Numerical inversion of the Laplace transform

A generalization of Doetsch's formula [Math. Z. 42, 263 (1937)] is derived to develop a stable numerical inversion of the one-sided Laplace transform (C) over cap (beta). The necessary input is only the values of C ( b) on th e positive real axis. The method is applicable provided that the functions (C) over cap (beta) belong to the function space L-alpha(2) defined by the condition that G(x) = e(x alpha)(C) over cap(e(x)), alpha>0 has to be squar e integrable. The inversion algorithm consists of two sequential Fourier tr ansforms where the second Fourier integration requires a cutoff, whose magn itude depends on the accuracy of the data. For high accuracy data, the cuto ff tends to infinity and the inversion is very accurate. The presence of no ise in the signal causes a lowering of the cutoff and a lowering of the acc uracy of the inverted data. The optimal cutoff value is shown to be one whi ch leads to an inversion which remains consistent with the original data an d its noise level. The method is demonstrated for some model problems: a ha rmonic partition function, resonant transmission through a barrier, noisy c orrelation functions, and noisy Monte Carlo generated data for tunneling co efficients obtained via the recently introduced quantum transition state th eory (QTST). (C) 1999 American Institute of Physics. [S0021-9606(99)00421-3 ].