## Scattering for Schrödinger operators with potentials concentrated near a subspace

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- by Adam Black and Tal Malinovitch PDF
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**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.

## Additional Information

**Adam Black**- Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
- Email: adam.black@yale.edu
**Tal Malinovitch**- Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
- ORCID: 0000-0002-2904-095X
- Email: tal.malinovitch@yale.edu
- Received by editor(s): May 17, 2022
- Published electronically: January 23, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 2525-2555 - MSC (2000): Primary 47A40, 35J10, 35Q40, 81U05
- DOI: https://doi.org/10.1090/tran/8854
- MathSciNet review: 4557873