Perturbation theory series in quantum mechanics: Phase transition and exact asymptotic forms for the expansion coefficients
Tóm tắt
We consider the model of a harmonic oscillator with a power-law potential and derive new asymptotic formulas for the coefficients of the perturbation theory series in powers of the coupling constant in the case of a power-law perturbing potential |x|p, p > 0. We prove the existence of a critical value p
0
and compute it. It is a threshold in the sense that the asymptotic forms of the studied coefficients for 0 < p < p
0
and for p > p
0
differ qualitatively. We note that the considered physical system undergoes a phase transition at p = p
0
. The analysis uses the Laplace method for functional integrals with Gaussian measures.
Tài liệu tham khảo
J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer, New York (1981).
H.-O. Georgii, Gibbs Measures and Phase Transitions (De Gruyter Stud. Math., Vol. 9), de Gruyter, Berlin (2011).
V. I. Piterbarg and V. R. Fatalov, Russ. Math. Surveys, 50, 1151–1239 (1995).
V. R. Fatalov, Theor. Math. Phys., 168, 1112–1149 (2011).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. 1, 2, and 4, Acad. Press, New York (1978).
R. P. Feynman, “Space-time approach to nonrelativistic quantum mechanics,” Rev. Modern Phys., 20, 367–387 (1948).
V. R. Fatalov, “On the Laplace method for Gaussian measures in a Banach space,” Theory Probab. Appl. (to appear).
V. R. Fatalov, Izv. Math., 75, 837–868 (2011).
V. R. Fatalov, Theory Probab. Appl., 53, 13–36 (2009).
S. Albeverio, V. Fatalov, and V. Piterbarg, J. Math. Sci. Univ. Tokyo, 16, 55–93 (2009).
V. R. Fatalov, Izv. Math., 75, 837–868 (2011).
V. R. Fatalov, “Exact asymptotics of probability distributions and functional integrals: Laplace method [in Russian],” Doctoral dissertation, Moscow State Univ., Moscow (2009).
B. Simon, Functional Integration and Quantum Physics (Pure Appl. Math., Vol. 86), Acad. Press, New York (1979).
W. Janke and A. Pelster, eds., Path Integrals: New Trends and Perspectives (Proc. 9th Intl. Conf., Dresden, 23–28 September 2007), World Scientific, Hackensack, N. J. (2008).
J. Zinn-Justin, “Functional integrals in physics: The main achievements,” in: Path Integrals: New Trends and Perspectives (Proc. 9th Intl. Conf., Dresden, 23–28 September 2007, W. Janke and A. Pelster, eds.), World Scientific, Hackensack, N. J. (2008), pp. 251–260.
I. M. Koval’chik and L. A. Yanovich, The Generalized Wiener Integral and Some of its Applications [in Russian], Nauka i Tekhnika, Minsk (1989).
R. S. Ellis and J. S. Rosen, Comm. Math. Phys., 82, 153–181 (1981).
R. S. Ellis and J. S. Rosen, Trans. Amer. Math. Soc., 273, 447–481 (1982).
C. M. Bender and T. T. Wu, Phys. Rev., 184, 1231–1260 (1969).
C. M. Bender and T. T. Wu, Phys. Rev. Lett., 27, 461–465 (1971).
C. M. Bender and T. T. Wu, Phys. Rev. D, 7, 1620–1636 (1972).
T. Banks, C. M. Bender, and T. T. Wu, Phys. Rev. D, 8, 3346–3366 (1973).
C. M. Bender and T. T. Wu, Phys. Rev. Lett., 37, 117–120 (1976).
L. N. Lipatov, Soviet Phys. JETP, 45, 216–223 (1977).
L. N. Lipatov, Lett. JETP, 25, 104–107 (1977).
E. Brézin, J. C. Le Guillou, and J. Zinn-Justin, Phys. Rev. D, 15, 1544–1557 (1977).
E. Brézin, J. C. Le Guillou, and J. Zinn-Justin, Phys. Rev. D, 15, 1558–1564 (1977).
J. C. Le Guillou and J. Zinn-Justin, eds., Large Order Behaviour of Perturbation Theory, North-Holland, Amsterdam (1990).
