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2017 | 132 | 6 | 1677-1682
Article title

Asymptotics of Resonances Induced by Point Interactions

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Abstracts
EN
We consider the resonances of the self-adjoint three-dimensional Schrödinger operator with point interactions of constant strength supported on the set X={xₙ}_{n=1}^{N}. The size of X is defined by V_{X} = max_{π ∈ Π_{N}} ∑_{n=1}^{N} |xₙ - x_{π(n)}|, where Π_{N} is the family of all the permutations of the set {1,2,...,N}. We prove that the number of resonances counted with multiplicities and lying inside the disc of radius R behaves asymptotically linear W_{X}/πR + O(1) as R → ∞, where the constant W_{X} ∈ [0,V_{X}] can be seen as the effective size of X. Moreover, we show that there exist a configuration of any number of points such that W_{X}=V_{X}. Finally, we construct an example for N=4 with W_{X} < V_{X}, which can be viewed as an analogue of a quantum graph with non-Weyl asymptotics of resonances.
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EN
Year
Volume
132
Issue
6
Pages
1677-1682
Physical description
Dates
published
2017-12
References
  • [1] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Vol. 350, 2nd ed., AMS Chelsea Publ., Providence (RI) 2005, doi: 10.1090/chel/350
  • [2] S. Albeverio, I. Karabash, Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions, to appear in Oper. Matrices
  • [3] J.F. Brasche, R. Figari, A. Teta, Potent. Anal. 8, 163 (1998), doi: 10.1023/A:1008654423238
  • [4] P. Exner, R. Gawlista, P. Šeba, M. Tater, Ann. Phys. 252, 133 (1996), doi: 10.1006/aphy.1996.0127
  • [5] G. Dell'Antonio, R. Figari, A. Teta, in: Inverse Problems and Imaging, lectures given at the C.I.M.E. Summer School held in Martina Franca (Italy), 2002, Springer, Berlin 2008, p. 171, doi: 10.1007/978-3-540-78547-7_7
  • [6] R. Langer, Bull. Am. Math. Soc. 37, 213 (1931), doi: 10.1090/S0002-9904-1931-05133-8
  • [7] R. Bellman, K.L. Cooke, Differential-Difference Equations, Academic Press, New York 1963
  • [8] C.A. Berenstein, R. Gay, Complex Analysis and Special Topics in Harmonic Analysis, Springer, New York 1995, doi: 10.1007/978-1-4613-8445-8
  • [9] E.B. Davies, P. Exner, J. Lipovský, J. Phys. A Math. Theor. 43, 474013 (2010), doi: 10.1088/1751-8113/43/47/474013
  • [10] E.B. Davies, A. Pushnitski, Anal. PDE 4, 729 (2011), doi: 10.2140/apde.2011.4.729
  • [11] M. Zworski, J. Funct. Anal. 73, 277 (1987), doi: 10.1016/0022-1236(87)90069-3
  • [12] T.J. Christiansen, P.D. Hislop, J. Équ. Dériv. Partielles 3, 18 (2008), doi: 10.5802/jedp.47
  • [13] G. Pólya, Münch. Sitzungsber. 50, 285 (1920)
  • [14] E. Schwengeler, Ph.D. Thesis, Techn. Hochsch., Zürich 1925, doi: 10.3929/ethz-a-000092005
  • [15] C.J. Moreno, Compos. Math. 26, 69 (1973)
  • [16] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge 2000
  • [17] F.A. Berezin, L.D. Faddeev, Sov. Math. Dokl. 2, 372 (1961); translation from Dokl. Akad. Nauk SSSR 137, 1011 (1961)
  • [18] S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators. Solvable Schrödinger Type Operators, London Mathematical Society Lecture Note Series, Vol. 271, Cambridge University Press, Cambridge 1999, doi: 10.1017/CBO9780511758904
  • [19] N. Goloshchapova, M. Malamud, V. Zastavnyi, Math. Nachr. 285, 1839 (2012), doi: 10.1002/mana.201100132
  • [20] A. Posilicano, Oper. Matrices 2, 483 (2008), doi: 10.7153/oam-02-30
  • [21] A. Teta, Publ. Res. Inst. Math. Sci. 26, 803 (1990), doi: 10.2977/prims/1195170735
  • [22] R. Band, J.M. Harrison, C.H. Joyner, J. Phys. A Math. Theor. 45, 325204 (2012), doi: 10.1088/1751-8113/45/32/325204
  • [23] J. Lipovský, Acta Phys. Pol. A 128, 968 (2015), doi: 10.12693/aphyspola.128.968
  • [24] J. Lipovský, J. Phys. A Math. Theor. 49, 375202 (2016), doi: 10.1088/1751-8113/49/37/375202
  • [25] M. Bóna, Combinatorics of Permutations, Chapman and Hall/CRC, Boca Raton 2004, doi: 10.1201/9780203494370
  • [26] R. Brualdi, D. Cvetković, A Combinatorial Approach to Matrix Theory and Its Applications, CRC Press, Boca Raton 2009, doi: 10.1201/9781420082241
  • [27] S. Deus, P.M. Koch, L. Sirko, Phys. Rev. E 52, 1146 (1995), doi: 10.1103/PhysRevE.52.1146
Document Type
Publication order reference
YADDA identifier
bwmeta1.element.bwnjournal-article-appv132n6p05kz
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