Boundary and scattering rigidity problems in the presence of a magnetic field and a potential

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November 2015, 9(4): 935-950. doi: 10.3934/ipi.2015.9.935

Boundary and scattering rigidity problems in the presence of a magnetic field and a potential

Yernat M. Assylbekov ^1, and Hanming Zhou ^2,

1.	Department of Mathematics, University of Washington, Seattle, WA 98195-4350
2.	DPMMS, Centre for Mathematical Sciences, Cambridge CB3 0WB, United Kingdom

Received February 2015 Revised June 2015 Published October 2015

Abstract
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In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$ and a potential $U$. For brevity, this type of systems are called $\mathcal{MP}$-systems. On simple $\mathcal{MP}$-systems, we consider both the boundary rigidity problem and scattering rigidity problem. Unlike the cases of geodesic or magnetic systems, knowing boundary action functions or scattering relations for only one energy level is insufficient to uniquely determine a simple $\mathcal{MP}$-system up to natural obstructions, even under the assumption that the boundary restriction of the system is given, and we provide some counterexamples. By reducing an $\mathcal{MP}$-system to the corresponding magnetic system and applying the results of [6] on simple magnetic systems, we prove rigidity results for metrics in a given conformal class, for simple real analytic $\mathcal{MP}$-systems and for simple two-dimensional $\mathcal{MP}$-systems.

Keywords: magnetic field, action., gauge invariance, Boundary rigidity, potential.

Mathematics Subject Classification: Primary: 53C24; Secondary: 35R3.

Citation: Yernat M. Assylbekov, Hanming Zhou. Boundary and scattering rigidity problems in the presence of a magnetic field and a potential. Inverse Problems and Imaging, 2015, 9 (4) : 935-950. doi: 10.3934/ipi.2015.9.935

References:

[1]	D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian], Uspekhi Mat. Nauk, 22 (1967), 107-172.
[2]	V. I. Arnold, Some remarks on flows of line elements and frames, Sov. Math. Dokl., 138 (1961), 255-257.
[3]	V. I. Arnold and A. B. Givental, Symplectic Geometry, Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, Springer Verlag, Berlin, 1990.
[4]	C. B. Croke, Rigidity and distance between boundary points, J. Diff. Geom., 33 (1991), 445-464.
[5]	C. B. Croke, Rigidity theorems in Riemannian geometry, in Geometric Methods in Inverse Problems and PDE Control, IMA Vol. Math. Appl., 137, Springer, New York, 2004, 47-72. doi: 10.1007/978-1-4684-9375-7_4.
[6]	N. S. Dairbekov, G. P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014.
[7]	N. S. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation, Inverse Problems and Imaging, 4 (2010), 397-409. doi: 10.3934/ipi.2010.4.397.
[8]	M. L. Gerver and N. S. Nadirashvili, Inverse problem of mechanics at high energies, (Russian) Comput. Seismology, 15 (1983), 118-125.
[9]	P. Herreros, Scattering boundary rigidity in the presence of a magnetic field, Comm. Anal. Geom., 20 (2012), 501-528. doi: 10.4310/CAG.2012.v20.n3.a3.
[10]	P. Herreros and J. Vargo, Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field, J. Geom. Anal., 21 (2011), 641-664. doi: 10.1007/s12220-010-9162-z.
[11]	A. Jollivet, On inverse problems in electromagnetic field in classical mechanics at fixed energy, J. Geom. Anal., 17 (2007), 275-319. doi: 10.1007/BF02930725.
[12]	V. V. Kozlov, Calculus of variations in the large and classical mechanics, (Russian) Uspekhi Mat. Nauk, 40 (1985), 33-60, 237.
[13]	R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., 65 (1981), 71-83. doi: 10.1007/BF01389295.
[14]	R. G. Mukhometov, On a problem of reconstructing Riemannian metrics, (Russian) Sibirsk. Mat. Zh., 22 (1981), 119-135, 237.
[15]	R. G. Mukhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in $n$-dimensional space, (Russian) Dokl. Akad. Nauk SSSR, 243 (1978), 41-44.
[16]	S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 37-52, 96.
[17]	S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.
[18]	S. P. Novikov and I. Shmel'tser, Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in a liquid, and the extended Lyusternik-Schnirelmann-Morse theory. I, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 54-66.
[19]	R. G. Novikov, Small angle scattering and X-ray transform in classical mechanics, Ark. Mat., 37 (1999), 141-169. doi: 10.1007/BF02384831.
[20]	G. P. Paternain and M. Paternain, Anosov geodesic flows and twisted symplectic structures, in International Conference on Dynamical Systems (Montevideo, 1995) (eds. F. Ledrappier, J. Lewowicz and S. Newhouse), Pitman Research Notes in Math., 362, Longman, Harlow, 1996, 132-145.
[21]	L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.
[22]	V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994. doi: 10.1515/9783110900095.
[23]	P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. doi: 10.1215/S0012-7094-04-12332-2.
[24]	P. Stefanov and G.Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7.
[25]	P. Stefanov and G. Uhlmann, Recent progress on the boundary rigidity problem, Electron. Res. Announc. Amer. Math. Soc., 11 (2005), 64-70. doi: 10.1090/S1079-6762-05-00148-4.
[26]	P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data,, preprint, (). doi: 10.1090/jams/846.

