Book contents
- Frontmatter
- Contents
- Preface
- 1 Propagator and Resolvent
- 2 The WKB Method and Nonperturbative Effects
- 3 The Phase Space Formulation of Quantum Mechanics
- 4 The Path Integral Formulation of Quantum Mechanics
- 5 Metastable States
- Appendix A Asymptotic Series and Borel Resummation
- Appendix B Special Functions
- Appendix C Gaussian Integration
- References
- Index
- References
References
Published online by Cambridge University Press: 02 December 2021
Book contents
- Frontmatter
- Contents
- Preface
- 1 Propagator and Resolvent
- 2 The WKB Method and Nonperturbative Effects
- 3 The Phase Space Formulation of Quantum Mechanics
- 4 The Path Integral Formulation of Quantum Mechanics
- 5 Metastable States
- Appendix A Asymptotic Series and Borel Resummation
- Appendix B Special Functions
- Appendix C Gaussian Integration
- References
- Index
- References
Summary
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
- Type
- Chapter
- Information
- Advanced Topics in Quantum Mechanics , pp. 254 - 260Publisher: Cambridge University PressPrint publication year: 2021
References
Álvarez, G. 1988. Coupling-constant behavior of the resonances of the cubic anharmonic oscillator. Phys. Rev., A37(11), 4079.Google Scholar
Álvarez, G. 1995. Bender-Wu branch points in the cubic oscillator. J. Phys., A28(16), 4589.Google Scholar
Álvarez, G. 2004. Langer–Cherry derivation of the multi-instanton expansion for the symmetric double well. J. Math. Phys., 45(8), 3095–3108.Google Scholar
Álvarez, G., and Casares, C. 2000a. Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator. J. Phys., A33(29), 5171.Google Scholar
Álvarez, G., and Casares, C. 2000b. Uniform asymptotic and JWKB expansions for anharmonic oscillators. J. Phys., A33(13), 2499.Google Scholar
Álvarez, G., Howls, C. J., and Silverstone, H. J. 2002. Anharmonic oscillator discontinuity formulae up to second-exponentially-small order. J. Phys., A35(18), 4003.Google Scholar
Andreassen, A., Farhi, D., Frost, W., and Schwartz, M. D. 2017. Precision decay rate calculations in quantum field theory. Phys. Rev., D95(8), 085011.Google Scholar
Baaquie, B. E. 2014. Path integrals and Hamiltonians. Principles and methods. Cambridge University Press.CrossRefGoogle Scholar
Babelon, O., Bernard, D., and Talon, M. 2003. Introduction to classical integrable systems. Cambridge Monographs on Mathematical Physics. Cambridge University Press.Google Scholar
Balian, R., Parisi, G., and Voros, A. 1978. Discrepancies from asymptotic series and their relation to complex classical trajectories. Phys. Rev. Lett., 41, 1141–1144.Google Scholar
Balian, R., Parisi, G., and Voros, A. 1979. Quartic oscillator. Pages 337–360 of: Feynman Path Integrals, vol. 106. Springer-Verlag.Google Scholar
Bartlett, M. S., and Moyal, J. E. 1949. The exact transition probabilities of quantum-mechanical oscillators calculated by the phase-space method. Math. Proc. Camb. Philos. Soc., 45, 545–553.Google Scholar
Başar, G., and Dunne, G. V. 2015. Resurgence and the Nekrasov–Shatashvili limit: Connecting weak and strong coupling in the Mathieu and Lamé systems. J. High Energy Phys., 02, 160.Google Scholar
Başar, G., Dunne, G. V., and Ünsal, M. 2017. Quantum geometry of resurgent perturbative/non-perturbative relations. J. High Energy Phys., 05, 087.Google Scholar
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D. 1978. Deformation theory and quantization. 2. Physical applications. Annals Phys., 111, 111.Google Scholar
Bender, C. M., and Orszag, S. A. 2013. Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer.Google Scholar
