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 Hallstromin 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), 923-929.  Google Scholar

[2]

R. Adams and J. Fournier, Sobolev Spaces, Pure and Applied Mathematics Series, Elsevier, 2003.  Google Scholar

[3]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 2 (1975), 151-218.  Google Scholar

[4]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math., 30 (1976), 1-38. doi: 10.1007/BF02786703.  Google Scholar

[5]

E. Blåsten, L. Päivärinta and J. Sylvester, Corners always scatter, Commun. Math. Phys., 331 (2014), 725-753. 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), 237-255. doi: 10.1137/090769338.  Google Scholar

[7]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications, Inside Out II (ed. G. Uhlmann) MSRI Publications, Cambridge University Press, 60 (2013), 529-580.  Google Scholar

[8]

F. Cakoni and H. Haddar, Transmission eigenvalues, Inverse Problems, 29 (2013), 100201, 3PP. 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), 1472-1483. doi: 10.1137/0520096.  Google Scholar

[10]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, 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), 97-125. 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), 13-28. 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), 14-24.  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), 409-446. 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), 1675-1690. 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), 2965-2986. 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), 103-116. 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, Springer, 2005.  Google Scholar

[19]

H. Isozaki and H. Morioka, A Rellich type theorem for discrete Schrödinger operators, Inverse Probl. Imaging, 8 (2014), 475-489. doi: 10.3934/ipi.2014.8.475.  Google Scholar

[20]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, 1995.  Google Scholar

[21]

A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37 (1986), 213-225. 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), 104003, 19PP. 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), 449-459. 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), 403-407. 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), 11-20.  Google Scholar

[26]

M. Murata, A theorem of Liouville type for partial differential equations with constant coefficients, Journal of the Faculty of Science, the University of Tokyo, Section IA Mathematics, 21 (1974), 395-404.  Google Scholar

[27]

M. Murata, Asymptotic behaviors at infinity of solutions to certain partial differential equations, Journal of the Faculty of Science, the University of Tokyo, Section IA Mathematics, 23 (1976), 107-148.  Google Scholar

[28]

R. G. Newton, Construction of potentials from the phase shifts at fixed energy, J. Math. Phys., 3 (1962), 75-82. 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), 1771-1798. doi: 10.1016/j.jfa.2010.06.002.  Google Scholar

[30]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753. 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), 951-976. 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), 57-65.  Google Scholar

[34]

L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104001, 28PP. doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

[35]

W. Rudin, Real and Complex Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, 1986. Google Scholar

[36]

M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries. Background Analysis and Advanced Topics, Pseudo-Differential Operators, Theory and Applications, 2, Birkhäuser, 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), 1515-1531. doi: 10.1063/1.1705062.  Google Scholar

[38]

V. Serov, Transmission eigenvalues for non-regular cases, Commun. Math. Anal., 14 (2013), 129-142.  Google Scholar

[39]

V. Serov and J. Sylvester, Transmission eigenvalues for degenerate and singular cases, Inverse Problems, 28 (2012), 065004, 8PP. 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), 525-556. doi: 10.1080/00036810108841007.  Google Scholar

[41]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354. 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), 153-169. doi: 10.2307/1971291.  Google Scholar

[43]

F. Trèves, Differential polynomials and decay at infinity, Bull. Amer. Math. Soc. (N.S.), 66 (1960), 184-186. doi: 10.1090/S0002-9904-1960-10423-5.  Google Scholar

[44]

I. N. Vekua, Metaharmonic functions, Trudy Tbilisskogo matematicheskogo instituta, 12 (1943), 105-174.  Google Scholar

[45]

E. V. Vesalainen, Transmission eigenvalues for a class of non-compactly supported potentials, Inverse Problems, 29 (2013), 104006, 11PP. 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), 923-929.  Google Scholar

[2]

R. Adams and J. Fournier, Sobolev Spaces, Pure and Applied Mathematics Series, Elsevier, 2003.  Google Scholar

[3]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 2 (1975), 151-218.  Google Scholar

[4]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math., 30 (1976), 1-38. doi: 10.1007/BF02786703.  Google Scholar

[5]

E. Blåsten, L. Päivärinta and J. Sylvester, Corners always scatter, Commun. Math. Phys., 331 (2014), 725-753. 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), 237-255. doi: 10.1137/090769338.  Google Scholar

[7]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications, Inside Out II (ed. G. Uhlmann) MSRI Publications, Cambridge University Press, 60 (2013), 529-580.  Google Scholar

[8]

F. Cakoni and H. Haddar, Transmission eigenvalues, Inverse Problems, 29 (2013), 100201, 3PP. 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), 1472-1483. doi: 10.1137/0520096.  Google Scholar

[10]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, 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), 97-125. 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), 13-28. 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), 14-24.  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), 409-446. 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), 1675-1690. 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), 2965-2986. 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), 103-116. 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, Springer, 2005.  Google Scholar

[19]

H. Isozaki and H. Morioka, A Rellich type theorem for discrete Schrödinger operators, Inverse Probl. Imaging, 8 (2014), 475-489. doi: 10.3934/ipi.2014.8.475.  Google Scholar

[20]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, 1995.  Google Scholar

[21]

A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37 (1986), 213-225. 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), 104003, 19PP. 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), 449-459. 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), 403-407. 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), 11-20.  Google Scholar

[26]

M. Murata, A theorem of Liouville type for partial differential equations with constant coefficients, Journal of the Faculty of Science, the University of Tokyo, Section IA Mathematics, 21 (1974), 395-404.  Google Scholar

[27]

M. Murata, Asymptotic behaviors at infinity of solutions to certain partial differential equations, Journal of the Faculty of Science, the University of Tokyo, Section IA Mathematics, 23 (1976), 107-148.  Google Scholar

[28]

R. G. Newton, Construction of potentials from the phase shifts at fixed energy, J. Math. Phys., 3 (1962), 75-82. 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), 1771-1798. doi: 10.1016/j.jfa.2010.06.002.  Google Scholar

[30]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753. 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), 951-976. 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), 57-65.  Google Scholar

[34]

L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104001, 28PP. doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

[35]

W. Rudin, Real and Complex Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, 1986. Google Scholar

[36]

M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries. Background Analysis and Advanced Topics, Pseudo-Differential Operators, Theory and Applications, 2, Birkhäuser, 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), 1515-1531. doi: 10.1063/1.1705062.  Google Scholar

[38]

V. Serov, Transmission eigenvalues for non-regular cases, Commun. Math. Anal., 14 (2013), 129-142.  Google Scholar

[39]

V. Serov and J. Sylvester, Transmission eigenvalues for degenerate and singular cases, Inverse Problems, 28 (2012), 065004, 8PP. 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), 525-556. doi: 10.1080/00036810108841007.  Google Scholar

[41]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354. 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), 153-169. doi: 10.2307/1971291.  Google Scholar

[43]

F. Trèves, Differential polynomials and decay at infinity, Bull. Amer. Math. Soc. (N.S.), 66 (1960), 184-186. doi: 10.1090/S0002-9904-1960-10423-5.  Google Scholar

[44]

I. N. Vekua, Metaharmonic functions, Trudy Tbilisskogo matematicheskogo instituta, 12 (1943), 105-174.  Google Scholar

[45]

E. V. Vesalainen, Transmission eigenvalues for a class of non-compactly supported potentials, Inverse Problems, 29 (2013), 104006, 11PP. 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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