# American Institute of Mathematical Sciences

August  2014, 8(3): 865-883. doi: 10.3934/ipi.2014.8.865

## Rellich type theorems for unbounded domains

 1 Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hallstromin katu 2b), FI-00014 University of Helsinki, Finland

Received  January 2014 Revised  May 2014 Published  August 2014

We give several generalizations of Rellich's classical uniqueness theorem to unbounded domains. We give a natural half-space generalization for super-exponentially decaying inhomogeneities using real variable techniques. We also prove under super-exponential decay a discrete generalization where the inhomogeneity only needs to vanish in a suitable cone.
The more traditional complex variable techniques are used to prove the half-space result again, but with less exponential decay, and a variant with polynomial decay, but with supports exponentially thin at infinity. As an application, we prove the discreteness of non-scattering energies for non-compactly supported potentials with suitable asymptotic behaviours and supports.
Citation: Esa V. Vesalainen. Rellich type theorems for unbounded domains. Inverse Problems & Imaging, 2014, 8 (3) : 865-883. doi: 10.3934/ipi.2014.8.865
##### References:
 [1] R. Adams, Capacity and compact imbeddings,, Journal of Mathematics and Mechanics, 19 (1970), 923. Google Scholar [2] R. Adams and J. Fournier, Sobolev Spaces,, Pure and Applied Mathematics Series, (2003). Google Scholar [3] S. Agmon, Spectral properties of Schrödinger operators and scattering theory,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 2 (1975), 151. Google Scholar [4] S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics,, J. Anal. Math., 30 (1976), 1. doi: 10.1007/BF02786703. Google Scholar [5] E. Blåsten, L. Päivärinta and J. Sylvester, Corners always scatter,, Commun. Math. Phys., 331 (2014), 725. doi: 10.1007/s00220-014-2030-0. Google Scholar [6] F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237. doi: 10.1137/090769338. Google Scholar [7] F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory,, in Inverse Problems and Applications, 60 (2013), 529. Google Scholar [8] F. Cakoni and H. Haddar, Transmission eigenvalues,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/100201. Google Scholar [9] D. Colton, A. Kirsch and L. Päivärinta, Far field patterns for acoustic waves in an inhomogeneous medium,, SIAM J. Math. Anal., 20 (1989), 1472. doi: 10.1137/0520096. Google Scholar [10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Applied Mathematical Sciences, (2013). doi: 10.1007/978-1-4614-4942-3. Google Scholar [11] D. Colton and P. Monk, The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium,, Quart. J. Mech. Appl. Math., 41 (1988), 97. doi: 10.1093/qjmam/41.1.97. Google Scholar [12] D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Probl. Imaging, 1 (2007), 13. doi: 10.3934/ipi.2007.1.13. Google Scholar [13] P. G. Grinevich and S. V. Manakov, The inverse scattering problem for the two-dimensional Schrödinger operator, the $\overline\partial$-method and non-linear equations,, Funct. Anal. Appl., 20 (1986), 14. Google Scholar [14] P. G. Grinevich and R. G. Novikov, Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials,, Commun. Math. Phys., 174 (1995), 409. doi: 10.1007/BF02099609. Google Scholar [15] K. Hickmann, Interior transmission eigenvalue problem with refractive index having $C^2$-transition to the background medium,, Appl. Anal., 91 (2012), 1675. doi: 10.1080/00036811.2011.577741. Google Scholar [16] M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Anal., 42 (2010), 2965. doi: 10.1137/100793748. Google Scholar [17] L. Hörmander, Lower bounds at infinity for solutions of differential equations with constant coefficients,, Israel J. Math., 16 (1973), 103. doi: 10.1007/BF02761975. Google Scholar [18] L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients,, Classics in Mathematics, (2005). Google Scholar [19] H. Isozaki