for Q(n) and P(n) respectively; diagonalizing the

Fourier transformation UPSILON(n).

In [Section IX], the matrix elements of the unitary diagonalizing transformation XI(n) for Q(n) where found. Q(n) -> XI!(n) Q(n) XI(n) so |q(n, k)> = XI!(n) |n, k> |n, k> = XI(n) |q(n, k)> where |q(n, k)> are the eigenvectors of Q(n).The matrix XI(n) is real, so the diagonalizing transformation is actually orthogonal. The diagonalizing transformation PI(n) for P(n), is then quite naturally related by the Fourier transform Fr(n): Define PI(n) as the transformation that maps the eigenbasis of N(n) to the eigenbasis of P(n) so,

P(n) -> PI!(n) P(n) PI(n) (10.1a) so P(n) is diagonalized and |p(n, k)> = PI!(n) |n, k> (10.1b) |n, k> = PI(n) |p(n, k)> (10.1c) where |p(n, k)> are the eigenvectors of P(n) and PI(n) |p(n, k)> = p(n, k) |p(n, k)> (10.1d) p(n, k) = q(n, k) (10.1e) Then using the properties of XI(n) and Fr(n), it is not difficult to see that PI(n) = Fr(n) XI(n) Fr!(n) (10.2a) is the diagonalizing transformation for P(n), and then PI!(n) = Fr(n) XI!(n) Fr!(n) (10.2b) Although XI(n) is orthogonal, PI(n) is not, yet it is, of course, unitary.

The second Fourier transform UPSILON(n) arises as the diagonalizing
transformation of C(n) the cyclic operator used in the proper
polar decomposition of B(n) and B!(n).
Diagonalizing UPSILON(n) is apparently not so simple an
enterprise,
even though the eigenvalues, __sans__ multiplicity, are obvious.
The following are pertinent results of [Mehta 1987]:

Let f_jk = f_jk(n) = infinity SIGMA exp( - (pi/n)(mn + j)^2 ) H_k(sqrt(2 pi/n)(mn + j)) (10.3) m=-infinity where H_k(z) is the kth unnormalized Hermite polynomial as defined by the Rodrigues formula (9.4). Then, (n-1) SIGMA <n, j| UPSILON(n) |n, m> f_mk(n) = i^k f_jk(n) (10.4) m=0 So the second index of f_jk labels eigenvectors of <n, j| UPSILON(n) |n, m> that are attached to the eigenvalues i^k. f_jk is a periodic function of both indicies with period n, so f_0k = f_nk, and f_j0 = f_jn. More generally: f_jk(n) = (-1)^k f_(n-jk)(n) = f_(j+nk)(n) (10.5) Further, (n-1) SIGMA f_(jm)(n) f_(mk)(n) = 0, i^j not= i^k (10.6) m=0 Eigenvectors with different eigenvalues are orthogonal. The f_jk(n) provide an infinite number of eigenvectors; the problem is to choose a linearly independent set of them. Mehta makes the following conjecture based on the numerical evidence of special cases. The f_jk(n) are linearly independent for: k = 0, 1, ..., n-1 if n is odd k = 0, 1, ..., n-2, n if n is even

A bothersome and somewhat surprising (to me) event is that the eigenvectors are expressed as a doubly sided infinite summation. It would be considerably more convenient to have a finite linearly independent set expressed in closed form.

These three diagonalizing transformations are all outside the group that preserve the form of G(n). They will all however, preserve the matrix elements <psi|phi> and <psi|G(n)|phi>.

We wish to establish relative to the indexing implicitly chosen in [Section IX], which eigenvalue belongs to the index k of q(n, k). By inspection, as k increases, the real value of the eigenvalue decreases. This can be rigorized; it is messy. The operator Fr^2(n) maps Q(n) to -Q(n), and therefore reverses the signs of the spectral values of Q(n). Therefore, the matrix elements of Fr^2(n) in the basis

|q(n, k)> = X^(t(n) |n, k> must be, <q(n, k)| Fr^2(n) |q(n, j)> = delta_k(n-j) (10.7) When diagonalizing Q(n) by XI(n) q(n, 0) is always the maximal eigenvalue; further, if n is odd, q(n, n-1/2) = 0. If n is even, we've seen that there is no zero eigenvalue for Q(n). For any n, the eigenvalues of Q(n) when diagonalized by XI(n), are uniformly decreasing down the diagonal. The above is then also true for P(n) diagonalized by PI(n). Since the second Fourier transform UPSILON(n) diagonalizes the cyclic operator C(n) and, C(n) = exp( i omega_T t(n) ) = exp( i2 pi/n F(n) ) where F(n) is the "phase matrix" and t(n) is the time operator for a harmonic oscillator, then UPSILON(n) also diagonalizes t(n) and F(n).

Go to Table of Contents

Go to Previous Chapter (IX)

Go to Next Chapter (XI)

Go to Physics Pages

Go to Home Page

Email me, Bill Hammel at:

bhammel@graham.main.nc.us READ WARNING BEFORE SENDING E-MAIL

The URL for this document is:

http://graham.main.nc.us/~bhammel/FCCR/X.html

Created: August 1997

Last Updated: May 28, 2000