WKB methods for difference equations II

Dingle, R.; Morgan, G J

doi:10.1007/bf00382349

Cited by 26 publications

(26 citation statements)

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“…Despite the resemblance between equation (1.2) and (1.12), one can see, after a careful comparison, that they are not of the same type. There are two older papers by Dingle and Morgan [6,7] on WKB approximations for second-order linear difference equations, where they have briefly discussed the connection formulas for linking exponential and trigonometrical regions and the behavior near turning points. However, their argument is too sketchy and non-rigorous.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic expansions for second-order linear difference equations with a turning point

Wang

Wong

2003

Numerische Mathematik

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Section: Introductionmentioning

confidence: 99%

Asymptotic expansions for second-order linear difference equations with a turning point

Wang

Wong

2003

Numerische Mathematik

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“…As in paper I, our analysis is based on the discrete phase integral (DPI), or Wentzel-Kramers-Brillouin method [15]. This method is semiclassical in character, with 1/J playing the same role ash in the continuum phase integral method.…”

Section: Introduction a The Story So Farmentioning

confidence: 99%

Quenched spin tunneling and diabolical points in magnetic molecules. I. Symmetric configurations

Garg

2001

Phys. Rev. B

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The perfect quenching of spin tunneling first predicted for a model with biaxial symmetry, and recently observed in the magnetic molecule Fe 8 , is further studied using the discrete phase integral (or Wentzel-Kramers-Brillouin) method. The analysis of the previous paper is extended to the case where the magnetic field has both hard and easy components, so that the Hamiltonian has no obvious symmetry. Herring's formula is now inapplicable, so the problem is solved by finding the wavefunction and using connection formulas at every turning point. A general formula for the energy surface in the vicinity of the diabolo is obtained in this way. This formula gives the tunneling apmplitude between two wells unrelated by symmetry in terms of a small number of action integrals, and appears to be generally valid, even for problems where the recursion contains more than five terms. Explicit results are obtained for the diabolical points in the model for Fe 8 . These results exactly parallel the experimental observations. It is found that the leading semiclassical results for the diabolical points appear to be exact, and the points themselves lie on a perfect centered rectangular lattice in the magnetic field space. A variety of evidence in favor of this perfect lattice hypothesis is presented. 75.10Dg, 03.65.Sq, 75.50Xx, 75.45.+j

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“…A similar method (which we may call the discrete WKB method) has been applied to the solution of difference equations [5], [8], [12], [38] and it is currently being extended [11], [35], [36], [37], to include difference equations with turning points. Another type of analysis, based on perturbation techniques, was considered by C. Lange and R. Miura in [17], [18], [19], [20], [21], and [22].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of the Hermite polynomials from their differential–difference equation

Dominici

2007

Journal of Difference Equations and Applications

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WKB methods for difference equations II

Cited by 26 publications

References 1 publication

Asymptotic expansions for second-order linear difference equations with a turning point

Asymptotic expansions for second-order linear difference equations with a turning point

Quenched spin tunneling and diabolical points in magnetic molecules. I. Symmetric configurations

Asymptotic analysis of the Hermite polynomials from their differential–difference equation

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