and H. Morioka, A Rellich type theorem for discrete Schrödinger operators,, Inverse Probl. Imaging, 8 (2014), 475. doi: 10.3934/ipi.2014.8.475. Google Scholar [20] T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995). Google Scholar [21] A. Kirsch, The denseness of the far field patterns for the transmission problem,, IMA J. Appl. Math., 37 (1986), 213. doi: 10.1093/imamat/37.3.213. Google Scholar [22] E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104003. Google Scholar [23] W. Littman, Decay at infinity of solutions to partial differential equations with constant coefficients,, Trans. Amer. Math. Soc., 123 (1966), 449. doi: 10.1090/S0002-9947-1966-0197951-7. Google Scholar [24] W. Littman, Decay at infinity of solutions to partial differential equations; removal of the curvature assumption,, Israel J. Math., 8 (1970), 403. doi: 10.1007/BF02798687. Google Scholar [25] W. Littman, Maximal rates of decay of solutions of partial differential equations,, Arch. Ration. Mech. Anal., 37 (1970), 11. Google Scholar [26] M. Murata, A theorem of Liouville type for partial differential equations with constant coefficients,, Journal of the Faculty of Science, 21 (1974), 395. Google Scholar [27] M. Murata, Asymptotic behaviors at infinity of solutions to certain partial differential equations,, Journal of the Faculty of Science, 23 (1976), 107. Google Scholar [28] R. G. Newton, Construction of potentials from the phase shifts at fixed energy,, J. Math. Phys., 3 (1962), 75. doi: 10.1063/1.1703790. Google Scholar [29] L. Päivärinta, M. Salo and G. Uhlmann, Inverse scattering for the magnetic Schrödinger operator,, J. Funct. Anal., 259 (2010), 1771. doi: 10.1016/j.jfa.2010.06.002. Google Scholar [30] L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525. Google Scholar [31] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness,, Academid Press, (1975). Google Scholar [32] T. Regge, Introduction to complex orbital moments,, Il Nuovo Cimento, 14 (1959), 951. doi: 10.1007/BF02728177. Google Scholar [33] F. Rellich, Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u=0$ im unendlichen Gebieten,, Jahresber. Dtsch. Math.-Ver., 53 (1943), 57. Google Scholar [34] L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104001. Google Scholar [35] W. Rudin, Real and Complex Analysis,, International Series in Pure and Applied Mathematics, (1986). Google Scholar [36] M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries. Background Analysis and Advanced Topics,, Pseudo-Differential Operators, (2010). doi: 10.1007/978-3-7643-8514-9. Google Scholar [37] P. C. Sabatier, Asymptotic properties of the potentials in the inverse-scattering problem at fixed energy,, J. Math. Phys., 7 (1966), 1515. doi: 10.1063/1.1705062. Google Scholar [38] V. Serov, Transmission eigenvalues for non-regular cases,, Commun. Math. Anal., 14 (2013), 129. Google Scholar [39] V. Serov and J. Sylvester, Transmission eigenvalues for degenerate and singular cases,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/6/065004. Google Scholar [40] W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators,, Appl. Anal., 80 (2001), 525. doi: 10.1080/00036810108841007. Google Scholar [41] J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341. doi: 10.1137/110836420. Google Scholar [42] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. Google Scholar [43] F. Trèves, Differential polynomials and decay at infinity,, Bull. Amer. Math. Soc. (N.S.), 66 (1960), 184. doi: 10.1090/S0002-9904-1960-10423-5. Google Scholar [44] I. N. Vekua, Metaharmonic functions,, Trudy Tbilisskogo matematicheskogo instituta, 12 (1943), 105. Google Scholar [45] E. V. Vesalainen, Transmission eigenvalues for a class of non-compactly supported potentials,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104006. Google Scholar [46] M. W. Wong, An Introduction to Pseudo-Differential Operators,, World Scientific, (1999). doi: 10.1142/4047. Google Scholar

