The lower part of the spectrum of the Hamiltonian of the spinless PauliFierz model (A twocomponent Bose field interacting with a charged particle)
Author:
R. A. Minlos
Translated by:
E. Khukhro
Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom 69 (2008).
Journal:
Trans. Moscow Math. Soc. 2008, 209253
MSC (2000):
Primary 81Q10; Secondary 47A10, 47A40, 47A55, 81T10, 81U99, 81V10
Published electronically:
November 19, 2008
MathSciNet review:
2549448
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We investigate a model of a vector massive spinless Bose field in the space interacting with a nonrelativistic particle, where the interaction parameter is assumed to be sufficiently small. We study the ground state of the Hamiltonian for a fixed total momentum of the system and show that such a state is nondegenerate and exists only for a bounded domain of values of this momentum. We also show that, apart from the ground state, the operator has no other eigenvalues below the continuous spectrum. Furthermore, the next two, ``oneboson'', branches of the spectrum of are constructed, which describe the scattering of one boson (with two possible polarization values) on the ground state.
 1.
Nicolae
Angelescu, Robert
A. Minlos, and Valentin
A. Zagrebnov, Lower spectral branches of a particle coupled to a
Bose field, Rev. Math. Phys. 17 (2005), no. 10,
1111–1142. MR 2187291
(2007b:81072), http://dx.doi.org/10.1142/S0129055X05002509
 2.
Asao
Arai, Spectral analysis of a quantum harmonic oscillator coupled to
infinitely many scalar bosons, J. Math. Anal. Appl.
140 (1989), no. 1, 270–288. MR 997857
(90j:81035), http://dx.doi.org/10.1016/0022247X(89)90108X
 3.
V.
I. Arnol′d, S.
M. GuseĭnZade, and A.
N. Varchenko, Singularities of differentiable maps. Vol. II,
Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc.,
Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from
the Russian by Hugh Porteous; Translation revised by the authors and James
Montaldi. MR
966191 (89g:58024)
 4.
Volker
Bach, Jürg
Fröhlich, and Alessandro
Pizzo, Infraredfinite algorithms in QED: the groundstate of an
atom interacting with the quantized radiation field, Comm. Math. Phys.
264 (2006), no. 1, 145–165. MR 2212219
(2007d:81248), http://dx.doi.org/10.1007/s0022000514783
 5.
V. Bach, J. Fröhlich, and I.M. Sigal,
 1.
Volker
Bach, Jürg
Fröhlich, and Israel
Michael Sigal, Spectral analysis for systems of atoms and molecules
coupled to the quantized radiation field, Comm. Math. Phys.
207 (1999), no. 2, 249–290. MR 1724854
(2001j:81256), http://dx.doi.org/10.1007/s002200050726
 2.
Volker
Bach, Jürg
Fröhlich, and Israel
Michael Sigal, Quantum electrodynamics of confined nonrelativistic
particles, Adv. Math. 137 (1998), no. 2,
299–395. MR 1639713
(99e:81051b), http://dx.doi.org/10.1006/aima.1998.1734
 1.
 N. Angelescu, R.A. Minlos, and V.A. Zagrebnov, Lower spectral branches of a particle coupled to a Bose field, Rev. Math. Phys. 17 (2005), no.9, 132. MR 2187291 (2007b:81072)
 2.
 A. Arai, Spectral analysis of a quantum harmonic oscillator coupled to infinitely many scalar bosons, J. Math. Anal. Appl. 140 (1989), 270288. MR 0997857 (90j:81035)
 3.
 V.I. Arnol'd, S.M. GuseĭnZade, and A.N. Varchenko, Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals, Nauka, Moscow, 1984; English transl., Birkhäuser, Boston, MA, 1988. MR 0966191 (89g:58024)
 4.
 V. Bach, J. Fröhlich, and A. Pizzo, Infraredfinite algorithms in QED: the groundstate in an atom interacting with the quantized radiation field, Commun. Math. Phys. 264 (2006), no.1, 145165. MR 2212219 (2007d:81248)
 5.
 V. Bach, J. Fröhlich, and I.M. Sigal,
 1.
 Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field, Commun. Math. Phys. 207 (1999), no.2, 249290. MR 1724854 (2001j:81256)
 2.
 Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137 (1998), no.2, 299395. MR 1639713 (99e:81051b)
 6.
 Ju.M. Berezanskiı, Expansions in eigenfunctions of selfadjoint operators, Naukova Dumka, Kiev, 1965; English transl., Translations of Mathematical Monographs, Vol. 17, Amer. Math. Soc., Providence, RI, 1968. MR 0222718 (36:5768)
 7.
 F.A. Berezin, The method of second quantization, Nauka, Moscow, 1965; English transl., Pure and Applied Physics, Vol. 24 Academic Press, New York, 1966. MR 0208930 (34:8738)
 8.
 M.Sh. Birman and M.Z. Solomyak, Spectral theory of selfadjoint operators in Hilbert space, Leningrad Univ., Leningrad, 1980; English transl., Mathematics and its Applications (Soviet Series). D. Reidel, Dordrecht, 1987. MR 1192782 (93g:47001)
 9.
 J. Dereziński and C. Gérard, Asymptotic completeness in quantum field theory. Massive PauliFierz Hamiltonians, Rev. Math. Phys. 11 (1999), no.4, 383450. MR 1682684 (2000h:81141)
 10.
 H. Fröhlich, Electrons in lattice fields, Adv. Phys. 3 (1954), 325364.
 11.
 J. Fröhlich,
 1.
 On the infrared problem in a model of scalar electrons and massless, scalar bosons, Ann. Inst. H. Poincaré Sect. A (N.S.) 19 (1973), 1103. MR 0368649 (51:4890)
 2.
 Existence of dressed oneelectron states in a class of persistent models, Fortschr. Phys. 22 (1974), 159198.
 12.
 J. Fröhlich, M. Griesemer, and B. Schlein, Asymptotic completeness for Compton scattering, Commun. Math. Phys. 252 (2004), no.13, 415476. MR 2104885 (2006d:81331)
 13.
 C. Gérard,
 1.
 On the existence of ground states for massless PauliFierz Hamiltonians, Ann. Henri Poincaré 1 (2000), no.3, 443459. MR 1777307 (2001g:81065)
 2.
 On the scattering theory of massless Nelson models, Rev. Math. Phys. 14 (2002), no.11, 11651280. MR 1943190 (2004g:81137)
 14.
 I.M. Gel'fand and G.E. Shilov, Generalized functions, Vol. 3, Nauka, Moscow, 1984; English transl. of 1st ed.: Generalized functions. Vol. 3: Theory of differential equations, Academic Press, New York, 1967. MR 0217416 (36:506)
 15.
 B. Gerlach and H. Löwen, Analytical properties of polaron systems or: Do polaronic phase transitions exist or not? Rev. Modern Phys. 63 (1991), no.1, 6390. MR 1102193 (92j:82021)
 16.
 M. Hirokawa, F. Hiroshima, and H. Spohn, Ground state for point particles interacting through a massless scalar Bose field, Adv. Math. 191 (2005), no.2, 339392. MR 2103217 (2005h:81206)
 17.
 F. Hiroshima and H. Spohn, Ground state degeneracy of the PauliFierz Hamiltonian with spin, Adv. Theor. Math. Phys. 5 (2001), no.6, 10911104. MR 1926665 (2003m:81294)
 18.
 F. Hiroshima, Ground states and spectrum of quantum electrodynamics of nonrelativistic particles, Trans. Amer. Math. Soc. 353 (2001), no. 11, 44974528 (electronic). MR 1851181 (2002f:81014)
 19.
 T.D. Lee, F.E. Low, and D. Pines, The motion of slow electrons in a polar crystal, Phys. Rev. (2) 90 (1953) 297302. MR 0103072 (21:1856)
 20.
 E.H. Lieb and L.E. Thomas, Exact ground state energy of the strongcoupling polaron, Commun. Math. Phys. 183 (1997), no.3, 511519; Erratum: ``Exact ground state energy of the strongcoupling polaron'', ibid. 188 (1997), no.2, 499500. MR 1462224 (99a:81034a); MR 1471825 (99a:81034b)
 21.
 V.A. Malyshev and R.A. Minlos, Linear operators in infiniteparticle systems, Nauka, Moscow, 1994; English transl.: Linear infiniteparticle operators, Translations of Mathematical Monographs, Vol. 143, Amer. Math. Soc., Providence, RI, 1995. MR 1317349 (96d:82038)
 22.
 R.A. Minlos, On the lower branch of the spectrum of a fermion interacting with a boson gas (a polaron), Teoret. Mat. Fiz. 92 (1992), no.2, 255268; English transl. in Theoret. Math. Phys. 92 (1993), no.2, 869878. MR 1226014 (94b:81153)
 23.
 J.S. Møller, The translation invariant massive Nelson model. I. The bottom of the spectrum, Ann. Henri Poincaré 6 (2005), no.6, 10911135. MR 2189378 (2006m:81168)
 24.
 M. Reed and B. Simon, Methods of modern mathematical physics. III. Scattering theory, Academic Press, New York, 1979; IV. Analysis of operators, Academic Press, New York, 1978.MR 0529429 (80m:81085) MR 0493421 (58:12429c)
 25.
 H. Spohn,
 1.
 Ground state of a quantum particle coupled to a scalar Bose field, Lett. Math. Phys. 44 (1998), no.1, 916. MR 1623746 (99c:81278)
 2.
 The polaron at large total momentum, J. Phys. A 21 (1988), no.5, 11991211. MR 0939707 (89m:81207)
 3.
 Dynamics of charged particles and their radiation field, Cambridge Univ. Press, Cambridge, 2004. MR 2097788 (2005g:81355)
 26.
 D.R. Yafaev, Mathematical scattering theory. General theory, S.Peterburg. Univ., 1994; English transl., Transl. of Math. Monographs, Vol. 105. Amer. Math. Soc., Providence, RI, 1992. MR 1180965 (94f:47012)
Similar Articles
Retrieve articles in Transactions of the Moscow Mathematical Society
with MSC (2000):
81Q10,
47A10,
47A40,
47A55,
81T10,
81U99,
81V10
Retrieve articles in all journals
with MSC (2000):
81Q10,
47A10,
47A40,
47A55,
81T10,
81U99,
81V10
Additional Information
R. A. Minlos
Affiliation:
Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia
Email:
minl@iitp.ru
DOI:
http://dx.doi.org/10.1090/S0077155408001702
PII:
S 00771554(08)001702
Keywords:
Nonrelativistic charged particle,
Bose field,
PauliFierz model,
Hamiltonian,
spectrum,
ground state,
scattering
Published electronically:
November 19, 2008
Additional Notes:
The author thanks Professor H. Spohn and Dr. E. A. Zhizhina for useful discussions on questions relating to this paper. The author thanks the Mathematics Centre of the Munich Technical University, where the plan for this research emerged for the first time, for its warm hospitality and financial support. The author also thanks the following organisations for financial support: the Russian Foundation for Basic Research (grant #050100449), the President Foundation for Support of Scientific Schools of Russia, and the American Foundation CRDF, grant RUM12603MO05.
Article copyright:
© Copyright 2008
American Mathematical Society