Bender, C. M., and Turbiner, A. 1993. Analytic continuation of eigenvalue problems. Phys. Lett., A173, 442–446.Google Scholar
Bender, C. M., and Wu, T. T. 1973. Anharmonic oscillator. 2: A study of perturbation theory in large order. Phys. Rev., D7, 1620–1636.Google Scholar
Bender, C. M., Olaussen, K., and Wang, P. S. 1977. Numerological analysis of the WKB approximation in large order. Phys. Rev., D16, 1740–1748.Google Scholar
Berezin, F. A., and Shubin, M. 2012. The Schrödinger equation. Vol. 66. Springer Science & Business Media.Google Scholar
Berry, M. V. 1977. Semi-classical mechanics in phase space: A study of Wigner’s function. Phil. Trans. Roy. Soc. Lond., A287, 237–271.Google Scholar
Berry, M. V. 1983. Semiclassical mechanics of regular and irregular motion. Les Houches lecture series, 36, 171–271.Google Scholar
Berry, M. V., and Mount, K. E. 1972. Semiclassical approximations in wave mechanics. Rep. Prog. Phys., 35(1), 315.Google Scholar
Brezin, E., Parisi, G., and Zinn-Justin, J. 1977. Perturbation theory at large orders for potential with degenerate minima. Phys. Rev., D16, 408–412.Google Scholar
Brown, L. M., ed. 2005. Feynman’s thesis: a new approach to quantum theory. World Scientific.Google Scholar
Caliceti, E., Graffi, S., and Maioli, M. 1980. Perturbation theory of odd anharmonic oscillators. Commun. Math. Phys., 75(1), 51–66.Google Scholar
Çevik, D., Gadella, M., Kuru, Ş., and Negro, J. 2016. Resonances and antibound states for the Pöschl–Teller potential: Ladder operators and SUSY partners. Phys. Lett., A380, 1600–1609.Google Scholar
Cherry, T. M. 1950. Uniform asymptotic formulae for functions with transition points. Trans. Am. Math. Soc., 68(2), 224–257.Google Scholar
Codesido, S., and Mariño, M. 2018. Holomorphic anomaly and quantum mechanics. J. Phys., A51(5), 055402.Google Scholar
Cohen-Tannoudji, C., Dupont-Roc, J., and Grynberg, G. 2012. Processus d’interaction entre photons et atomes. EDP Sciences.Google Scholar
Colin de Verdière, Y. 2005. Bohr-Sommerfeld rules to all orders. Ann. Henri Poincaré, 6, 925–936.CrossRefGoogle Scholar
Collins, J. C., and Soper, D. E. 1978. Large order expansion in perturbation theory. Ann. Phys., 112, 209–234.Google Scholar
Curtright, T., Fairlie, D., and Zachos, C. K. 1998. Features of time independent Wigner functions. Phys. Rev., D58, 025002.Google Scholar
Curtright, T. L., Fairlie, D. B., and Zachos, C. K. 2014. A concise treatise on quantum mechanics in phase space. World Scientific and Imperial College Press.Google Scholar
Curtright, T., Uematsu, T., and Zachos, C. K. 2001. Generating all Wigner functions. J. Math. Phys., 42, 2396.CrossRefGoogle Scholar
Curtright, T., and Zachos, C. K. 2001. Negative probability and uncertainty relations. Mod. Phys. Lett., A16, 2381–2385.Google Scholar
de Almeida, A. M. O. 1984. Semiclassical matrix elements. Revista Brasileira de Física, 14(1), 62–85.Google Scholar
de Almeida, A. M. O. 1990. Hamiltonian systems: Chaos and quantization. Cambridge University Press.Google Scholar
de Almeida, A. M. O, and Hannay, J. H. 1982. Geometry of two dimensional tori in phase space: Projections, sections and the Wigner function. Ann. Phys., 138(1), 115–154.Google Scholar
de la Madrid, R., and Gadella, M. 2002. A pedestrian introduction to Gamow vectors. Am.J.Phys., 70(6), 626–638.Google Scholar
Delabaere, E. 2006. Effective resummation methods for an implicit resurgent function. arXiv:math-ph/0602026Google Scholar
Delabaere, E, Dillinger, H., and Pham, F. 1997. Exact semiclassical expansions for one-dimensional quantum oscillators. J. Math. Phys., 38(12), 6126–6184.Google Scholar
