Search Advanced

Mathematics in Engineering

2021, Volume 3, Issue 2: 1-14. doi: 10.3934/mine.2021017
Previous Article Next Article
Research article Special Issues

On real resonances for three-dimensional Schrödinger operators with point interactions

  • 1.
    Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany
  • 2.
    Gran Sasso Science Institute - GSSI, Viale Francesco Crispi 7, 67100 L'Aquila, Italy
  • †This contribution is part of the Special Issue: Qualitative Analysis and Spectral Theory for Partial Differential Equations
    Guest Editor: Veronica Felli
    Link: www.aimspress.com/mine/article/5511/special-articles
  • Received: 18 February 2020 Accepted: 27 May 2020 Published: 29 June 2020

  • MSC : 35B25, 35B34, 35J10, 47A75, 81Q10

  • We prove the absence of positive real resonances for Schrödinger operators with finitely many point interactions in $\mathbb{R}^3$ and we discuss such a property from the perspective of dispersive and scattering features of the associated Schrodinger propagator.

    Citation: Alessandro Michelangeli, Raffaele Scandone. On real resonances for three-dimensional Schrödinger operators with point interactions[J]. Mathematics in Engineering, 2021, 3(2): 1-14. doi: 10.3934/mine.2021017

    Related Papers:

  • We prove the absence of positive real resonances for Schrödinger operators with finitely many point interactions in $\mathbb{R}^3$ and we discuss such a property from the perspective of dispersive and scattering features of the associated Schrodinger propagator.


