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3
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6244259501
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for original path integral approach to Van Vleck's result. A rigorous asymptotic result with phase index, for global noncaustic times, however, appears originally in the work by (c)
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(1962)
USSR Comp. Math. Math. Phys.
, vol.3
, pp. 744
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Maslov, V.P.1
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4
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84926606175
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see also (d) V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation in Quantum Mechanics (Reidel, Boston, MA, 1981).
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9
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33744584849
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the review by (d) M. V. Berry and K. E. Mount
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See also the review by (d) M. V. Berry and K. E. Mount, Rep. Prog. Phys. 35, 315(1972).
-
(1972)
Rep. Prog. Phys.
, vol.35
, pp. 315
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31
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0000955101
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Asymptotic evaluation of multidimensional integrals for the S matrix in the semiclassical theory of inelastic and reactive molecular collisions
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(1973)
Molecular Physics
, vol.25
, pp. 181
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-
Connor, J.N.L.1
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35
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84926607223
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See also Refs. citecatas,ref9.
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-
-
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36
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84926604773
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Throughout we use the obvious notation wherein the coordinates as functions of initial conditions q(t0)=q0,p(t0)=p0 and time t-t0 are denoted explicitly by q(q0,p0;t-t0) and likewise for the momenta.
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37
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84926562745
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(a) H. Goldstein, Classical Mechanics, 2nd ed. (Addison Wesley, Reading, 1980); (b) V. I. Arnold, Mathematical Methods in Classical Mechanics (North Holland, New York, 1980); (c) E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. (Cambridge University, Cambridge, 1937).
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38
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84926584253
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M. Morse, Variational Analysis (Wiley, New York, 1973); J. W. Milnor, Morse Theory, Annals of Mathematical Studies Vol. 51 (Princeton University Press, Princeton, N. J., 1963).
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-
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40
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84926536433
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This does not imply, however, that wave functions propagated via Eq. ( refeq:eq1) are not accurate in the asymptotic limit hbar → 0 for fixed t. See theorem 1.3 of Ref. [1(c)].
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-
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41
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84926570250
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A uniform semiclassical treatment to the coordinate path integral is briefly discussed in Ref. citeSchulman. See also Ref. citeLevit for applications of canonical maps to the phase space path integral.
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-
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42
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84980174402
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Oscillatory integrals, lagrange immersions and unfolding of singularities
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In catastrophe theory the mapping Eq. ( refeq:eq11) defines an equivalence of Z and varphi as unfoldings of functions of x depending on parameters boldmathβ. The function Z is called the canonical exponent or unfolding of the singularity. See, for example
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(1974)
Communications on Pure and Applied Mathematics
, vol.27
, pp. 207
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Duistermaat, J.J.1
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44
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84926608719
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The stationary phase method is the simplest example of such a mapping with only one parameter β0= varphi(x(k)) and Z is simply a nondegenerate quadratic function about z=0 given by Morse's Lemma citeMorse for each separate domain of an x(k) (e.g., like Taylor expansion of varphi to second order in x - x(k)).
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-
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45
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84926592493
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For a clear explanation on the concepts of asymptotic equivalence, asymptotic sequences, and expansions see, for example, N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986); Erdélyi, Asymptotic Expansions (Dover, New York, 1956).
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47
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84926534669
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This result, particularly its momentum representation counterpart, is the propagator generalization of the familiar rainbow scattering amplitude formula; see, for example, Ref. [3(d)], and references therein.
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-
-
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48
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84926550932
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The factor (-2π i hbar)-N/2 arises simply from the choice of normalization =(2π i hbar)-N/2 eiq cdotp/hbar.
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