Symmetry of bound and antibound states in the semiclassical limit for a general class of potentials

Dyatlov, Semyon; Ghosh, Subhroshekhar

doi:10.1090/S0002-9939-2010-10519-1

Symmetry of bound and antibound states in the semiclassical limit for a general class of potentials
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by Semyon Dyatlov and Subhroshekhar Ghosh PDF

Proc. Amer. Math. Soc. 138 (2010), 3203-3210 Request permission

Abstract:

We consider the Schrödinger operator $-h^2\partial _x^2+V(x)$ on a half-line, where $V$ is a compactly supported potential which is positive near the endpoint of its support. We prove that the eigenvalues and the purely imaginary resonances are symmetric up to an error $Ce^{-\delta /h}$.

References

A. A. Abramov, A. Aslanyan, and E. B. Davies, Bounds on complex eigenvalues and resonances, J. Phys. A 34 (2001), no. 1, 57–72. MR 1819914, DOI 10.1088/0305-4470/34/1/304
D. Bindel, MatScat: MATLAB Codes for 1D Potential Scattering, http://cims.nyu.edu/ ~dbindel/matscat.
David Bindel and Maciej Zworski, Symmetry of bound and antibound states in the semiclassical limit, Lett. Math. Phys. 81 (2007), no. 2, 107–117. MR 2336226, DOI 10.1007/s11005-007-0178-7
P. Briet, J.-M. Combes, and P. Duclos, On the location of resonances for Schrödinger operators in the semiclassical limit. II. Barrier top resonances, Comm. Partial Differential Equations 12 (1987), no. 2, 201–222. MR 876987, DOI 10.1080/03605308708820488
Richard Froese, Asymptotic distribution of resonances in one dimension, J. Differential Equations 137 (1997), no. 2, 251–272. MR 1456597, DOI 10.1006/jdeq.1996.3248
Michael Hitrik, Bounds on scattering poles in one dimension, Comm. Math. Phys. 208 (1999), no. 2, 381–411. MR 1729092, DOI 10.1007/s002200050763
Evgeny Korotyaev, Inverse resonance scattering on the real line, Inverse Problems 21 (2005), no. 1, 325–341. MR 2146179, DOI 10.1088/0266-5611/21/1/020
L. D. Landau and E. M. Lifshitz, Quantum mechanics: non-relativistic theory. Course of Theoretical Physics, Vol. 3, Addison-Wesley Series in Advanced Physics, Pergamon Press, Ltd., London-Paris; for U.S.A. and Canada: Addison-Wesley Publishing Company, Inc., Reading, Mass., 1958. Translated from the Russian by J. B. Sykes and J. S. Bell. MR 0093319
L. Nedelec, Asymptotics of resonances for a Schrödinger operator with matrix values, math.SP/0509391.
T. Regge, Analytic properties of the scattering matrix, Nuovo Cimento (10) 8 (1958), 671–679. MR 95702
Barry Simon, Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178 (2000), no. 2, 396–420. MR 1802901, DOI 10.1006/jfan.2000.3669
S.-H. Tang and M. Zworski, Potential Scattering on the Real Line, lecture notes, http: //math.berkeley.edu/~zworski/tz1.pdf.
Maciej Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (1987), no. 2, 277–296. MR 899652, DOI 10.1016/0022-1236(87)90069-3