Formal series

This means that the discrete solution nearly conserves the Hamiltonian H and, thus, conserves H up to 0 t ). If H is analytic, then the truncation index N in (2) is arbitrary. In general, however, the above formal series diverges as jV —> 00. The term exponentially close may be specified by the following theorem. 

In fact, it is too much to require the convergence in the mathematical sense of the formal series of p.m. In all practical problems only their first several terms are calculated, and they may ultimately be divergent, or even possessed of only finite terms, without being unavailable. Thus we are lead to regard them as asymptotic rather than power series, and it is natural to expect that the range of applicability of p.m. be much extended by this new interpretation. 

These circumstances will easily be understood if we note that even the existence of the coefficients of the formal series given in 3 is not generally guaranteed. For instance, the coefficient — QSIlo isecond order term in the expression (3.15) of the eigenvalue might be meaningless if does not exist, i.e., [Pg.30]

On the other hand, if we take for instance q> — const. e x" in Ex. 3, we can cal-culate all coefficients of the formal series without encountering anything like divergence difficulty. But in reality the eigenvalue is non-existent as was shown before ( 6.1). Thus the formal series thus obtained has no meaning as an eigenvalue (cf., however, the next chapter). 

T n any case it is rather usual that wo can construct the formal series given in 3 representing the eigenvalues and eigenvectors at least to some finite order. Let us therefore assume that these series up to the order + K J +. .. +, ... 

Formal series. Next we shall derive the solution in the form of a power series of k. For the moment we proceed quite formally and discuss the validity of the results in later stages. We put... 

This is in remarkable contrast with p.m. of the first kind discussed in Chap. I. There the convergence of the formal series was established under the assumption of regular perturbation, but this is not the case with p.m. of the second kind now under consideration. Even if we assume the simplest case of regular perturbation where... 

If we multiply each equation by the corresponding power of A and write t/(A, t) = 2 (/)A", where the summation goes from n = 1 to infinity, (this is a formal series and no question of convergence arises), we have an equation for U... 

Exploiting the above identity repeatedly, using Eqs. (5.2) and (5.1), we obtain the formal series expansion... 

The cumulative generating function that is actually a formal series can be obtained from the moment generating function by division through 1 — exp(x). It is easily seen that the partial sums are obtained when the divisor is expanded in a geometric series and resolved term by term, namely... 

However, the operators involved are not equivalent to the operators that would arise from any stochastic differential equation (see [403]), so we must give up the direct analogy between the formal series and continuous SDE models (which is available in the deterministic setting). Nonetheless, it is possible to analyze the resulting operator expansion in a formal way. 

The formal series expansion of the solution is somewhat more complicated than the type (2.6), and may contain polynomials in t. The problem of the convergence of this series when the spectrum is infinite remains unsolved. The best result obtainable to date is of an asymptotic nature. We now describe it briefly. 

---