Delabaere, E., and Pham, F. 1999. Resurgent methods in semi-classical asymptotics. Ann. Inst. Henri Poincaré, 71, 1–94.Google Scholar
Dillinger, H., Delabaere, E., and Pham, F. 1993. Résurgence de Voros et périodes des courbes hyperelliptiques. Ann. Inst. Fourier, 43, 163.CrossRefGoogle Scholar
Dirac, P. A. M. 1933. The Lagrangian in quantum mechanics. Physikalische Zeitschirift der Sowjetunion, 3, 312–320.Google Scholar
Dorey, P., Dunning, C., and Tateo, R. 2007. The ODE/IM correspondence. J. Phys., A40, R205.Google Scholar
Dorey, P., and Tateo, R. 1999. Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations. J. Phys., A32, L419–L425.Google Scholar
Dunham, J. L. 1932. The Wentzel–Brillouin–Kramers method of solving the wave equation. Phys. Rev., 41, 713–720.CrossRefGoogle Scholar
Dunne, G. V. 2008. Functional determinants in quantum field theory. J. Phys., A41, 304006.Google Scholar
Dunne, G. V., and Ünsal, M. 2014. Uniform WKB, multi-instantons, and resurgent trans-series. Phys. Rev., D89(10), 105009.Google Scholar
Faddeev, L. D., and Takhtajan, L. A. 2015. On the spectral theory of a functional-difference operator in conformal field theory. Izvestiya: Mathematics, 79(2), 388–410Google Scholar
Fairlie, D. B. 1964. The formulation of quantum mechanics in terms of phase space functions. Proc. Cambridge Phil. Soc., 60, 581–586.Google Scholar
Feynman, R. P. 1948. Space-time approach to nonrelativistic quantum mechanics. Rev. Mod. Phys., 20, 367–387.Google Scholar
Feynman, R. P., Hibbs, A. R., and Styer, D. F. 2010. Quantum mechanics and path integrals. Courier Corporation.Google Scholar
Gaiotto, D., Moore, G. W., and Neitzke, A. 2009. Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403.Google Scholar
Gaiotto, D., Moore, G. W., and Neitzke, A. 2010. Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys., 299, 163–224.CrossRefGoogle Scholar
Galitski, V., Karnakov, B., and Kogan, V. 2013. Exploring quantum mechanics: A collection of 700+ solved problems for students, lecturers, and researchers. Oxford University Press.Google Scholar
Gaudin, M., and Pasquier, V. 1992. The periodic Toda chain and a matrix generalization of the Bessel function’s recursion relations. J. Phys., A25, 5243.Google Scholar
Giannopoulou, K. S., and Makrakis, G. N. 2017. An approximate series solution of the semiclassical Wigner equation. arXiv:1705.06754 [math-ph]Google Scholar
Grammaticos, B., and Voros, A. 1979. Semiclassical approximations for nuclear Hamiltonians. 1. Spin independent potentials. Ann. Phys., 123, 359.Google Scholar
Grassi, A., Mariño, M., and Zakany, S. 2015. Resumming the string perturbation series. J. High Energy Phys., 1505, 038.Google Scholar
Gross, D. J., Perry, M. J., and Yaffe, L. G. 1982. Instability of flat space at finite temperature. Phys. Rev., D25, 330–355.Google Scholar
Gutzwiller, M. C. 1981. The quantum mechanical Toda lattice, II. Ann. Phys., 133, 304.Google Scholar
Hillery, M., O’Connell, R. F., Scully, M. O., and Wigner, E. P. 1984. Distribution functions in physics: Fundamentals. Phys. Rept., 106, 121–167.Google Scholar
Hislop, P. D., and Sigal, I. M. 2012. Introduction to spectral theory: With applications to Schrödinger operators. Springer-Verlag.Google Scholar
Hoe, N., D’Etat, B., Grumberg, J., Caby, M., Leboucher, E., and Coulaud, G. 1982. Stark effect of hydrogenic ions. Phys. Rev., A25(Feb), 891–906.Google Scholar
Hudson, R. L. 1974. When is the Wigner quasi-probability density non-negative? Rep. Math. Phys., 6(2), 249–252.CrossRefGoogle Scholar
Ito, K., Mariño, M., and Shu, H. 2019. TBA equations and resurgent quantum mechanics. J. High Energy Phys., 01, 228.Google Scholar
