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  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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020243
References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

M. B. Erdoğan, W. R. Green and E. Toprak, Dispersive estimates for Dirac operators in dimension three with obstructions at threshold energies, Amer. J. Math., 141 (2019), 1217–1258, arXiv: 1609.05164. doi: 10.1353/ajm.2019.0031.  Google Scholar

[41]

M. B. Erdoğan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: Ⅰ, Dyn. PDE, 1 (2004), 359-379.  doi: 10.4310/DPDE.2004.v1.n4.a1.  Google Scholar

[42]

M. B. Erdoğan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: Ⅱ, J. Anal. Math., 99 (2006), 199-248.  doi: 10.1007/BF02789446.  Google Scholar

[43]

L. Erdős and J. P. Solovej, The kernel of Dirac operators on $ {\mathbb{S}}^3$ and $ {\mathbb{R}}^3$, Rev. Math. Phys., 13 (2001), 1247-1280.  doi: 10.1142/S0129055X01000983.  Google Scholar

[44]

G. B. Folland, Real Analysis. Modern Techniques and Their Applications, 2nd ed., Wiley, New York, 1999.  Google Scholar

[45]

J. FröhlichE. H. Lieb and M. Loss, Stability of Coulomb systems with magnetic fields. I. The one-electron atom, Commun. Math. Phys., 104 (1986), 251-270.  doi: 10.1007/BF01211593.  Google Scholar

[46]

F. Gesztesy and H. Holden, A unified approach to eigenvalues and resonances of Schrödinger operators using Fredholm determinants, J. Math. Anal. Appl., 123 (1987), 181-198.  doi: 10.1016/0022-247X(87)90303-9.  Google Scholar

[47]

F. GesztesyH. Holden and R. Nichols, On factorizations of analytic operator-valued functions and eigenvalue multiplicity questions, Erratum, 85 (2016), 301-302.  doi: 10.1007/s00020-016-2290-5.  Google Scholar

[48]

F. GesztesyY. LatushkinM. Mitrea and M. Zinchenko, Nonselfadjoint operators, infinite determinants, and some applications, Russ. J. Math. Phys., 12 (2005), 443-471.   Google Scholar

[49]

F. GesztesyM. MalamudM. Mitrea and S. Naboko, Generalized polar decompositions for closed operators in Hilbert spaces and some applications, Integral Eq. Operator Th., 64 (2009), 83-113.  doi: 10.1007/s00020-009-1678-x.  Google Scholar

[50]

A. Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in $L^2({\mathbb{R}}^m)$, $m \geqslant 5$, Duke Math. J., 47 (1980), 57-80.  doi: 10.1215/S0012-7094-80-04706-7.  Google Scholar

[51]

A. Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in $L^2({\mathbb{R}}^4)$, J. Math. Anal. Appl., 101 (1984), 397-422.  doi: 10.1016/0022-247X(84)90110-0.  Google Scholar

[52]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.   Google Scholar

[53]

A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys., 16 (2004), 675-677.  doi: 10.1142/S0129055X04002102.  Google Scholar

[54]

H. Kalf, T. Okaji and O. Yamada, The Dirac operator with mass $m_0 \geqslant 0$: Non-existence of zero modes and of threshold eigenvalues, Doc. Math., 20 (2015), 37–64; Addendum, Doc. Math. (to appear).  Google Scholar

[55]

H. Kalf and O. Yamada, Essential self-adjointness of $n$-dimensional Dirac operators with a variable mass term, J. Math. Phys., 42 (2001), 2667-2676.  doi: 10.1063/1.1367331.  Google Scholar

[56]

M. Klaus, Some applications of the Birman–Schwinger principle, Helv. Phts. Acta, 55 (1982), 49-68.   Google Scholar

[57]

M. Klaus, On coupling constant thresholds and related eigenvalue properties of Dirac operators, J. reine angew. Math., 362 (1985), 197-212.   Google Scholar

[58]

M. Klaus, On the Levinson theorem for Dirac operators, J. Math. Phys., 31 (1990), 182-190.  doi: 10.1063/1.528858.  Google Scholar

[59]

