December  2020, 13(12): 3427-3460. doi: 10.3934/dcdss.2020243

On absence of threshold resonances for Schrödinger and Dirac operators

1. 

Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, USA

2. 

Department of Mathematics, The University of Tennessee at Chattanooga, 415 EMCS Building, Dept. 6956

3. 

615 McCallie Avenue, Chattanooga, TN 37403-2504, USA

* Corresponding author: Fritz Gesztesy

Received  January 2019 Published  December 2020 Early access  January 2020

Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrödinger operators with sufficiently short-range interactions in general space dimensions.

More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions $ n \geqslant 3 $, the absence of resonances at $ \pm m $ for massive Dirac operators (with mass $ m > 0 $) in dimensions $ n \geqslant 5 $, and recall the well-known case of absence of zero-energy resonances for Schrödinger operators in dimension $ n \geqslant 5 $.

Citation: Fritz Gesztesy, Roger Nichols. On absence of threshold resonances for Schrödinger and Dirac operators. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3427-3460. doi: 10.3934/dcdss.2020243
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Y. Aharonov and A. Casher, Ground state of a spin-$1/2$ charged particle in a two-dimensional magnetic field, Phys. Rev. A (3), 19 (1979), 2461-2462.  doi: 10.1103/PhysRevA.19.2461.

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D. Aiba, Absence of zero resonances of massless Dirac operators, Hokkaido Math. J., 45 (2016), 263-270. 

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S. Albeverio, F. Gesztesy and R. Høegh-Krohn, The low energy expansion in nonrelativistic scattering theory, Ann. Inst. H. Poincaré, 37 (1982), 1–28.

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A. A. Balinsky and W. D. Evans, On the zero modes of Pauli operators, J. Funct. Anal., 179 (2001), 120-135.  doi: 10.1006/jfan.2000.3670.

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A. A. Balinsky and W. D. Evans, On the zero modes of Weyl-Dirac operators and their multiplicity, Bull. London Math. Soc., 34 (2002), 236-242.  doi: 10.1112/S0024609301008736.

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A. Balinsky and W. D. Evans, Zero modes of Pauli and Weyl–Dirac operators, Contemp. Math., 327 (2003), 1-9.  doi: 10.1090/conm/327/05800.

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A. Balinsky, W. D. Evans and Y. Saito, Dirac–Sobolev inequalities and estimates for the zero modes of massless Dirac operators, J. Math. Phys., 49 (2008), 043514, 10pp. doi: 10.1063/1.2912229.

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J. BehrndtF. GesztesyH. Holden and R. Nichols, Dirichlet-to Neumann maps, abstract Weyl–Titchmarsh $M$-functions, and a generalized index of unbounded meromorphic operator-valued functions, J. Diff. Eq., 261 (2016), 3551-3587.  doi: 10.1016/j.jde.2016.05.033.

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D. BolléF. Gesztesy and S. F. J. Wilk, New results for scattering on the line, Phys. Lett., 97A (1983), 30-34.  doi: 10.1016/0375-9601(83)90094-4.

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A. Carey, F. Gesztesy, G. Levitina, R. Nichols, F. Sukochev and D. Zanin, On the limiting absorption principle for massless Dirac operators and properties of spectral shift functions, work in progress.

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A. CareyF. GesztesyG. LevitinaD. PotapovF. Sukochev and D. Zanin, On index theory for non-Fredholm operators: A $(1+1)$-dimensional example, Math. Nachrichten, 289 (2016), 575-609.  doi: 10.1002/mana.201500065.

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A. CareyF. GesztesyG. Levitina and F. Sukochev, On the index of a non-Fredholm model operator, Operators and Matrices, 10 (2016), 881-914.  doi: 10.7153/oam-10-50.

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M. B. ErdoğanM. Goldberg and W. R. Green, Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions, Commun. Math. Phys., 367 (2019), 241-263.  doi: 10.1007/s00220-018-3231-8.

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M. B. ErdoğanM. Goldberg and W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $ {\mathbb{R}}^3$, J. Eur. Math. Soc., 10 (2008), 507-531.  doi: 10.4171/JEMS/120.

