Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Numerical analysis of spectral properties of coupled oscillator Schrödinger operators. I. Single and double well anharmonic oscillators


Authors: D. Isaacson, E. L. Isaacson, D. Marchesin and P. J. Paes-Leme
Journal: Math. Comp. 37 (1981), 273-292
MSC: Primary 65N25; Secondary 81C05
DOI: https://doi.org/10.1090/S0025-5718-1981-0628695-4
MathSciNet review: 628695
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We describe several methods for computing many eigenvalues and eigenfunctions of a single anharmonic oscillator Schrödinger operator whose potential may have one or two minima. One of the methods requires the solution of an ill-conditioned generalized eigenvalue problem. This method has the virtue of using a bounded amount of work to achieve a given accuracy in both the single and double well regions. We give rigorous bounds, and we prove that the approximations converge faster than any inverse power of the size of the matrices needed to compute them.

We present the results of our computations for the $ g:{\phi ^4}{:_1}$ theory. These results indicate that the methods actually converge exponentially fast. We conjecture why this is so.


References [Enhancements On Off] (What's this?)

  • [1] D. Isaacson, D. Marchesin & P. J. Paes-Leme, Second Internat. Conf. on Computational Methods in Nonlinear Mechanics, T.I.C.O.M., 1979, p. 185; Internat. J. Engrg. Sci., v. 18, 1980, pp. 341-349. MR 661275
  • [2] N. W. Bazley & D. W. Fox, "Lower bounds for eigenvalues of Schrödinger's equation," Phys. Rev. (2), v. 124, 1961, pp. 483-492. MR 0142898 (26:465)
  • [3] J. L. Richardson & R. Blankenbecler, "Moment recursions and the Schrödinger problem," Phys. Rev. D, v. 19 (3), 1979, pp. 496-502. MR 518729 (80d:81015)
  • [4] D. Isaacson, "Singular perturbations and asymptotic eigenvalue degeneracy," Comm. Pure Appl. Math., v. 29, 1976, pp. 531-551. MR 0422792 (54:10778)
  • [5] D. Marchesin, "The scaling limit of the $ {\varphi ^2}$ field in the anharmonic oscillator," J. Math. Phys., v. 20, 1979, pp. 830-836. MR 531285 (80k:81115)
  • [6] D. Isaacson, "The critical behavior of $ \phi _1^4$," Comm. Math. Phys., v. 53, 1977, pp. 257-275. MR 0438956 (55:11858)
  • [7] A. H. Stroud & D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, N. J., 1966. MR 0202312 (34:2185)
  • [8] B. T. Smith et al., Matrix Eigensystem Routines, Eispack Guide, Springer-Verlag, Berlin and New York, 1974.
  • [9] G. Temple, "The theory of Rayleigh's principle as applied to continuous systems," Proc. Roy. Soc. Ser. A, v. 119, 1928, pp. 276-293.
  • [10] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin and New York, 1972.
  • [11] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. MR 0184422 (32:1894)
  • [12] J. D. P. Donnelly, "Bounds for the eigenvalues of self-adjoint operators," SIAM J. Numer. Anal., v. 7, 1970, pp. 458-478. MR 0290545 (44:7725)
  • [13] P. Dirac, Quantum Mechanics, Clarendon Press, Oxford, 1935.
  • [14] E. M. Harrell, "On the rate of asymptotic eigenvalue degeneracy," Comm. Math. Phys., v. 60, 1978, pp. 73-95. MR 0486764 (58:6464)
  • [15] S. G. Mikhlin, The Numerical Performance of Variational Methods, Wolters-Noordhoff, Groningen, 1971. MR 0278506 (43:4236)
  • [16] W. Ritz, "Über eine neue Methode zur Lösunggewisser Variationsprobleme der Mathematischen Physik," J. Reine Angew. Math., v. 135, 1908, pp. 1-61; "Theorie der Transversalschwingungen einer quadratischen Platte mit freien Rändern," Ann. Physik, v. 28, 1909, pp. 737-786.
  • [17] S. G. Mikhlin, Variationsmethoden der Mathematischen Physik, Akademie-Verlag, Berlin, 1962. MR 0141248 (25:4658)
  • [18] M. A. Krasnosel'skiĭ, G. M. Vainikko et al., Approximate Solution of Operator Equations, Wolters-Noordhoff, Groningen, 1972. MR 0385655 (52:6515)
  • [19] G. Strang & G. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, N. J., 1973. MR 0443377 (56:1747)
  • [20] E. C. Titchmarsh, Eigenfunction Expansions Associated With Second-Order Differential Equations, Vols. 1 and 2, Clarendon Press, Oxford, 1958. MR 0094551 (20:1065)
  • [21] A. Jaffe, Dynamics of a Cutoff $ \lambda {\phi ^4}$ Field Theory, Princeton thesis, 1965.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N25, 81C05

Retrieve articles in all journals with MSC: 65N25, 81C05


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1981-0628695-4
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society