M. Klaus and B. Simon, Coupling constant thresholds in nonrelativistic quantum mechanics. Ⅰ. Short-range two-body case, Ann. Phys., 130 (1980), 251-281.  doi: 10.1016/0003-4916(80)90338-3.  Google Scholar

[60]

M. Loss and H.-T. Yau, Stability of Coulomb systems with magnetic fields. Ⅲ. Zero energy bound states of the Pauli operator, Commun. Math. Phys., 104 (1986), 283-290.  doi: 10.1007/BF01211595.  Google Scholar

[61]

R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Commun. Pure Appl. Math., 32 (1979), 783-795.  doi: 10.1002/cpa.3160320604.  Google Scholar

[62]

M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56.  doi: 10.1016/0022-1236(82)90084-2.  Google Scholar

[63]

R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, New York, 2002.  Google Scholar

[64]

L. Nirenberg and H. F. Walker, The null spaces of elliptic partial differential operators on $ {\mathbb{R}}^n$, J. Math. Anal. Appl., 42 (1973), 271-301.  doi: 10.1016/0022-247X(73)90138-8.  Google Scholar

[65]

M. Persson, Zero modes for the magnetic Pauli operator in even-dimensional Euclidean space, Lett. Math. Phys., 85 (2008), 111-128.  doi: 10.1007/s11005-008-0265-4.  Google Scholar

[66]

G. Rozenblum and N. Shirokov, Infiniteness of zero modes for the Pauli operator with singular magnetic field, J. Funct. Anal., 233 (2006), 135-172.  doi: 10.1016/j.jfa.2005.08.001.  Google Scholar

[67]

Y. Saitō and T. Umeda, The zero modes and zero resonances of massless Dirac operators, Hokkaido Math. J., 37 (2008), 363-388.  doi: 10.14492/hokmj/1253539560.  Google Scholar

[68]

Y. Saitō and T. Umeda, The asymptotic limits of zero modes of massless Dirac operators, Lett. Math. Phys., 83 (2008), 97-106.  doi: 10.1007/s11005-007-0207-6.  Google Scholar

[69]

Y. Saitō and T. Umeda, Eigenfunctions at the threshold energies of magnetic Dirac operators, Rev. Math. Phys., 23 (2011), 155-178.  doi: 10.1142/S0129055X11004254.  Google Scholar

[70]

Y. Saitō and T. Umeda, A sequence of zero modes of Weyl–Dirac operators and an associated sequence of solvable polynomials, in Spectral theory, Function Spaces and Inequalities. New Techniques and Recent Trends, B. M. Brown, J. Lang, and I. G. Wood (eds.), Operator Theory: Advances and Applications, Birkhäuser, Springer, Basel, 219 (2012), 197–209. doi: 10.1007/978-3-0348-0263-5_11.  Google Scholar

[71]

K. M. Schmidt, Spectral properties of rotationally symmetric massless Dirac operators, Lett. Math. Phys., 92 (2010), 231-241.  doi: 10.1007/s11005-010-0393-5.  Google Scholar

[72]

K. M. Schmidt and T. Umeda, Spectral properties of massless Dirac operators with real-valued potentials, RIMS Kôkyûroku Bessatsu, B45 (2014), 25–30.  Google Scholar

[73]

K. M. Schmidt and T. Umeda, Schnol's theorem and spectral properties of massless Dirac operators with scalar potentials, Lett. Math. Phys., 105 (2015), 1479-1497.  doi: 10.1007/s11005-015-0799-1.  Google Scholar

[74] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.   Google Scholar
[75]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.  Google Scholar

[76]

E. Toprak, A weighted estimate for two dimensional Schrödinger, matrix Schrödinger, and wave equations with resonance of the first kind at zero energy, J. Spectral Th., 7 (2017), 1235-1284.  doi: 10.4171/JST/189.  Google Scholar

[77]

K. Yajima, Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalues, Commun. Math. Phys., 259 (2005), 475-509.  doi: 10.1007/s00220-005-1375-9.  Google Scholar

[78]

Y. Zhong and G. L. Gao, Some new results about the massless Dirac operator, J. Math. Phys., 54 (2013), 043510, 25pp. doi: 10.1063/1.4799936.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