show all references

References:

[1]	D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian], Uspekhi Mat. Nauk, 22 (1967), 107-172.
[2]	V. I. Arnold, Some remarks on flows of line elements and frames, Sov. Math. Dokl., 138 (1961), 255-257.
[3]	V. I. Arnold and A. B. Givental, Symplectic Geometry, Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, Springer Verlag, Berlin, 1990.
[4]	C. B. Croke, Rigidity and distance between boundary points, J. Diff. Geom., 33 (1991), 445-464.
[5]	C. B. Croke, Rigidity theorems in Riemannian geometry, in Geometric Methods in Inverse Problems and PDE Control, IMA Vol. Math. Appl., 137, Springer, New York, 2004, 47-72. doi: 10.1007/978-1-4684-9375-7_4.
[6]	N. S. Dairbekov, G. P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014.
[7]	N. S. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation, Inverse Problems and Imaging, 4 (2010), 397-409. doi: 10.3934/ipi.2010.4.397.
[8]	M. L. Gerver and N. S. Nadirashvili, Inverse problem of mechanics at high energies, (Russian) Comput. Seismology, 15 (1983), 118-125.
[9]	P. Herreros, Scattering boundary rigidity in the presence of a magnetic field, Comm. Anal. Geom., 20 (2012), 501-528. doi: 10.4310/CAG.2012.v20.n3.a3.
[10]	P. Herreros and J. Vargo, Scattering rigidity for analytic Riemannian manifolds with a possible magnetic field, J. Geom. Anal., 21 (2011), 641-664. doi: 10.1007/s12220-010-9162-z.
[11]	A. Jollivet, On inverse problems in electromagnetic field in classical mechanics at fixed energy, J. Geom. Anal., 17 (2007), 275-319. doi: 10.1007/BF02930725.
[12]	V. V. Kozlov, Calculus of variations in the large and classical mechanics, (Russian) Uspekhi Mat. Nauk, 40 (1985), 33-60, 237.
[13]	R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., 65 (1981), 71-83. doi: 10.1007/BF01389295.
[14]	R. G. Mukhometov, On a problem of reconstructing Riemannian metrics, (Russian) Sibirsk. Mat. Zh., 22 (1981), 119-135, 237.
[15]	R. G. Mukhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in $n$-dimensional space, (Russian) Dokl. Akad. Nauk SSSR, 243 (1978), 41-44.
[16]	S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 37-52, 96.
[17]	S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.
[18]	S. P. Novikov and I. Shmel'tser, Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in a liquid, and the extended Lyusternik-Schnirelmann-Morse theory. I, (Russian) Funktsional. Anal. i Prilozhen., 15 (1981), 54-66.
[19]	R. G. Novikov, Small angle scattering and X-ray transform in classical mechanics, Ark. Mat., 37 (1999), 141-169. doi: 10.1007/BF02384831.
[20]	G. P. Paternain and M. Paternain, Anosov geodesic flows and twisted symplectic structures, in International Conference on Dynamical Systems (Montevideo, 1995) (eds. F. Ledrappier, J. Lewowicz and S. Newhouse), Pitman Research Notes in Math., 362, Longman, Harlow, 1996, 132-145.
[21]	L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.
[22]	V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994. doi: 10.1515/9783110900095.
[23]	P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. doi: 10.1215/S0012-7094-04-12332-2.
[24]	P. Stefanov and G.Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7.
[25]	P. Stefanov and G. Uhlmann, Recent progress on the boundary rigidity problem, Electron. Res. Announc. Amer. Math. Soc., 11 (2005), 64-70. doi: 10.1090/S1079-6762-05-00148-4.
[26]	P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data,, preprint, (). doi: 10.1090/jams/846.

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