Jentschura, U. D., Surzhykov, A., and Zinn-Justin, J. 2010. Multi-instantons and exact results. III: Unification of even and odd anharmonic oscillators. Ann. Phys., 325, 1135–1172.Google Scholar
Kashaev, R., and Mariño, M. 2016. Operators from mirror curves and the quantum dilogarithm. Commun. Math. Phys., 346(3), 967.Google Scholar
Kawai, T., and Takei, Y. 2005. Algebraic analysis of singular perturbation theory. American Mathematical Society.Google Scholar
Kenfack, A., and Życzkowski, K. 2004. Negativity of the Wigner function as an indicator of non-classicality. J. Opt. B: Quantum Semiclass. Opt., 6(10), 396–404.Google Scholar
Khandekar, D. C., and Lawande, S. V. 1986. Feynman path integrals: Some exact results and applications. Phys. Rep., 137, 115–229.Google Scholar
Kleinert, H. 2009. Path integrals in quantum mechanics, statistics, polymer physics, and financial markets. World Scientific.Google Scholar
Konishi, K., and Paffuti, G. 2009. Quantum mechanics: A new introduction. Oxford University Press.Google Scholar
Kozlowski, K. K., and Teschner, J. 2010. TBA for the Toda chain. New Trends in Quantum Integrable Systems, Word Scientific, 195–210.Google Scholar
Kurchan, J., Leboeuf, P., and Saraceno, M. 1989. Semiclassical approximations in the coherent-state representation. Phys. Rev., A40(Dec), 6800–6813.Google Scholar
Lam, C. S. 1968. Behavior of very high order perturbation diagrams. Nuovo Cim., A55, 258–274.Google Scholar
Langer, R. E. 1949. The asymptotic solutions of ordinary linear differential equations of the second order, with special reference to a turning point. Trans. Am. Math. Soc., 67(2), 461–490.Google Scholar
Lebedev, N. N. 1972. Special functions and their applications. Revised ed., Dover PublicationsGoogle Scholar
Lee, H. W. 1995. Theory and application of the quantum phase-space distribution functions. Phys. Rep., 259(3), 147–211.Google Scholar
Mariño, M. 2015. Instantons and large N: An introduction to non-perturbative methods in quantum field theory. Cambridge University Press.Google Scholar
Miller, P. D. 2006. Applied asymptotic analysis. Vol. 75. American Mathematical Society.Google Scholar
Miller, S. C., Jr., and Good, R. H., Jr. 1953. A WKB-type approximation to the Schrödinger equation. Phys. Rev., 91(1), 174.Google Scholar
Moshinsky, M., and Quesne, C. 1971. Linear canonical transformations and their unitary representations. J. Math. Phys., 12(8), 1772–1780.Google Scholar
Moyal, A. 2006. Maverick mathematician. The life and science of J. E. Moyal. Australian National University Press.Google Scholar
Nekrasov, N. A., and Shatashvili, S. L. 2009. Quantization of integrable systems and four dimensional gauge theories. In the 16th International Congress on Mathematical Physics, Prague, August 2009, 265–289, World Scientific 2010.Google Scholar
Nikolaev, N. 2020. Exact solutions for the singularly perturbed Riccati Equation and exact WKB Analysis. arXiv:2008.06492 [math.CA]Google Scholar
Reinhardt, W. P. 1982. Complex coordinates in the theory of atomic and molecular structure and dynamics. Annu. Rev. Phys. Chem., 33(1), 223–255.Google Scholar
Robnik, M., and Romanovski, V. G. 2000. Some properties of the WKB series. J. Phys., A33(28), 5093.Google Scholar
Robnik, M., and Salasnich, L. 1997. WKB to all orders and the accuracy of the semiclassical quantization. J. Phys., A30(5), 1711.Google Scholar
Schulman, L. S. 2012. Techniques and applications of path integration. Courier Corporation.Google Scholar
Serone, M., Spada, G., and Villadoro, G. 2017. The power of perturbation theory. J. High Energy Phys., 05, 056.Google Scholar
