Skip to main content
SpringerLink
Log in

Functional Relations in Stokes Multipliers—Fun with x6+αx2 Potential

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider eigenvalue problems in quantum mechanics in one dimension. Hamiltonians contain a class of double well potential terms, x\(^6\)+αx\(^2\), for example. The space coordinate is continued to a complex plane and the connection problem of fundamental system of solutions is considered. A hidden U\(_q\)(\(\widehat{gl}\)(2 ∣ 1)) structure arises in “fusion relations” of Stokes multipliers. With this observation, we derive coupled nonlinear integral equations which characterize the spectral properties of both ±α potentials simultaneously.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. Voros, The return of the quartic oscillator, Ann. Inst. H. Poincare A 39:211–338 (1983).

    Google Scholar 

  2. A. Voros, Spectral Zeta functions, Adv. Stud. Pure. Math. 21:327–358 (1992).

    Google Scholar 

  3. A. Voros, Exact quantization condition for anharmonic oscillators (in one dimension), J. Phys. A 27:4653–4661 (1994).

    Google Scholar 

  4. A. Voros, Exact anharmonic quantization condition (in one dimension), in Quasiclassical Method (IMA Proceedings, Minneapolis, 1995), J. Rauch and B. Simon, eds., IMA Series 95 (Springer, 1997), pp. 189–224.

  5. A. Voros, Airy function (exact WKB result for potentials of odd degree), J. Phys. A 32:1301–1311 (1999).

    Google Scholar 

  6. A. Voros, Exact resolution method for general 1D polynomial Schrödinger equation, J. Phys. A 32:5993–6007 (1999).

    Google Scholar 

  7. A. Voros, Exact quantization method for the polynomial 1D Schrödinger equation, in Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear, T. Kawai et al., eds. (Proceedings, Kyoto, 1998), to be published by Kyoto University Press.

  8. T. Kawai and Y. Takei, Algebraic Analysis on Singular Perturbations (Iwanami, 1999) (in Japanese).

  9. P. Dorey and R. Tateo, Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations, J. Phys. A 32:L419–L426 (1999) (hep-th/9812211).

    Google Scholar 

  10. P. Dorey and R. Tateo, On the relation between Stokes multipliers and the T-Q systems of conformal field theory, Nucl. Phys. B 563:573–602 (1999) (hep-th/9906219).

    Google Scholar 

  11. V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Spectral Determinants for Schrödinger Equation and Q-Operators of Conformal Field Theory, hep-th/9812247.

  12. J. Suzuki, Anharmonic oscillators, spectral determinant and short exact sequence of \(U_q \left( {\widehat{\mathfrak{s}}{\mathfrak{l}_2 }} \right)\), J. Phys. A 32:L183–L188 (1999) (hep-th/9902053).

    Google Scholar 

  13. Y. Sibuya, Global Theory of Second Order Linear Ordinary Differential Operator with a Polynomial Coefficient, Mathematics Studies 18 (North-Holland, 1975). See Chapter 5, especially Eq. (27.6).

  14. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press).

  15. V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Comm. Math. Phys. 177:381–398 (1997).

    Google Scholar 

  16. P. Dorey and R. Tateo, Differential equations and integrable models: SU(3) case, Nucl. Phys. B 571:583–606 (2000) (hep-th/9910102).

    Google Scholar 

  17. J. Suzuki, Functional relations in Stokes multipliers and solvable models related to \(U_q \left( {A_n^{\left( 1 \right)} } \right)\), J. Phys. A 33:3507–3521 (2000) (hep-th/9910215).

    Google Scholar 

  18. M. V. Fedoryuk, Asymptotic Analysis (Springer, 1993).

  19. A. Klümper and P. A. Pearce, Conformal weights of RSOS lattice models and their fusion hierarchies, Physica A 183:304–350 (1992).

    Google Scholar 

  20. A. Klümper, M. T. Batchelor, and P. A. Pearce, Central charges of the 6–and 19–vertex models with twisted boundary conditions, J. Phys. A 24:3111–3133 (1991).

    Google Scholar 

  21. C. Destri and H. J. de Vega, New thermodynamic Bethe ansatz equations without strings, Phys. Rev. Lett. 69:2313–2317 (1992).

    Google Scholar 

  22. A. Klümper, Free energy and correlation lengths of quantum chains related to restricted solid-on-solid lattice models, Ann. Physik 1:540 (1992).

    Google Scholar 

  23. V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Integrable structure of conformal field theory II, Q-operator and DDV equation, Comm. Math. Phys. 190:247 (1997).

    Google Scholar 

  24. V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Integrable quantum field theories in finite volume: excited state energies, Nucl. Phys. B 489:487–532 (1997).

    Google Scholar 

  25. G. Jüttner, A. Klümper, and J. Suzuki, Exact thermodynamics and Luttinger liquid properties of the integrable t - J model, Nucl. Phys. B 486:650–674 (1997).

    Google Scholar 

  26. G. Jüttner, A. Klümper, and J. Suzuki, The Hubbard chain at finite temperatures: ab-initio calculations of Tomonaga-Luttinger liquid properties, Nucl. Phys. B 522:471–502 (1998).

    Google Scholar 

  27. J. Suzuki, Spinons in magnetic chains of arbitrary spins at finite temperatures, J. Phys. A 32:2341–2359 (1999).

    Google Scholar 

  28. J. H. H. Perk and C. Schulz, New families of commuting transfer matrices in q-state vertex models, Phys. Lett. A 84:407–410 (1981).

    Google Scholar 

  29. C. L. Schulz, Solvable q-state models in lattice statistics and quantum field theory, Phys. Rev. Lett. 46:629–633 (1981).

    Google Scholar 

  30. C. L. Schulz, Eigenvectors of the multi-component generalization of the six-vertex model, Physica A 122:71–88 (1983).

    Google Scholar 

  31. P-F Hsieh and Y. Sibuya, On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients, J. Math. Analysis and Applications 16:84–103 (1966).

    Google Scholar 

  32. G. Jüttner, A. Klümper, and J. Suzuki, From fusion hierarchy to excited state TBA, Nucl. Phys. B 512:581–600 (1998).

    Google Scholar 

  33. V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, On non-equilibrium states in QFT model with boundary interaction, Nucl. Phys. B 5489:529–545 (1999).

    Google Scholar 

  34. J. Oscar Rosas-Ortiz, Exactly solvable hydrogen-like potentials and the factorization method, J. Phys. A 31:10163–10179 (1998).

    Google Scholar 

  35. F. Cannata, G. Junker, and J. Trost, Schrödinger operators with complex potential but real spectrum, Phys. Lett. A 246:219–226 (1998). See also discussion. on quasi-solvable sextic potentials, B. Bagchi, F. Cannata, C. Quesne, PT-symmetric sextic potentials, Phys. Lett. A 269:79–82 (2000) (quant-ph/0003085).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Faculty of Science, Shizuoka University, 836 Ohya, Shizuoka, 422, Japan

    J. Suzuki

Authors
  1. J. Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Suzuki, J. Functional Relations in Stokes Multipliers—Fun with x6+αx2 Potential. Journal of Statistical Physics 102, 1029–1047 (2001). https://doi.org/10.1023/A:1004823608260

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004823608260

Search

Navigation