Scattering for Schrödinger operators with potentials concentrated near a subspace

Black, Adam; Malinovitch, Tal

doi:10.1090/tran/8854

Scattering for Schrödinger operators with potentials concentrated near a subspace
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by Adam Black and Tal Malinovitch PDF

Trans. Amer. Math. Soc. 376 (2023), 2525-2555 Request permission

Abstract:

We study the scattering properties of Schrödinger operators with bounded potentials concentrated near a subspace of $\mathbb {R}^d$. For such operators, we show the existence of scattering states and characterize their orthogonal complement as a set of surface states, which consists of states that are confined to the subspace (such as pure point states) and states that escape it at a sublinear rate, in a suitable sense. We provide examples of surface states for different systems including those that propagate along the subspace and those that escape the subspace arbitrarily slowly. Our proof uses a novel interpretation of the Enss method [Comm. Math. Phys. 61 (1978), pp. 285–291] in order to obtain a dynamical characterization of the orthogonal complement of the scattering states.

References

Shmuel Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218. MR 397194
W. O. Amrein and V. Georgescu, On the characterization of bound states and scattering states in quantum mechanics, Helv. Phys. Acta 46 (1973/74), 635–658. MR 363267
René Carmona, One-dimensional Schrödinger operators with random or deterministic potentials: new spectral types, J. Funct. Anal. 51 (1983), no. 2, 229–258. MR 701057, DOI 10.1016/0022-1236(83)90027-7
Monique Combescure and Didier Robert, Coherent states and applications in mathematical physics, Theoretical and Mathematical Physics, Springer, Dordrecht, 2012. MR 2952171, DOI 10.1007/978-94-007-0196-0
E. B. Davies, Quantum theory of open systems, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0489429
E. B. Davies, Scattering from infinite sheets, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 2, 327–334. MR 449311, DOI 10.1017/S0305004100053962
E. B. Davies, On Enss’ approach to scattering theory, Duke Math. J. 47 (1980), no. 1, 171–185. MR 563374, DOI 10.1215/S0012-7094-80-04713-4
E. B. Davies and B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Comm. Math. Phys. 63 (1978), no. 3, 277–301. MR 513906, DOI 10.1007/BF01196937
Anne Boutet de Monvel, Galina Kazantseva, and Marius Mantoiu, Some anisotropic Schrödinger operators without singular spectrum, Helv. Phys. Acta 69 (1996), no. 1, 13–25. MR 1398558
Anne Boutet de Monvel and Peter Stollmann, Dynamical localization for continuum random surface models, Arch. Math. (Basel) 80 (2003), no. 1, 87–97. MR 1968291, DOI 10.1007/s000130300009
Jan Dereziński and Christian Gérard, Scattering theory of classical and quantum $N$-particle systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. MR 1459161, DOI 10.1007/978-3-662-03403-3
Volker Enss, Asymptotic completeness for quantum mechanical potential scattering. I. Short range potentials, Comm. Math. Phys. 61 (1978), no. 3, 285–291. MR 523013, DOI 10.1007/BF01940771
David W. Fox, Spectral measures and separation of variables, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 3, 347–351. MR 425661, DOI 10.6028/jres.080B.036
Rupert L. Frank, On the scattering theory of the Laplacian with a periodic boundary condition. I. Existence of wave operators, Doc. Math. 8 (2003), 547–565. MR 2029173, DOI 10.4171/dm/150
V. Grinshpun, Localization for random potentials supported on a subspace, Lett. Math. Phys. 34 (1995), no. 2, 103–117. MR 1335579, DOI 10.1007/BF00739090
Dirk Hundertmark and Werner Kirsch, Spectral theory of sparse potentials, Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999) CMS Conf. Proc., vol. 28, Amer. Math. Soc., Providence, RI, 2000, pp. 213–238. MR 1803388, DOI 10.1007/bf02178350
W. Hunziker, I. M. Sigal, and A. Soffer, Minimal escape velocities, Comm. Partial Differential Equations 24 (1999), no. 11-12, 2279–2295. MR 1720738, DOI 10.1080/03605309908821502
Vojkan Jakšić and Yoram Last, Corrugated surfaces and a.c. spectrum, Rev. Math. Phys. 12 (2000), no. 11, 1465–1503. MR 1809458, DOI 10.1142/S0129055X00000563
Vojkan Jakšić and Yoram Last, Spectral structure of Anderson type Hamiltonians, Invent. Math. 141 (2000), no. 3, 561–577. MR 1779620, DOI 10.1007/s002220000076
Vojkan Jak ić and Yoram Last, Surface states and spectra, Comm. Math. Phys. 218 (2001), no. 3, 459–477. MR 1828849, DOI 10.1007/PL00005560
Peter A. Perry, Scattering theory by the Enss method, Mathematical Reports, vol. 1, Harwood Academic Publishers, Chur, 1983. Edited by B. Simon. MR 752694
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Michael Reed and Barry Simon, Methods of modern mathematical physics - IV: analysis of operators, vol. 4, Elsevier, 1978.
Michael Reed and Barry Simon, Methods of modern mathematical physics - III: scattering theory, vol. 3, Elsevier, 1979.
Serge Richard, Spectral and scattering theory for Schrödinger operators with Cartesian anisotropy, Publ. Res. Inst. Math. Sci. 41 (2005), no. 1, 73–111. MR 2115968, DOI 10.2977/prims/1145475405
D. Ruelle, A remark on bound states in potential-scattering theory, Nuovo Cimento A (10) 61 (1969), 655–662 (English, with Italian summary). MR 246603, DOI 10.1007/BF02819607
A. W. Sáenz, Quantum-mechanical scattering by impenetrable periodic surfaces, J. Math. Phys. 22 (1981), no. 12, 2872–2884. MR 638093, DOI 10.1063/1.525168
Barry Simon, Phase space analysis of simple scattering systems: extensions of some work of Enss, Duke Math. J. 46 (1979), no. 1, 119–168. MR 523604
D. R. Yafaev, On the break-down of completeness of wave operators in potential scattering, Comm. Math. Phys. 65 (1979), no. 2, 167–179. MR 528189, DOI 10.1007/BF01225147
D. R. Yafaev, Scattering theory: some old and new problems, Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 2007.