M. B. Erdoğan, W. R. Green and E. Toprak, Dispersive estimates for Dirac operators in dimension three with obstructions at threshold energies, Amer. J. Math., 141 (2019), 1217–1258, arXiv: 1609.05164. doi: 10.1353/ajm.2019.0031.  Google Scholar

[41]

M. B. Erdoğan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: Ⅰ, Dyn. PDE, 1 (2004), 359-379.  doi: 10.4310/DPDE.2004.v1.n4.a1.  Google Scholar

[42]

M. B. Erdoğan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: Ⅱ, J. Anal. Math., 99 (2006), 199-248.  doi: 10.1007/BF02789446.  Google Scholar

[43]

L. Erdős and J. P. Solovej, The kernel of Dirac operators on $ {\mathbb{S}}^3$ and $ {\mathbb{R}}^3$, Rev. Math. Phys., 13 (2001), 1247-1280.  doi: 10.1142/S0129055X01000983.  Google Scholar

[44]

G. B. Folland, Real Analysis. Modern Techniques and Their Applications, 2nd ed., Wiley, New York, 1999.  Google Scholar

[45]

J. FröhlichE. H. Lieb and M. Loss, Stability of Coulomb systems with magnetic fields. I. The one-electron atom, Commun. Math. Phys., 104 (1986), 251-270.  doi: 10.1007/BF01211593.  Google Scholar

[46]

F. Gesztesy and H. Holden, A unified approach to eigenvalues and resonances of Schrödinger operators using Fredholm determinants, J. Math. Anal. Appl., 123 (1987), 181-198.  doi: 10.1016/0022-247X(87)90303-9.  Google Scholar

[47]

F. GesztesyH. Holden and R. Nichols, On factorizations of analytic operator-valued functions and eigenvalue multiplicity questions, Erratum, 85 (2016), 301-302.  doi: 10.1007/s00020-016-2290-5.  Google Scholar

[48]

F. GesztesyY. LatushkinM. Mitrea and M. Zinchenko, Nonselfadjoint operators, infinite determinants, and some applications, Russ. J. Math. Phys., 12 (2005), 443-471.   Google Scholar

[49]

F. GesztesyM. MalamudM. Mitrea and S. Naboko, Generalized polar decompositions for closed operators in Hilbert spaces and some applications, Integral Eq. Operator Th., 64 (2009), 83-113.  doi: 10.1007/s00020-009-1678-x.  Google Scholar

[50]

A. Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in $L^2({\mathbb{R}}^m)$, $m \geqslant 5$, Duke Math. J., 47 (1980), 57-80.  doi: 10.1215/S0012-7094-80-04706-7.  Google Scholar

[51]

A. Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in $L^2({\mathbb{R}}^4)$, J. Math. Anal. Appl., 101 (1984), 397-422.  doi: 10.1016/0022-247X(84)90110-0.  Google Scholar

[52]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.   Google Scholar

[53]

A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys., 16 (2004), 675-677.  doi: 10.1142/S0129055X04002102.  Google Scholar

[54]

H. Kalf, T. Okaji and O. Yamada, The Dirac operator with mass $m_0 \geqslant 0$: Non-existence of zero modes and of threshold eigenvalues, Doc. Math., 20 (2015), 37–64; Addendum, Doc. Math. (to appear).  Google Scholar

[55]

H. Kalf and O. Yamada, Essential self-adjointness of $n$-dimensional Dirac operators with a variable mass term, J. Math. Phys., 42 (2001), 2667-2676.  doi: 10.1063/1.1367331.  Google Scholar

[56]

M. Klaus, Some applications of the Birman–Schwinger principle, Helv. Phts. Acta, 55 (1982), 49-68.   Google Scholar

[57]

M. Klaus, On coupling constant thresholds and related eigenvalue properties of Dirac operators, J. reine angew. Math., 362 (1985), 197-212.   Google Scholar

[58]

M. Klaus, On the Levinson theorem for Dirac operators, J. Math. Phys., 31 (1990), 182-190.  doi: 10.1063/1.528858.  Google Scholar

[59]