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M. B. Erdoğan and W. R. Green, Dispersive estimates for the Schrödinger equation for $C^{\frac{n-3}{2}}$ potentials in odd dimensions, Int. Math. Res. Notices, 2010 (2010), 2532-2565. 

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show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1966.

[2]

C. Adam, B. Muratori and C. Nash, Zero modes of the Dirac operator in three dimensions, Phys. Rev. D (3), 60 (1999), 125001, 8pp. doi: 10.1103/PhysRevD.60.125001.

[3]

C. AdamB. Muratori and C. Nash, Degeneracy of zero modes of the Dirac operator in three dimensions, Phys. Lett. B, 485 (2000), 314-318.  doi: 10.1016/S0370-2693(00)00701-2.

[4]

C. Adam, B. Muratori and C. Nash, Multiple zero modes of the Dirac operator in three dimensions, Phys. Rev. D (3), 62 (2000), 085026, 9pp. doi: 10.1103/PhysRevD.62.085026.

[5]

C. AdamB. Muratori and C. Nash, Zero modes in finite range magnetic fields, Modern Phys. Lett. A, 15 (2000), 1577-1581.  doi: 10.1142/S0217732300001948.

[6]

Y. Aharonov and A. Casher, Ground state of a spin-$1/2$ charged particle in a two-dimensional magnetic field, Phys. Rev. A (3), 19 (1979), 2461-2462.  doi: 10.1103/PhysRevA.19.2461.

[7]

D. Aiba, Absence of zero resonances of massless Dirac operators, Hokkaido Math. J., 45 (2016), 263-270. 

[8]

S. Albeverio, F. Gesztesy and R. Høegh-Krohn, The low energy expansion in nonrelativistic scattering theory, Ann. Inst. H. Poincaré, 37 (1982), 1–28.

[9]

A. A. Balinsky and W. D. Evans, On the zero modes of Pauli operators, J. Funct. Anal., 179 (2001), 120-135.  doi: 10.1006/jfan.2000.3670.

[10]

A. A. Balinsky and W. D. Evans, On the zero modes of Weyl-Dirac operators and their multiplicity, Bull. London Math. Soc., 34 (2002), 236-242.  doi: 10.1112/S0024609301008736.

[11]

A. Balinsky and W. D. Evans, Zero modes of Pauli and Weyl–Dirac operators, Contemp. Math., 327 (2003), 1-9.  doi: 10.1090/conm/327/05800.

[12] A. Balinsky and W. D. Evans, Spectral Analysis of Relativistic Operators, Imperial College Press, London, 2011. 
[13]

A. Balinsky, W. D. Evans and Y. Saito, Dirac–Sobolev inequalities and estimates for the zero modes of massless Dirac operators, J. Math. Phys., 49 (2008), 043514, 10pp. doi: 10.1063/1.2912229.

[14]

J. BehrndtF. GesztesyH. Holden and R. Nichols, Dirichlet-to Neumann maps, abstract Weyl–Titchmarsh $M$-functions, and a generalized index of unbounded meromorphic operator-valued functions, J. Diff. Eq., 261 (2016), 3551-3587.  doi: 10.1016/j.jde.2016.05.033.

[15]

R. D. Benguria and H. Van Den Bosch, A criterion for the existence of zero modes for the Pauli operator with fastly decaying fields, J. Math. Phys., 56 (2015), 052104, 7pp. doi: 10.1063/1.4920924.

[16]

H. BlancarteB. Grebert and R. Weder, High- and low-energy estimates for the Dirac equation, J. Math. Phys., 36 (1995), 991-1015.  doi: 10.1063/1.531138.

[17]

D. BolléF. Gesztesy and S. F. J. Wilk, New results for scattering on the line, Phys. Lett., 97A (1983), 30-34.  doi: 10.1016/0375-9601(83)90094-4.

[18]

D. BolléF. Gesztesy and S. F. J. Wilk, A complete treatment of low-energy scattering in one dimension, J. Operator Theory, 13 (1985), 3-31. 