Shen, H., and Silverstone, H. J. 2004. JWKB method as an exact technique. Int. J. Quantum Chem., 99(4), 336–352.Google Scholar
Shen, H., Silverstone, H. J, and Álvarez, G. 2005. On the bidirectionality of the JWKB connection formula at a linear turning point. Collect. Czech. Chem. Commun., 70(6), 740–754.Google Scholar
Silverstone, H. J. 1985. JWKB connection-formula problem revisited via Borel summation. Phys. Rev. Lett., 55(23), 2523.Google Scholar
Silverstone, H. J., Harris, J. G., Čížek, J., and Paldus, J. 1985a. Asymptotics of high-order perturbation theory for the one-dimensional anharmonic oscillator by quasisemiclassical methods. Phys. Rev., A32(4), 1965.Google Scholar
Silverstone, H. J., Nakai, S., and Harris, J. G. 1985b. Observations on the summability of confluent hypergeometric functions and on semiclassical quantum mechanics. Phys. Rev., A32, 1341–1345.Google Scholar
Simon, B., and Dicke, A. 1970. Coupling constant analyticity for the anharmonic oscillator. Ann. Phys., 58(1), 76 – 136.Google Scholar
Sulejmanpasic, T., and Ünsal, M. 2018. Aspects of perturbation theory in quantum mechanics: The BenderWu Mathematica R○ package. Comput. Phys. Commun., 228, 273–289.Google Scholar
Takhtajan, L. A. 2008. Quantum mechanics for mathematicians. American Mathematical Society.Google Scholar
Tatarskii, V. I. 1983. The Wigner representation of quantum mechanics. Physics-Uspekhi, 26(4), 311–327.Google Scholar
Temme, N. M. 1990. Asymptotic estimates for Laguerre polynomials. Z. Angew. Math. Phys., 41(1), 114–126.Google Scholar
Van Vleck, J. H. 1928. The correspondence principle in the statistical interpretation of quantum mechanics. Proc. Natl. Acad. Sci. U.S.A., 14(2), 178–188.Google Scholar
Veble, G., Robnik, M., and Romanovski, V. 2002. Semiclassical analysis of Wigner functions. J. Phys., A35(18), 4151.Google Scholar
Voros, A. 1977. Asymptotic -expansions of stationary quantum states, Ann. Inst. H. Poincaré, A26, 343–403.Google Scholar
Voros, A. 1980. The zeta function of the quartic oscillator. Nucl. Phys., B165, 209–236.Google Scholar
Voros, A. 1981. Spectre de l’équation de Schrödinger et méthode BKW. Publications Mathématiques d’Orsay.Google Scholar
Voros, A. 1983. The return of the quartic oscillator. The complex WKB method. Ann. Inst. Henri Poincaré, 39(3), 211–338.Google Scholar
Voros, A. 1989. Wentzel-Kramers-Brillouin method in the Bargmann representation. Phys. Rev., A40(Dec), 6814–6825.Google Scholar
Wigner, E. P. 1932. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40, 749–760.Google Scholar
Yaris, R., Bendler, J., Lovett, R. A., Bender, C. M., and Fedders, P. A. 1978. Resonance calculations for arbitrary potentials. Phys. Rev., A18, 1816.Google Scholar
Zachos, C. K., Fairlie, D. B., and Curtright, T. L. 2005. Quantum mechanics in phase space: an overview with selected papers. World Scientific.Google Scholar
Zinn-Justin, J. 1981. Multi-instanton contributions in quantum mechanics. Nucl. Phys., B192, 125–140.Google Scholar
Zinn-Justin, J. 1983. Multi-instanton contributions in quantum mechanics. 2. Nucl. Phys., B218, 333–348.Google Scholar
Zinn-Justin, J. 2012. Intégrale de chemin en mécanique quantique: Introduction. EDP sciences.Google Scholar
Zinn-Justin, J., and Jentschura, U. D. 2004a. Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions. Ann. Phys., 313, 197–267.Google Scholar
Zinn-Justin, J., and Jentschura, U. D. 2004b. Multi-instantons and exact results II: Specific cases, higher-order effects, and numerical calculations. Ann. Phys., 313, 269–325.Google Scholar