Skip to main content
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
/content/aip/journal/jmp/57/8/10.1063/1.4960469
1.
Albeverio, S. , Gesztesy, F. , and Holden, H. , Solvable Models in Quantum Mechanics (Springer Berlin Heidelberg, 1988).
2.
Albeverio, S. and Kurasov, P. , Singular Perturbations of Differential Operators (Cambridge University Press, 2000), Cambridge Books Online.
3.
Colin de Verdière, Y. , “Ergodicité et fonctions propres du laplacien,” Commun. Math. Phys. 102(3), 497502 (1985).
http://dx.doi.org/10.1007/BF01209296
4.
Exner, P. and Seba, P. , “Point interaction in dimension two and three as model of small scatterers,” Phys. Lett. A 222, 1 (1996).
http://dx.doi.org/10.1016/0375-9601(96)00640-8
5.
Heath-Brown, D. R. , “Lattice points in the sphere,” in Number Theory in Progress, Vol. 2 (Zakopane-Kościelisko, 1997) (de Gruyter, Berlin, 1999), pp. 883892.
6.
Kronig, R. de L. and Penney, W. G. , “Quantum mechanics of electrons in crystal lattices,” Proc. R. Soc. A 130(814), 499513 (1931).
http://dx.doi.org/10.1098/rspa.1931.0019
7.
Kurlberg, P. and Rosenzweig, L. , “Scarred eigenstates for arithmetic toral point scatterers,” e-print arXiv:1508.02978 (2015).
8.
Kurlberg, P. and Ueberschär, H. , “Quantum ergodicity for point scatterers on arithmetic tori,” Geom. Funct. Anal. 24(5), 15651590 (2014).
http://dx.doi.org/10.1007/s00039-014-0275-6
9.
Marklof, J. , “Pair correlation densities of inhomogeneous quadratic forms. II,” Duke Math. J. 115(3), 409434 (2002).
http://dx.doi.org/10.1215/S0012-7094-02-11531-2
10.
Parnovski, L. and Sobolev, A. V. , “Bethe-sommerfeld conjecture for periodic operators with strong perturbations,” Inventiones Math. 181(3), 467540 (2010).
http://dx.doi.org/10.1007/s00222-010-0251-1
11.
Rudnick, Z. and Ueberschär, H. , “Statistics of wave functions for a point scatterer on the torus,” Commun. Math. Phys. 316, 763782 (2012).
http://dx.doi.org/10.1007/s00220-012-1556-2
12.
Shigehara, T. , “Conditions for the appearance of wave chaos in quantum singular systems with a pointlike scatterer,” Phys. Rev. E 50, 43574370 (1994).
http://dx.doi.org/10.1103/PhysRevE.50.4357
13.
Shigehara, T. and Cheon, T. , “Wave chaos in quantum billiards with small but finite-size scatterer,” J. Phys. E 54, 13211331 (1996).
http://dx.doi.org/10.1103/physreve.54.1321
14.
Shigehara, T. and Cheon, T. , “Spectral properties of three-dimensional quantum billiards with a pointlike scatterer,” Phys. Rev. E 55, 6832 (1997).
http://dx.doi.org/10.1103/physreve.55.6832
15.
Shnirel’man, A. I. , “Ergodic properties of eigenfunctions,” Usp. Mat. Nauk 29(6), 181182 (1974).
16.
Sěba, P. , “Wave chaos in singular quantum billiard,” Phys. Rev. Lett. 64, 18551858 (1990).
http://dx.doi.org/10.1103/PhysRevLett.64.1855
17.
Ueberschaer, H. and Kurlberg, P. , “Superscars in the Seba billiard,” J. Eur. Math. Soc. (JEMS) (to appear); e-print arXiv:1409.6878 (2014).
18.
Yesha, N. , “Eigenfunction statistics for a point scatterer on a three-dimensional torus,” Ann. Henri Poincaré 14(7), 18011836 (2013).
http://dx.doi.org/10.1007/s00023-013-0232-1
19.
Yesha, N. , “Quantum ergodicity for a point scatterer on the three-dimensional torus,” Ann. Henri Poincaré 16(1), 114 (2015).
http://dx.doi.org/10.1007/s00023-014-0318-4
20.
Zelditch, S. , “Uniform distribution of eigenfunctions on compact hyperbolic surfaces,” Duke Math. J. 55(4), 919941 (1987).
http://dx.doi.org/10.1215/S0012-7094-87-05546-3
http://aip.metastore.ingenta.com/content/aip/journal/jmp/57/8/10.1063/1.4960469
Loading
/content/aip/journal/jmp/57/8/10.1063/1.4960469
Loading

Data & Media loading...

Loading

Article metrics loading...

/content/aip/journal/jmp/57/8/10.1063/1.4960469
2016-08-09
2016-09-28

Abstract

Consider the 3-dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for Floquet-Bloch modes with fixed quasi-momentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Ueberschaer and Kurlberg, J. Eur. Math. Soc. (JEMS) (to appear); [e-print arXiv:1409.6878 (2014)] who show momentum localisation for zero quasi-momentum in 2-dimensions and is the first result in this direction in 3-dimensions.

Loading

Full text loading...

/deliver/fulltext/aip/journal/jmp/57/8/1.4960469.html;jsessionid=2BhoUyCNbQiSd6lkJVV9772a.x-aip-live-03?itemId=/content/aip/journal/jmp/57/8/10.1063/1.4960469&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/jmp
true
true

Access Key

  • FFree Content
  • OAOpen Access Content
  • SSubscribed Content
  • TFree Trial Content
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
/content/realmedia?fmt=ahah&adPositionList=
&advertTargetUrl=//oascentral.aip.org/RealMedia/ads/&sitePageValue=jmp.aip.org/57/8/10.1063/1.4960469&pageURL=http://scitation.aip.org/content/aip/journal/jmp/57/8/10.1063/1.4960469'
Right1,Right2,Right3,