N. N. Bogolyubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields [in Russian], Nauka, Moscow (1984); English transl. prev. ed. (Interscience Monogr. Phys. Astronomy, Vol. 3), Interscience, New York (1959).
V. V. Belokurov and D. V. Shirkov, Theory of the Interaction of Particles [in Russian], Nauka, Moscow (1986).
V. V. Belokurov, Yu. P. Solov’ev, and E. T. Shavgulidze, “Method for constructing a quantum field theory of perturbations with convergent series [in Russian],” Preprint Sci.-Res. Inst. Nucl. Phys. 95-31/395, Moscow State Univ., Moscow (1995).
J. Zinn-Justin, Phys. Rep., 70, 109–167 (1981).
S. G. Krein, ed., Functional Analysis [in Russian], Nauka, Moscow (1972); English transl., Wolters-Noordhoff, Groningen (1972).
B. Simon, The P(φ)2 Euclidean (Quantum) Field Theory, Princeton Univ. Press, Princeton, N. J. (1974).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995).
B. Simon, Ann. Phys., 58, 76–136 (1970).
E. Lieb, Bull. Amer. Math. Soc., 82, 751–753 (1976).
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland Math. Library, Vol. 24), North-Holland, Amsterdam (1981).
T. Hida, Brownian Motion (Appl. Math., Vol. 11), Springer, New York (1980).
A. N. Borodin and P. Salminen, Handbook of Brownian motion [in Russian], Lan’, St. Petersburg (2000); English transl., Birkhäuser, Basel (2002).
H.-H. Kuo, Gaussian Measures in Banach Spaces (Lect. Notes Math., Vol. 463), Springer, Berlin (1975).
N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions on Banach Spaces [in Russian], Nauka, Moscow (1985); English transl. (Math. Appl. Soviet Ser., Vol. 14), Reidel, Dordrecht (1987).
M. A. Lifshits, Gaussian Random Functions [in Russian], TViMS, Kiev (1995); English transl. (Math. Appl., Vol. 322), Kluwer, Dordrecht (1995).
V. I. Bogachev, Gaussian Measures [in Russian], Fizmatlit, Moscow (1997); English transl. (Math. Surv. Monogr., Vol. 62), Amer. Math. Soc., Providence, R. I. (1998).
S. R. S. Varadhan, Comm. Pure Appl. Math., 19, 261–286 (1996).
S. R. S. Varadhan, Lett. Math. Phys., 88, 175–185 (2009).
E. Olivieri and M. E. Vares, Large Deviations and Metastability (Encycl. Math. Its Appl., Vol. 100), Cambridge Univ. Press, Cambridge (2005).
R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, Springer, Berlin (2006).
E. Bolthausen, Prob. Theory Relat. Fields, 72, 305–318 (1986).
I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes [in Russian], Vol. 1, Nauka, Moscow (1971); English transl., Springer, New York (1974).
N. Dunford and J. T. Schwartz, Linear Operators: Part I. General Theory, Wiley, New York (1988).
M. A. Krasnosel’skij, P. P. Zabrejko, E. I. Pustylnik, and P. E. Pustil’nik, Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966); English transl., Noordhoff International, Leiden (1976).
M. M. Vainberg, Variational Method and Method of Monotone Operators [in Russian], Nauka, Moscow (1972); English transl.: Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Wiley, New York (1973).
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1979); English transl., Consultants Bureau, New York (1987).
X. M. Fernique, “La régularité des trajectoires des fonctions aléatoires gaussiennes,” in: Ecole d’Ete de Probabilites de Saint-Flour IV — 1974 (Lect. Notes Math., Vol. 480, P.-L. Hennequin, ed.), Springer, Berlin (1975), pp. 1–96.
E. Kamke, Differentialgleichungen, Lösunsmethoden, und Lösungen, Academie, Leipzig (1959).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series: Elementary Functions [in Russian], Nauka, Moscow (1981); English transl., Gordon and Breach, New York (1986).