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Mathematics in Engineering

2021, Volume 3, Issue 2: 1-14. doi: 10.3934/mine.2021017
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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.


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    [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.
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