show all references

##### References:
 [1] R. Adams, Capacity and compact imbeddings,, Journal of Mathematics and Mechanics, 19 (1970), 923. Google Scholar [2] R. Adams and J. Fournier, Sobolev Spaces,, Pure and Applied Mathematics Series, (2003). Google Scholar [3] S. Agmon, Spectral properties of Schrödinger operators and scattering theory,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 2 (1975), 151. Google Scholar [4] S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics,, J. Anal. Math., 30 (1976), 1. doi: 10.1007/BF02786703. Google Scholar [5] E. Blåsten, L. Päivärinta and J. Sylvester, Corners always scatter,, Commun. Math. Phys., 331 (2014), 725. doi: 10.1007/s00220-014-2030-0. Google Scholar [6] F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237. doi: 10.1137/090769338. Google Scholar [7] F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory,, in Inverse Problems and Applications, 60 (2013), 529. Google Scholar [8] F. Cakoni and H. Haddar, Transmission eigenvalues,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/100201. Google Scholar [9] D. Colton, A. Kirsch and L. Päivärinta, Far field patterns for acoustic waves in an inhomogeneous medium,, SIAM J. Math. Anal., 20 (1989), 1472. doi: 10.1137/0520096. Google Scholar [10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Applied Mathematical Sciences, (2013). doi: 10.1007/978-1-4614-4942-3. Google Scholar [11] D. Colton and P. Monk, The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium,, Quart. J. Mech. Appl. Math., 41 (1988), 97. doi: 10.1093/qjmam/41.1.97. Google Scholar [12] D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Probl. Imaging, 1 (2007), 13. doi: 10.3934/ipi.2007.1.13. Google Scholar [13] P. G. Grinevich and S. V. Manakov, The inverse scattering problem for the two-dimensional Schrödinger operator, the $\overline\partial$-method and non-linear equations,, Funct. Anal. Appl., 20 (1986), 14. Google Scholar [14] P. G. Grinevich and R. G. Novikov, Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials,, Commun. Math. Phys., 174 (1995), 409. doi: 10.1007/BF02099609. Google Scholar [15] K. Hickmann, Interior transmission eigenvalue problem with refractive index having $C^2$-transition to the background medium,, Appl. Anal., 91 (2012), 1675. doi: 10.1080/00036811.2011.577741. Google Scholar [16] M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Anal., 42 (2010), 2965. doi: 10.1137/100793748. Google Scholar [17] L. Hörmander, Lower bounds at infinity for solutions of differential equations with constant coefficients,, Israel J. Math., 16 (1973), 103. doi: 10.1007/BF02761975. Google Scholar [18] L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients,, Classics in Mathematics, (2005). Google Scholar [19] H. Isozaki and H. Morioka, A Rellich type theorem for discrete Schrödinger operators,, Inverse Probl. Imaging, 8 (2014), 475. doi: 10.3934/ipi.2014.8.475. Google Scholar [20] T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995). Google Scholar [21] A. Kirsch, The denseness of the far field patterns for the transmission problem,, IMA J. Appl. Math., 37 (1986), 213. doi: 10.1093/imamat/37.3.213. Google Scholar [22] E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104003. Google Scholar [23] W. Littman, Decay at infinity of solutions to partial differential equations with constant coefficients,, Trans. Amer. Math. Soc., 123 (1966), 449. doi: 10.1090/S0002-9947-1966-0197951-7. Google Scholar [24] W. Littman, Decay at infinity of solutions to partial differential equations; removal of the curvature assumption,, Israel J. Math., 8 (1970), 403. doi: 10.1007/BF02798687. Google Scholar [25] W. Littman, Maximal rates of decay of solutions of partial differential equations,, Arch. Ration. Mech. Anal., 37 (1970), 11. Google Scholar [26] M. Murata, A theorem of Liouville type for partial differential equations with constant coefficients,, Journal of the Faculty of Science, 21 (1974), 395. Google Scholar [27] M. Murata, Asymptotic behaviors at infinity of solutions to certain partial differential equations,, Journal of the Faculty of Science, 23 (1976), 107. Google Scholar [28] R. G. Newton, Construction of potentials from the phase shifts at fixed energy,, J. Math. Phys., 3 (1962), 75. doi: 10.1063/1.1703790. Google Scholar [29] L. Päivärinta, M. Salo and G. Uhlmann, Inverse scattering for the magnetic Schrödinger operator,, J. Funct. Anal., 259 (2010), 1771. doi: 10.1016/j.jfa.2010.06.002. Google Scholar [30] L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525. Google Scholar [31] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness,, Academid Press, (1975). Google Scholar [32] T. Regge, Introduction to complex orbital moments,, Il Nuovo Cimento, 14 (1959), 951. doi: 10.1007/BF02728177. Google Scholar [33] F. Rellich, Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u=0$ im unendlichen Gebieten,, Jahresber. Dtsch. Math.-Ver., 53 (1943), 57. Google Scholar [34] L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104001. Google Scholar [35] W. Rudin, Real and Complex Analysis,, International Series in Pure and Applied Mathematics, (1986). Google Scholar [36] M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries. Background Analysis and Advanced Topics,, Pseudo-Differential Operators, (2010). doi: 10.1007/978-3-7643-8514-9. Google Scholar [37] P. C. Sabatier, Asymptotic properties of the potentials in the inverse-scattering problem at fixed energy,, J. Math. Phys., 7 (1966), 1515. doi: 10.1063/1.1705062. Google Scholar [38] V. Serov, Transmission eigenvalues for non-regular cases,, Commun. Math. Anal., 14 (2013), 129. Google Scholar [39] V. Serov and J. Sylvester, Transmission eigenvalues for degenerate and singular cases,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/6/065004. Google Scholar [40] W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators,, Appl. Anal., 80 (2001), 525. doi: 10.1080/00036810108841007. Google Scholar [41] J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341. doi: 10.1137/110836420. Google Scholar [42] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. Google Scholar [43] F. Trèves, Differential polynomials and decay at infinity,, Bull. Amer. Math. Soc. (N.S.), 66 (1960), 184. doi: 10.1090/S0002-9904-1960-10423-5. Google Scholar [44] I. N. Vekua, Metaharmonic functions,, Trudy Tbilisskogo matematicheskogo instituta, 12 (1943), 105. Google Scholar [45] E. V. Vesalainen, Transmission eigenvalues for a class of non-compactly supported potentials,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104006. Google Scholar [46] M. W. Wong, An Introduction to Pseudo-Differential Operators,, World Scientific, (1999). doi: 10.1142/4047. Google Scholar
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