M. Klaus and B. Simon, Coupling constant thresholds in nonrelativistic quantum mechanics. Ⅰ. Short-range two-body case, Ann. Phys., 130 (1980), 251-281.  doi: 10.1016/0003-4916(80)90338-3.  Google Scholar

[60]

M. Loss and H.-T. Yau, Stability of Coulomb systems with magnetic fields. Ⅲ. Zero energy bound states of the Pauli operator, Commun. Math. Phys., 104 (1986), 283-290.  doi: 10.1007/BF01211595.  Google Scholar

[61]

R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Commun. Pure Appl. Math., 32 (1979), 783-795.  doi: 10.1002/cpa.3160320604.  Google Scholar

[62]

M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56.  doi: 10.1016/0022-1236(82)90084-2.  Google Scholar

[63]

R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, New York, 2002.  Google Scholar

[64]

L. Nirenberg and H. F. Walker, The null spaces of elliptic partial differential operators on $ {\mathbb{R}}^n$, J. Math. Anal. Appl., 42 (1973), 271-301.  doi: 10.1016/0022-247X(73)90138-8.  Google Scholar

[65]

M. Persson, Zero modes for the magnetic Pauli operator in even-dimensional Euclidean space, Lett. Math. Phys., 85 (2008), 111-128.  doi: 10.1007/s11005-008-0265-4.  Google Scholar

[66]

G. Rozenblum and N. Shirokov, Infiniteness of zero modes for the Pauli operator with singular magnetic field, J. Funct. Anal., 233 (2006), 135-172.  doi: 10.1016/j.jfa.2005.08.001.  Google Scholar

[67]

Y. Saitō and T. Umeda, The zero modes and zero resonances of massless Dirac operators, Hokkaido Math. J., 37 (2008), 363-388.  doi: 10.14492/hokmj/1253539560.  Google Scholar

[68]

Y. Saitō and T. Umeda, The asymptotic limits of zero modes of massless Dirac operators, Lett. Math. Phys., 83 (2008), 97-106.  doi: 10.1007/s11005-007-0207-6.  Google Scholar

[69]

Y. Saitō and T. Umeda, Eigenfunctions at the threshold energies of magnetic Dirac operators, Rev. Math. Phys., 23 (2011), 155-178.  doi: 10.1142/S0129055X11004254.  Google Scholar

[70]

Y. Saitō and T. Umeda, A sequence of zero modes of Weyl–Dirac operators and an associated sequence of solvable polynomials, in Spectral theory, Function Spaces and Inequalities. New Techniques and Recent Trends, B. M. Brown, J. Lang, and I. G. Wood (eds.), Operator Theory: Advances and Applications, Birkhäuser, Springer, Basel, 219 (2012), 197–209. doi: 10.1007/978-3-0348-0263-5_11.  Google Scholar

[71]

K. M. Schmidt, Spectral properties of rotationally symmetric massless Dirac operators, Lett. Math. Phys., 92 (2010), 231-241.  doi: 10.1007/s11005-010-0393-5.  Google Scholar

[72]

K. M. Schmidt and T. Umeda, Spectral properties of massless Dirac operators with real-valued potentials, RIMS Kôkyûroku Bessatsu, B45 (2014), 25–30.  Google Scholar

[73]

K. M. Schmidt and T. Umeda, Schnol's theorem and spectral properties of massless Dirac operators with scalar potentials, Lett. Math. Phys., 105 (2015), 1479-1497.  doi: 10.1007/s11005-015-0799-1.  Google Scholar

[74] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.   Google Scholar
[75]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.  Google Scholar

[76]

E. Toprak, A weighted estimate for two dimensional Schrödinger, matrix Schrödinger, and wave equations with resonance of the first kind at zero energy, J. Spectral Th., 7 (2017), 1235-1284.  doi: 10.4171/JST/189.  Google Scholar

[77]

K. Yajima, Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalues, Commun. Math. Phys., 259 (2005), 475-509.  doi: 10.1007/s00220-005-1375-9.  Google Scholar

[78]

Y. Zhong and G. L. Gao, Some new results about the massless Dirac operator, J. Math. Phys., 54 (2013), 043510, 25pp. doi: 10.1063/1.4799936.  Google Scholar

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