Numerical analysis of spectral properties of coupled oscillator Schrödinger operators. I. Single and double well anharmonic oscillators
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- by D. Isaacson, E. L. Isaacson, D. Marchesin and P. J. Paes-Leme PDF
- Math. Comp. 37 (1981), 273-292 Request permission
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
- D. Isaacson, D. Marchesin, and P. J. Paes-Leme, Numerical methods for studying anharmonic oscillator approximations to the $\varphi ^{4}_{2}$ quantum field theory, Internat. J. Engrg. Sci. 18 (1980), no. 2, 341–349. Computational methods in nonlinear problems in mechanics and engineering science (Austin, Tex., 1979). MR 661275, DOI 10.1016/0020-7225(80)90055-5
- Norman W. Bazley and David W. Fox, Lower bounds for eigenvalues of Schrödinger’s equation, Phys. Rev. (2) 124 (1961), 483–492. MR 142898, DOI 10.1103/PhysRev.124.483
- John L. Richardson and Richard Blankenbecler, Moment recursions and the Schrödinger problem, Phys. Rev. D (3) 19 (1979), no. 2, 496–502. MR 518729, DOI 10.1103/PhysRevD.19.496
- David Isaacson, Singular perturbations and asymptotic eigenvalue degeneracy, Comm. Pure Appl. Math. 29 (1976), no. 5, 531–551. MR 422792, DOI 10.1002/cpa.3160290506
- Dan Marchesin, The scaling limit of the $\varphi ^{2}$ field in the anharmonic oscillator, J. Math. Phys. 20 (1979), no. 5, 830–836. MR 531285, DOI 10.1063/1.524155
- David Isaacson, The critical behavior of $\phi ^{4}_{1}$, Comm. Math. Phys. 53 (1977), no. 3, 257–275. MR 438956, DOI 10.1007/BF01609850
- A. H. Stroud and Don Secrest, Gaussian quadrature formulas, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0202312 B. T. Smith et al., Matrix Eigensystem Routines, Eispack Guide, Springer-Verlag, Berlin and New York, 1974. G. Temple, "The theory of Rayleigh’s principle as applied to continuous systems," Proc. Roy. Soc. Ser. A, v. 119, 1928, pp. 276-293. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin and New York, 1972.
- J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
- J. D. P. Donnelly, Bounds for the eigenvalues of self-adjoint operators, SIAM J. Numer. Anal. 7 (1970), 458–478. MR 290545, DOI 10.1137/0707038 P. Dirac, Quantum Mechanics, Clarendon Press, Oxford, 1935.
- Evans M. Harrell, On the rate of asymptotic eigenvalue degeneracy, Comm. Math. Phys. 60 (1978), no. 1, 73–95. MR 486764, DOI 10.1007/BF01609474
- S. G. Mikhlin, The numerical performance of variational methods, Wolters-Noordhoff Publishing, Groningen, 1971. Translated from the Russian by R. S. Anderssen. MR 0278506 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.
- Solomon G. Michlin, Variationsmethoden der mathematischen Physik, Akademie-Verlag, Berlin, 1962 (German). In deutscher Sprache herausgegeben von Eugen Heyn. MR 0141248
- M. A. Krasnosel′skiĭ, G. M. Vaĭnikko, P. P. Zabreĭko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximate solution of operator equations, Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the Russian by D. Louvish. MR 0385655, DOI 10.1007/978-94-010-2715-1
- Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
- E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Vol. 2, Oxford, at the Clarendon Press, 1958. MR 0094551, DOI 10.1063/1.3062231 A. Jaffe, Dynamics of a Cutoff $\lambda {\phi ^4}$ Field Theory, Princeton thesis, 1965.
Additional Information
- © Copyright 1981 American Mathematical Society
- 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