We can compute the density and potential response corresponding to the
dressed particle by integrating the perturbed distribution function
from equation (3) over velocities:
The Laplace transformed expansion coefficients of the biorthogonal basis are then
This computation is easily performed by noting that the Jacobian of the canonical transform of is unity and we are free to choose any set. This procedure is the motivation behind the biorthogonal expansion. If we choose and use equation (7), we can do the angle integration trivially. Then, noting that , we can do the integral in using the orthogonality of the rotation matrices:
(Edmonds 1960). We get:
Equation (15) has the form and describes the response of the
stellar system, , to the perturbation, .
The self-gravitating response, then, is the solution to :
If , the equation for the response becomes an eigenvalue problem. The eigenvalues s are the zeros of the dispersion relation . For the general stable spherical stellar system, there will be no modes for and only in special cases with restricted phase space does one find oscillatory modes,
To evaluate the coefficients as a function of time, we perform the
inverse Laplace transform, deforming the contour to the
Assuming that the background is stable with no oscillatory modes, has no poles in the half plane . The final term gives a pole at . Although there may be poles in the half plane , these will vanish for relative to the pure imaginary contribution. To perform the integral, one may take the s integration into the phase-space integral for the elements of . In addition to , the s-dependence is in two simple poles and one finds an integral of the form
For large values of t, this expression oscillates rapidly and we may extract the dominant coherent contribution. There are two cases: without and with a resonance in the phase space. The existence of a resonance in phase space is defined by for . For the non-resonant case, the integrand has no singularity and we can consider each term separately. The first term yields a contribution in phase with the perturbation while the second term in the brackets oscillates incoherently and makes no net contribution. The second term, therefore, can be ignored. For the resonant case, the contribution at large t has a sharp peak about as . Expanding about and retaining only dominant terms, one finds the contribution near the resonance is
We will adopt the latter asymptotic form here and see in the final computation that the will cancel leaving only the delta functions. Putting both cases together yields a simple expression
To simplify notation, we have explicitly noted in equations (21) and (22) that the solution takes the the form . The integrals in the matrix elements of may be analytically continued using the Landau prescription (e.g. Krall & Trivelpiece 1973) after a conformal mapping of the discrete interval in E. Notice that this expression takes both resonant and non-resonant cases into account; without a resonance, the delta function does not contribute and principal value of a non-singular integrand is the integrand itself. One recovers the non-self-gravitating but global response by setting .