[19]

D. BolléF. Gesztesy and C. Danneels, Threshold scattering in two dimensions, Ann. Inst. H. Poincaré, 48 (1988), 175-204. 

[20]

D. BolléF. GesztesyC. Danneels and S. F. J. Wilk, Threshold behavior and Levinson's theorem for two-dimensional scattering systems: a surprise, Phys. Rev. Lett., 56 (1986), 900-903. 

[21]

D. Bollé, F. Gesztesy and M. Klaus, Scattering theory for one-dimensional systems with $\int dx V(x) = 0$, J. Math. Anal. Appl., 122 (1987), 496–518; Errata, 130 (1988), 590. doi: 10.1016/0022-247X(87)90281-2.

[22]

A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys., 28 (2016), 1630002 (55 pages). doi: 10.1142/S0129055X16300028.

[23]

A. Carey, F. Gesztesy, G. Levitina, R. Nichols, F. Sukochev and D. Zanin, On the limiting absorption principle for massless Dirac operators and properties of spectral shift functions, work in progress.

[24]

A. CareyF. GesztesyG. LevitinaD. PotapovF. Sukochev and D. Zanin, On index theory for non-Fredholm operators: A $(1+1)$-dimensional example, Math. Nachrichten, 289 (2016), 575-609.  doi: 10.1002/mana.201500065.

[25]

A. CareyF. GesztesyG. Levitina and F. Sukochev, On the index of a non-Fredholm model operator, Operators and Matrices, 10 (2016), 881-914.  doi: 10.7153/oam-10-50.

[26]

A. CareyF. GesztesyD. PotapovF. Sukochev and Y. Tomilov, On the Witten index in terms of spectral shift functions, J. Analyse Math., 132 (2017), 1-61.  doi: 10.1007/s11854-017-0003-x.

[27]

N. Du Plessis, An Introduction to Potential Theory, Oliver & Boyd, Edinburgh, 1970.

[28]

D. M. Elton, New examples of zero modes, J. Phys. A, 33 (2000), 7297-7303.  doi: 10.1088/0305-4470/33/41/304.

[29]

D. M. Elton, Spectral properties of the equation $(\nabla + i e A) \times u = \pm m u$, Proc. Roy. Soc. Edinburgh, 131 A (2001), 1065-1089.  doi: 10.1017/S030821050000127X.

[30]

D. M. Elton, The local structure of zero mode producing magnetic potentials, Commun. Math. Phys., 229 (2002), 121-139.  doi: 10.1007/s00220-002-0679-2.

[31]

M. B. ErdoğanM. Goldberg and W. R. Green, Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy, Commun. PDE, 39 (2014), 1936-1964.  doi: 10.1080/03605302.2014.921928.

[32]

M. B. ErdoğanM. Goldberg and W. R. Green, Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions, Commun. Math. Phys., 367 (2019), 241-263.  doi: 10.1007/s00220-018-3231-8.

[33]

M. B. Erdoğan, M. Goldberg and W. R. Green, The massless Dirac equation in two dimensions: zero-energy obstructions and dispersive estimates, arXiv: 1807.00219.

[34]

M. B. ErdoğanM. Goldberg and W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $ {\mathbb{R}}^3$, J. Eur. Math. Soc., 10 (2008), 507-531.  doi: 10.4171/JEMS/120.

[35]

M. B. ErdoğanM. Goldberg and W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions, Forum Math., 21 (2009), 687-722. 

[36]

M. B. Erdoğan and W. R. Green, Dispersive estimates for the Schrödinger equation for $C^{\frac{n-3}{2}}$ potentials in odd dimensions, Int. Math. Res. Notices, 2010 (2010), 2532-2565. 

[37]

M. B. Erdoğan and W. R. Green, Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energies, Trans. Amer. Math. Soc., 365 (2013), 6403-6440.  doi: 10.1090/S0002-9947-2013-05861-8.

[38]

M. B. Erdoğan and W. R. Green, The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions, Commun. Math. Phys., 352 (2017), 719-757.  doi: 10.1007/s00220-016-2811-8.

[39]

M. B. ErdoğanW. R. Green and E. Toprak, Dispersive estimates for massive Dirac operators in dimension two, J. Diff. Eq., 264 (2018), 5802-5837.  doi: 10.1016/j.jde.2018.01.019.

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