Initial-value phase-space integral representations for a time-dependent propagator are obtained in the coordinate and momentum representations. To do so we 6rst derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical phase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, including cases where the Morse index theorem is inapplicable. The method of obtaining the indices is general and based simply on concatenating uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asymptotic-equivalence approach. The integrals reduce to the nonuniform semiclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly account for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canonical mapping procedure on the coordinate-space path integral. Initial-value integrals for time-dependent wave-function propagation are also given. Evaluation of the initial-value integral expressions do not require trajectory root searches for propagation. PACS number(s): 03.65.Sq 1050-2947/94/50(2)/997(22)/$06. 00 50 997 1994 The American Physical Society 998 G. CAMPOLIETI AND PAUL BRUMER 50are explicit for any type of possibly occuring caustics or focal points in the many-dimensional case and are readily implementable numerically. In addition, the general phase index formulas that we obtain recover as particular cases the special index formulas obtained based on phase-space path-integration [13].Given these results we obtain, in Sec. III 8, the initialvalue integrals which provide uniform semiclassical approximations to the quantum propagator, including the all important integrand phase contributions &om each trajectory. In particular, we derive these integrals by extending the uniform short-time integral representations obtained in Sec. IIIA to global time using asymptotic
Two contributions t? sem~classical mechanics, within the initial value representation, are present~. T~e first IS the mtroduction of an efficient integration scheme, based upon number th~retlc la~tlces, which allows the effective evaluation of S matrix elements despite highly ?scdla~ory mte~rands. Applications to collinear H + H2 and three-dimensional He + H2 melast~c scatte~~g sho~ good ~greement with quantum results for both classically allowed and tunnelmg transItions With relatively few trajectories. The second contribution is the derivation of a? initial v~lue integral.repr~se?tation for reactive scattering using the idea of asymptotic eqUIvalence With th~ clas~lcal-hmlt formulas. The result is a generally useful formula, requiring only real valued trajectones, for reactive scattering above and below the classical reaction threshold.
A time-dependent initial value semiclassical propagator is developed and applied to dissociation dynamics. Numerically implementable formulas are given for computing detailed dissociation dynamics and photofragmentation matrix elements. The method is applied to the study of two- and three-dimensional HOH/HOD photodissociation in the à state. In the two-dimensional case, results obtained by a grid-based numerical integration method using relatively few classical trajectories show very good agreement with known quantum results. The three-dimensional study uses a stationary-phase Monte Carlo approach to computing dissociation cross sections. In this case a comparison with exact quantum calculations shows only qualitative agreement.
This paper develops bridge sampling path integral algorithms for pricing path-dependent options under a new class of nonlinear state dependent volatility models. Path-dependent option pricing is considered within a new (dual) Bessel family of semimartingale diffusion models, as well as the constant elasticity of variance (CEV) diffusion model, arising as a particular case of these models. The transition p.d.f.s or pricing kernels are mapped onto an underlying simpler squared Bessel process and are expressed analytically in terms of modified Bessel functions. We establish precise links between pricing kernels of such models and the randomized gamma distributions, and thereby demonstrate how a squared Bessel bridge process can be used for exact sampling of the Bessel family of paths. A Bessel bridge algorithm is presented which is based on explicit conditional distributions for the Bessel family of volatility models and is similar in spirit to the Brownian bridge algorithm. A special rearrangement and splitting of the path integral variables allows us to combine the Bessel bridge sampling algorithm with either adaptive Monte Carlo algorithms, or quasi-Monte Carlo techniques for significant numerical efficiency improvement. The algorithms are illustrated by pricing Asian-style and lookback options under the Bessel family of volatility models as well as the CEV diffusion model.
A time-dependent initial value semiclassical propagator approach is developed and applied to the propagation of a two-dimensional quantum system whose classical counterpart is highly chaotic. The energy spectrum of a quartic oscillator, obtained from the propagated wavefunction, is shown to be accurately and simply computed by application of stationary-phase Monte Carlo integration. Chaotic trajectories are handled naturally, without giving rise to the singularities seen in other methods.
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