Eigenvalues, continuous distribution

The complex susceptibility components %Y(co) can be evaluated from Eq. (147) by calculation of the eigenvalues Xyk for normal rotational diffusion (see Section III.C). However, /.,( or) may be much more effectively calculated by using the continued fraction method (see Ref. 103 for detail). Let us first evaluate the longitudinal response. By expanding the distribution function W(i9, t) in a Fourier series (here W is independent of 9)... 

Equation 10.10 is known as the Schrodinger equation and is a very important equation in quantum mechanics. Although we have placed certain restrictions on wavefunctions (continuous, single-valued, and so on), up to now there has been no requirement that an acceptable wavefunction satisfy any particular eigenvalue equation. However, if P is a stationary state (that is, if its probability distribution does not depend on time), it must also satisfy the Schrodinger equation. Also note that equation 10.10 does not include the variable for time. Because of this, equation 10.10 is more specifically referred to as the time-independent Schrodinger equation. (The time-dependent Schrodinger equation will be discussed near the end of the chapter and represents another postulate of quantum mechanics.)... 

Thus, it has been revealed that a correlation structure similar to those observed in the eigenvalues of the random unitary matrices and in the Riemann zeros is embedded in the Fermi gas system as well. The case of v = rf/2 —> 1/2 in the Fermi gas system gives a special correlation structure already discussed in the random matrix and zeta function theories. Therefore, the behaviors of the multiparticle correlations of the Fermi gas system with the correlation kernel K (r v) at and around = jl may provide useful information about the random matrices and the zeta function by regarding T(r v) as a continuous function of v. Another challenge, which would be more ambitious, is to look for a family of functions or matrices whose zero or eigenvalue distributions are described by the correlation functions given by Eqs. 14.44 and 14.81 for arbitrary values of v. 

These in turn are used in the next nuclear calculation of the power distribution. Iterations between the thermohydraulic and nuclear parts are continued until satisfactory convergence of the nuclear eigenvalue is achieved. 

---