    加载中


    [1] Agmon S (1975) Spectral properties of Schrödinger operators and scattering theory. Ann Scuola Norm Sup Pisa Cl Sci 2: 151-218.
    [2] Albeverio S, Brzeźniak Z, Dabrowski L (1995) Fundamental solution of the heat and Schrödinger equations with point interaction. J Funct Anal 130: 220-254.
    [3] Albeverio S, Fenstad JE, Høegh-Krohn R (1979) Singular perturbations and nonstandard analysis. T Am Math Soc 252: 275-295.
    [4] Albeverio S, Gesztesy F, Høegh-Krohn R, et al. (1988) Solvable Models in Quantum Mechanics, New York: Springer-Verlag.
    [5] Albeverio S, Høegh-Krohn R (1981) Point interactions as limits of short range interactions. J Operat Theor 6: 313-339.
    [6] Albeverio S, Høegh-Krohn R, Streit L (1977) Energy forms, Hamiltonians, and distorted Brownian paths. J Math Phys 18: 907-917.
    [7] Albeverio S, Karabash IM (2017) Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions. Oper Matrices 11: 1097-1117.
    [8] Albeverio S, Karabash IM (2019) On the multilevel internal structure of the asymptotic distribution of resonances. J Differ Equations 267: 6171-6197.
    [9] Albeverio S, Karabash IM (2018) Generic asymptotics of resonance counting function for Schrödinger point interactions. arXiv: 1803.06039.
    [10] Alsholm P, Schmidt G (1971) Spectral and scattering theory for Schrödinger operators. Arch Rational Mech Anal 40: 281-311.
    [11] Arlinskiĭ Y, Tsekanovskiĭ E (2005) The von Neumann problem for nonnegative symmetric operators. Integr Equat Oper Th 51: 319-356.
    [12] Berezin F, Faddeev L (1961) A Remark on Schrodinger's equation with a singular potential. Sov Math Dokl 2: 372-375.
    [13] Cornean HD, Michelangeli A, Yajima K (2020) Two-dimensional Schrödinger operators with point interactions: Threshold expansions, zero modes and Lp-boundedness of wave operators. Rev Math Phys 32: 1950012.
    [14] Dabrowski L, Grosse H (1985) On nonlocal point interactions in one, two, and three dimensions. J Math Phys 26: 2777-2780.
    [15] D'Ancona P, Pierfelice V, Teta A (2006) Dispersive estimate for the Schrödinger equation with point interactions. Math Method Appl Sci 29: 309-323.
    [16] Dell'Antonio G, Michelangeli A, Scandone R, et al. (2018) Lp-Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction. Ann Henri Poincaré 19: 283-322.
    [17] Duchêne V, Marzuola JL, Weinstein MI (2011) Wave operator bounds for one-dimensional Schrödinger operators with singular potentials and applications. J Math Phys 52: 1-17.
    [18] Erdoğan MB, Schlag W (2004) Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three. I. Dyn Part Differ Eq 1: 359-379.
    [19] Galtbayar A, Yakima K (2019) On the approximation by regular potentials of Schrödinger operators with point interactions. arXiv: 1908.02936.
    [20] Georgiev V, Visciglia N (2007) About resonances for Schrödinger operators with short range singular perturbation, In: Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, Hackensack: World Sci. Publ., 74-84.
    [21] Goldberg M, Green WR (2016) The Lp boundedness of wave operators for Schrödinger operators with threshold singularities. Adv Math 303: 360-389.
    [22] Goloshchapova NI, Malamud MM, Zastavnyĭ VP (2011) Positive-definite functions and the spectral properties of the Schrödinger operator with point interactions. Mat. Zametki 90: 151-156.
    [23] Goloshchapova N, Malamud M, Zastavnyi V (2012) Radial positive definite functions and spectral theory of the Schrödinger operators with point interactions. Math Nachr 285: 1839-1859.
    [24] Grossmann A, Høegh-Krohn R, Mebkhout M (1980) A class of explicitly soluble, local, manycenter Hamiltonians for one-particle quantum mechanics in two and three dimensions. I. J Math Phys 21: 2376-2385.
    [25] Grossmann A, Høegh-Krohn R, Mebkhout M (1980) The one particle theory of periodic point interactions. Commun Math Phys 77: 87-110.
    [26] Iandoli F, Scandone R (2017) Dispersive estimates for Schrödinger operators with point interactions in $\mathbb{R}^3$, In: Advances in Quantum Mechanics: Contemporary Trends and Open Problems, Springer International Publishing, 187-199.
    [27] Ionescu AD, Jerison D (2003) On the absence of positive eigenvalues of Schrödinger operators with rough potentials. Geom Funct Anal 13: 1029-1081.
    [28] Jensen A, Kato T (1979) Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math J 46: 583-611.
    [29] Jerison D, Kenig CE (1985) Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann Math 121: 463-494.
    [30] Kato T (1959) Growth properties of solutions of the reduced wave equation with a variable coefficient. Commun Pure Appl Math 12: 403-425.
    [31] Koch H, Tataru D (2001) Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. Commun Pure Appl Math 54: 339-360.
    [32] Koch H, Tataru D (2006) Carleman estimates and absence of embedded eigenvalues. Commun Math Phys 267: 419-449.
    [33] Kuroda ST (1978) An Introduction to Scattering Theory, Aarhus: Aarhus Universitet, Matematisk Institut.
    [34] Lipovský J, Lotoreichik V (2017) Asymptotics of resonances induced by point interactions. Acta Phys Pol A 132: 1677-1682.
    [35] Nelson E (1977) Internal set theory: A new approach to nonstandard analysis. B Am Math Soc 83: 1165-1198.
    [36] Rauch J (1978) Local decay of scattering solutions to Schrödinger's equation. Commun Math Phys 61: 149-168.
    [37] Rodnianski I, Schlag W (2004) Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent Math 155: 451-513.
    [38] Scandone R (2019) Zero modes and low-energy resolvent expansion for three-dimensional Schrödinger operators with point interactions. arXiv: 1901.02449.
    [39] Scarlatti S, Teta A (1990) Derivation of the time-dependent propagator for the three-dimensional Schrödinger equation with one-point interaction. J Phys A 23: L1033-L1035.
    [40] Sjöstrand J (2002) Lectures on resonances, Available from: http://sjostrand.perso.math.cnrs.fr/Coursgbg.pdf.
    [41] Sjöstrand J, Zworski M (1991) Complex scaling and the distribution of scattering poles. J Am Math Soc 4: 729-769.
    [42] Yajima K (2005) Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue. Commun Math Phys 259: 475-509.
    [43] Yajima K (2016) Remarks on Lp-boundedness of wave operators for Schrödinger operators with threshold singularities. Doc Math 21: 391-443.
    [44] Yajima K (2016) On wave operators for Schrödinger operators with threshold singuralities in three dimensions. arXiv: 1606.03575.
    [45] Zorbas J (1980) Perturbation of self-adjoint operators by Dirac distributions. J Math Phys 21: 840-847.
    [46] zu Castell W, Filbir F, Szwarc R (2005) Strictly positive definite functions in $\mathbb{R}^d$. J Approx Theory 137: 277-280.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematics in Engineering
1.0 2.2

Metrics

Article views(3019) PDF downloads(663) Cited by(3)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog