Abstract
It has been known since the paper(26) and then due to a rigorous result(3) that the answer to the question in the title is negative for a three-dimensional “ideal gas of charged bosons”. The present paper adds a new rigorous result in this direction. We show that the answer to the question becomes positive, if this “ideal gas of charged bosons” is simultaneously embedded in an appropriate periodic external potential. We prove that it is true for the Perfect Bose Gas (PBG), as well as for the Imperfect Bose Gas with a Mean-Field repulsive particle interaction.
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REFERENCES
M. Abramowitz and I. Stegun, Handbook of mathematical functions with formulas, graphs,and mathematical tables (Dover Publications,Inc., New York, 1992).
S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equa-tions:Bounds on Eigenfunctions of N -body Schroedinger Operators, Mathematical Notes 29 (Princeton University Press, Princeton, 1982).
N. Angelescu and A. Corciovei, On free quantum gases in a homogeneous magnetic field, Rev.Roum.Phys. 20:661–671 (1975).
J. Bellissard, A. van Elst,and H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect, J.Math.Phys. 35:5373–5451 (1994).
F.A. Berezin and M.A. Shubin, The Schr ¨odinger Equation (Kluwer Academic Publ., Dordrecht-London, 1991).
K. Berg-Sørensen and K. Mølmer, Bose-Einstein condensates in spatially periodic potentials, Phys.Rev.A 58:1480–1484 (1998).
M.Sh. Birman and T.A. Suslina, Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity, St.Petersburg Math.J. 11 1–40 (2000).
E. Brown, Bloch electrons in a uniform magnetic field, Phys.Rev. 133:A1038–1044 (1964).
H.D. Cornean and G. Nenciu, On eigenfunction decay for two-dimensional magnetic Schrödinger operators, Commun.Math.Phys. 192:671–685 (1998).
J. Callaway, Quantum Theory of the Solid State, 2nd ed. (Academic Press, New York, 1991).
Ph. de Smedt and V.A. Zagrebnov, Van de Waals limit of an interacting Bose gas in a weak external field, Phys.Rev.A 35:4763–4769 (1987).
M. Dimassi, J.C. Guillot,and J. Ralston, Semiclassical asymptotics in magnetic Bloch bands, J.Phys.A:Math.Gen 35:7597–7605 (2002).
S.I. Doi, A. Iwatsuka,and T. Mine, The uniqueness of the integrated density of states for the Schr¨odinger operators with magnetic field, Math.Z. 237:335–371 (2001).
F. Germinet and A. Klein, Operator kernel estimates for functions of generalized Schrödinger operators, Proc.Am.Math.Soc. 131:911–920 (2002).
B. Helffer and J. Sjöstrand, Equation de Schr¨odinger avec champ magn ´etique et ´equation de Harper, Lect.Notes in Phys. 345:118–197 (1989).
K. Huang, Statistical Mechanics, (Kluwer Academic Publ., Dordrecht-London, 1963).
R. Joynt and R. Prange, Conditions for the quantum Hall effect, Phys.Rev.B 29:3303–3320 (1984).
W. Kirsch and B. Simon, Comparison theorems for the gap of Schr¨odinger operators, J.Funct.Anal. 75:396–410 (1987).
P. Kuchment, Floquet teory for partial differential equations (Birkh¨auser, Basel, 1993).
R.M. May, Magnetic properties of charged ideal quantum gases in n dimensions, J.Math.Phys. 6:1462–1468 (1965).
Vl.V. Papoyan and V.A. Zagrebnov, The ensemble equivalence problem for Bose systems (non-ideal Bose gas), Theor.Math.Phys. 69:1240–1253 (1986).
L.A. Pastur and A.L. Figotin, Spectra of Random and Almost Periodic Operators (Springer Verlag, Berlin-Heidelberg, 1992).
M. Reed and B. Simon, Method of Modern Analysis II:Fourier Analysis,Self-Adjointness (Academic Press, New York, 1975).
M. Reed and B. Simon, Method of Modern Analysis IV:Analysis of Operators (Academic Press, New York, 1978).
D. Ruelle, Statistical Mechanics.Rigorous Results (W.A.Benjamin, New York, 1969).
M.R. Schafroth, Superconductivity of a charged ideal bose gas, Phys.Rev. 100:463–475 (1955).
B. Simon,Schr¨odinger semigroups, Proc.Am.Math.Soc. 7:447–526 (1982).
B. Simon, Functional Integration and Quantum Physics (Academic Press, New York, 1979).
A.V. Sobolev, Absolute continuity of the periodic magnetic Schr¨odinger operator, Invent. Math. 137:85–112 (1999).
M. van den Berg, J.T. Lewis,and Ph. de Smedt, Condensation in the imperfect boson gas, J.Stat.Phys. 37:697–707 (1984).
V.A. Zagrebnov and J.-B. Bru, The Bogoliubov model of weakly imperfect Bose gas, Phys.Rep. 350:291–434 (2001).
J. Zak, Dynamics of electrons in solids in external fields, Phys.Rev. 138 686–695 (1968).
R.M. Ziff, G.E. Uhlenbeck,and M. Kac, The ideal Bose-Einstein gas,revisited, Phys. Rep. 32C:169–248 (1977).
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Briet, P., Cornean, H.D. & Zagrebnov, V.A. Do Bosons Condense in a Homogeneous Magnetic Field?. Journal of Statistical Physics 116, 1545–1578 (2004). https://doi.org/10.1023/B:JOSS.0000041748.02351.07
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DOI: https://doi.org/10.1023/B:JOSS.0000041748.02351.07