Recovery of interior eigenvalues from reduced near field data

References

• Cakoni F, Cossonnière A, Haddar H. Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Prob. Imaging. 2012;6:373–398.
   Web of Science ®Google Scholar
• Sylvester J. Discreteness of transmission eigenvalues via upper triangular compact operators. SIAM. J. Math. Anal. 2011;44:341–354.
   Web of Science ®Google Scholar
• Eskin G, Ralston J. The inverse backscattering problem in 3 dimension. Comm. Math. Phys. 1989;124:169–215.
   Web of Science ®Google Scholar
• Eskin G, Ralston J. Inverse backscattering in two dimensions. Comm. Math. Phys. 1991;138:451–486.
   Web of Science ®Google Scholar
• Eskin G, Ralston J. Inverse backscattering. J. dAnalyse Math. 1992;58:177–190.
   Web of Science ®Google Scholar
• Prosser RT. Formal solutions of inverse scattering problems. J. Math. Phys. 1982;23:2127–2130.
   Web of Science ®Google Scholar
• Rakesh, Uhlmann G. Uniqueness for the inverse backscattering problem for angularly controlled potentials. Inverse Prob. 2014;30:065005.
   Web of Science ®Google Scholar
• Wang JN. Inverse backscattering problem for Maxwell’s equations. Math. Methods Appl. Sci. 1998;21:1441–1465.
   Web of Science ®Google Scholar
• Ola P, Pivrinta L, Serov V. Recovering singularities from backscattering in two dimensions. Commun. Partial Differ. Equ. 2001;26:697–715.
   Web of Science ®Google Scholar
• Ruiz A, Vargas A. Partial recovery of a potential from backscattering data. Commun. Partial Differ. Eqs. 2005;30:67–96.
   Web of Science ®Google Scholar
• Rakesh, Uhlmann G. The point source inverse back-scattering problem. 2014, arXiv:1403.1766.
   Google Scholar
• Christiansen Tanya J. Inverse obstacle problems with backscattering or generalized backscattering data in one or two directions. Asymptotic Anal. 2013;81:315–335.
   Web of Science ®Google Scholar
• Zelditch S. The inverse spectral problem. Somerville (MA): International Press; 2004. With an appendix by Johannes, S., Maciej Z. Surv. Differ. Geom. IX:401467.
   Google Scholar
• Mclaughlin JR, Polyakov PL. On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues. J. Differ. Eqs. 1994;107:351–382.
   Web of Science ®Google Scholar
• Chen LH. An uniqueness result with some density theorems with interior transmission eigenvalues. Applicable Anal. 2015;94:1527–1544.
   Web of Science ®Google Scholar
• Cakoni F, Gintides D, Haddar H. The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Anal. 2010;42:237–255.
   Web of Science ®Google Scholar
• McLean W. Strongly elliptic systems and boundary integral equations. Cambridge: Cambridge University Press; 2000.
   Google Scholar
• Doron E, Smilansky U. Semiclassical quantization of chaotic billiards -- a scattering theory approach. Nonlinearity. 1992;5:1055–1084.
   Web of Science ®Google Scholar
• Smilansky U. Semiclassical quantization of chaotic billiards -- a scattering approach In: Akkermans E, Montambaux G, Pichard J-L, Zinn-Justin J, editors. Proceedings of the 1994 Les-Houches summer school on “Mesoscopic quantum Physics”. North-Holland: Elsevier Science; 1995, p. 373–434.
   Google Scholar
• Eckmann JP, Pillet C-A. Spectral duality for planar billiards. Commun. Math. Phys. 1995;170:283–313.
   Web of Science ®Google Scholar
• Eckmann JP, Pillet C-A. Zeta functions with Dirichlet and Neumann boundary conditions for exterior domains. Helv. Phys. Acta. 1997;70:44–65.
   Google Scholar
• Lakshtanov E, Vainberg B. Remarks on interior transmission eigenvalues, Weyl formula and branching billiards. J. Phys A: Math. Theor. 2012;45:125202–125211.
   Web of Science ®Google Scholar
• Kirsch A, Lechleiter A. The inside--outside duality for scattering problems by inhomogeneous media. Inverse Prob. 2013;29:104011.
   Google Scholar
• Lakshtanov E, Vainberg B. Sharp Weyl law for signed counting function of positive interior transmission eigenvalues. 2014, arXiv:1401.6213.
   Google Scholar
• Kirsch A. The Denseness of the far field patterns for the transmission problem. IMA J. Appl. Math. 1986;37:213–225.
   Web of Science ®Google Scholar
• Lechleiter A, Rennoch M. Inside--outside duality and the determination of electromagnetic interior transmission eigenvalues. SIAM J. Math. Anal. 2015;47:684–705.
   Web of Science ®Google Scholar
• Lechleiter A, Peters S. Determining transmission eigenvalues of anisotropic inhomogeneous media from far field data. Commun. Math. Sci. 2015;13:1803–1827.
   Web of Science ®Google Scholar
• Lechleiter A, Peters S. Analytical characterization and numerical approximation of interior eigenvalues for impenetrable scatterers from far fields. Inverse Prob. 2014;4:045006.
   Web of Science ®Google Scholar
• Vainberg BR, Grushin VV. Uniformly nonelliptic problems. II. Mat. Sb. (N.S.). 1967;73:126–154.
   Google Scholar
• Vainberg BR, Grushin VV. Uniformly nonelliptic problems. I. Mat. Sb. (N.S.). 1967;72:602–636.
   Google Scholar