverse problem of seimic waves propagation. Problem of Math. Physics. Leningrad University, 1966, 68–81 (in Russian).
Katchalov, A., Kurylev, Y.: Multidimensional inverse problem with incomplete boundary spectral data. Comm. Partial Differential Equations 23 (1998), 55–95.
Katchalov, A., Kurylev, Y., Lassas, M.: Inverse boundary spectral problems. To appear on CRC Press.
Kurylev, Y.: Admissible groups of transformations that preserve the bound-ary spectral data in multidimensional inverse problems. Dokl. Akad. Nauk 327 (1992), 322–325.
Kurylev, Y.: Inverse boundary problems on Riemannian manifolds, Geometry of the spectrum, Contemp. Math., 173 (1993), 181–192.
Kurylev, Y.: Multi-dimensional inverse boundary problems by BC-method: groups of transformations and uniqueness results. Math. Comput. Modelling 18 (1993), 33–45.
Kurylev, Y.: An inverse boundary problem for the Schrödinger operator with magnetic field. J. Math. Phys. 36 (1995), 2761–2776.
Kurylev Y.: A multidimensional Gel’fand-Levitan inverse boundary problem. Differential Equations and Mathematical Physics (ed. I. Knowles ), Intern. Press (1995), 117–131.
Kurylev Y.: Multidimensional Gel’fand inverse problem and boundary distance map. In: Inverse Problems Related with Geometry (ed. H. Soga). Proceedings of the Symposium at Tokyo Metropolitan University. Ibaraki University, Japan (1997), 1–15.
Kurylev, Y., Lassas, M.: The multidimensional Gel’fand inverse problem for non-self-adjoint operators. Inverse Problems 13 (1997), 1495–1501.
Kurylev, Y., Lassas, M.: Hyperbolic inverse problem with data on a part of the boundary. Differential eq. and math. physics (Birmingham), AMS/IP Stud. Adv. Math., 16 (2000), 259–272.
Kurylev Y., Lassas, M.: Gelf’and Inverse Problem for a Quadratic Operator Pencil. J. Funct. Anal., 176 (2000), 247–263.
Nachman, A. Sylvester,.1., Uhlmann, G. An n-dimensional Borg-Levinson theorem. Comm. Math. Phys. 115 (1988), 595–605.
Novikov, R.G. A multidimensional inverse spectral problem for the equation —JO+ (v(x) — Eu(x))» = 0. (Russian) Funk. Anal. i Priloz. 22 (1988), 11–22.
Petersen, P.: Riemannian geometry. Springer-Verlag, 1998.
Ralston, J.: Gaussian beams and the propagation of singularities. Studies in partial differential equations, 206–248, MAA Stud. Math., 23, 1982.
Sylvester J., Uhlmann G. A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. (2) 125 (1987), 153–169.
Tataru D.: Unique continuation for the solutions to PDE’s; between Hormander’s theorem and Holmgren’s theorem, Comm. Partial Differential Equations 20 (1995), 855–884.
Tataru D.: Boundary controllability for conservative PDE’s, Appl. Math. Op-tim. 31 (1995), 257–295.
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Katchalov, A., Lassas, M. (2004). Gaussian Beams and Inverse Boundary Spectral Problems. In: Bingham, K., Kurylev, Y.V., Somersalo, E. (eds) New Analytic and Geometric Methods in Inverse Problems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-08966-8_4
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DOI: https://doi.org/10.1007/978-3-662-08966-8_4
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