Perturbation theories unique problems

Resonances are common and unique features of elastic and inelastic collisions, photodissociation, unimolecular decay, autoionization problems, and related topics. Their general behavior and formal description are rather universal and identical for nuclear, electronic, atomic, or molecular scattering. Truhlar (1984) contains many examples of resonances in various fields of atomic and molecular physics. Resonances are particularly interesting if more than one degree of freedom is involved they reflect the quasi-bound states of the Hamiltonian and reveal a great deal of information about the multi-dimensional PES, the internal energy transfer, and the decay mechanism. A quantitative analysis based on time-dependent perturbation theory follows in the next section. 

This relation shows how the action of the antisymmetrizer can mix different orders in perturbation theory. Secondly, the projected functions AglO ) 0 > do not form an orthogonal set in the antisymmetric subspace of the Hilbert space L2(r3N) if we take all excited states a > and b > in order to obtain a complete set a > b >, the projections As a > b > form a linearly dependent set. Expanding a given (antisymmetric) function in this overcomplete set is always possible, but the expansion coefficients are not uniquely defined. How the different symmetry adapted perturbation theories that have been formulated since the original treatment by Eisenschitz and London in 1930 , actually deal with these two problems can be read in the following reviews Usually, the first order interaction... 

The applications of many-body perturbation theory in contemporary research in the molecular sciences are manifold and it is certainly not possible to describe more than a mere fraction of the enormous number of publications which have exploited this approach to the molecular structure problem over recent years. Calculations based on second order many-body perturbation theory or MP2 theory are particularly prevalent offering unique advantages in terms of efficiency and accuracy over many other theoretical and computational approaches. Here, we shall briefly describe the use of graphical user interfaces and then concentrate on two recent applications of the many-body perturbation theory which have established new levels of precision. 

We emphasize that the question of stability of a CA under small random perturbations is in itself an important unsolved problem in the theory of fluctuations [92-94] and the difficulties in solving it are similar to those mentioned above. Thus it is unclear at first glance how an analogy between these two unsolved problems could be of any help. However, as already noted above, the new method for statistical analysis of fluctuational trajectories [60,62,95,112] based on the prehistory probability distribution allows direct experimental insight into the almost deterministic dynamics of fluctuations in the limit of small noise intensity. Using this techique, it turns out to be possible to verify experimentally the existence of a unique solution, to identify the boundary condition on a CA, and to find an accurate approximation of the optimal control function. 

In mathematics we have a classical definition of the ill-posed problem a problem is ill-posed, according to Hadamard (1902), if the solution is not unique or if it is not a continuous function of the data (i.c., if to a small perturbation of data there corresponds an arbitrarily large perturbation of the solution). Unfortunately, from the point of view of classical theory, all geophysical inverse problems are ill-posed,... 

It is noteworthy however that recent work has questioned the mathematical validity of the perturbation method (see ref. 13). Another problem is that it is difficult to determine unambiguously a value of z experimentally and the unique and universal relationship between a and z has been sufficiently doubted, that in a recent review Yamakawa, one of the chief proponents of two-parameter theory, has remarked as for the detailed deductions about this effect, there is not yet a complete consensus of all polymer physical chemists and physicists . 

---