Poisson summation formula

Note that Eq. (4.15) is simply the linearized equation of motion for the classical upside-down barrier (55/5 = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15) describes how the trajectory escapes from the instanton solution, when deviated from it. The parameter A, referred to as the stability angle [Gutzwiller, 1967 Rajaraman, 1975], generalizes the harmonic oscillator phase a>,/3, which would stand in (4.16), if w, were constant. The fact that A is real is a reminder of the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det(-dj + w2) is a function of A only, and using the Poisson summation formula, we are able to write... 

The question raised by the quasicrystal debate is much deeper than whether they exist or not. To see this, we recall that the interpretation of diffraction experiments on all known translationally invariant crystals, however complicated, depends ultimately on the existence of the Poisson summation formula. This relation asserts that the Fourier trtinsform of the periodic delta function is itself a periodic delta function, whence the term reciprocal space. Explicitly, the Poisson summation formula is... 

Now the Poisson summation formula is at the core of all mathematical analysis [33]. It is equivalent in fact to the calculus, the Jacobi theta function transformations, and to a statement of the Riemann relation connecting the... 

There has been no basic formula analogous to the Poisson summation formula, characteristic of translational invariance, on which to base an analysis of quasicrystal diffraction patterns. Here successive values of reciprocal space have geometric ratios instead of the arithmetic spacing of the peaked functions observed with ordinary crystalline diffraction. Fig. 2.15 illustrates a two-dimensional section in reciprocal space of a diffraction pattern. The five-fold symmetry is exact, and typically six indices instead of three are required to index each point, with the choice of origin arbitrary, and for assigiunent of indices, ambiguous. The features of interest are ... 

All this formula instructs us to do is to take the contribution to the total stress of each dislocation using the results of eqn (8.38) and to add them up, dislocation by dislocation. We follow Landau and Lifshitz (1959) in exploiting the Poisson summation formula to evaluate the sums. Note that while it would be simple enough to merely quote the result, it is fun to see how such sums work out explicitly. If we use dimensionless variables a = x/D and yS = y/D, then the sum may be rewritten as... 

With these clever algebraic machinations behind us, the original problem has been reduced to that of evaluating the sum J a, ). Recall that the Poisson summation formula tells us... 

For the problem at hand, the Poisson summation formula allows us to rewrite our sum as... 

The semiclassical approximation is reached in three different steps (Berry and Mount, 1972). The first stage is to transform the summation over the discrete values of / into an integration over the continuous variable A = (/ + ). The proper way to do this is to use the Poisson summation formula which gives... 

The series given in (10.264) may be summed with the aid of the Poisson summation formula, which is given by... 

The summation over I can be carried out in (16 a) by means of Poisson s summation formula. The will play a role in (16 a) only for I for which... 

Now we are in position to recognize the Poisson summation (the comb) formula (see Appendix A. 1.2) linking the exponential fluctuations with the delta Dirac point-contributions... 

In Refs. [24,63] one can find a proof for this equation and an efficient way of calculating E/c for charge neutral systems. We do not want to go through the full derivation again it consists of applying Poisson s summation formula along both periodic coordinates and performing the limit 0 analytically. One obtains... 

In the practical matter of performing the summations indicated for the various formulas that must be evaluated, the question arises as to how many terms need to be included this question is analogous to the need to decide the limits of integration that was implicit in evaluating the analogous expressions for the Normal Distribution. In the case of the Poisson distribution this is one decision that is actually easier to make. The reason is... 

---