To test the spherical implementation, I assigned 
and 
to the Hernquist model (Hernquist 1990) and compared
the SLE solution with the analytic recursion relations (Hernquist &
Ostriker 1992) for radial order 
and 
. Performance of
the spherical algorithm is well-documented so a comparison of
potential pairs suffices. For m=0, the numerically determined
functions differed from the results of the recursion relation by one
part in 
near the center and one part in 
elsewhere. This
difference is due to the extrapolation of the cusp at r=0. Here,
the boundary condition for the cuspy profile fixes the asymptotic
value of ratio 
as 
. For m>0
the differences are obtained to the specified tolerance (one part in
for these tests). To recover the Clutton-Brock (1973) set, one
assigns and according to the Plummer law; in this
case, differences between the SLE solution and recursion relations are
obtained for all m to the desired tolerance. In all cases, the
orthogonality relation remains accurate and the potential density pair
is an accurate solution of the Poisson equation.
Figure 1: Potential-density pairs for l=m=0 labeled by order,
(upper and lower panels, resp.) whose lowest order
member (n=1) is the singular isothermal sphere. The density
eigenfunctions are multiplied by 
.
Figure 2: Potential-density pairs for l=m=0 labeled by order,
(upper and lower panels, resp.) whose lowest order
member (n=1) is the spherical deprojection of the 
surface brightness law with 
. To better represent the
cuspy density profile graphically, the density eigenfunctions are
shown relative to the deprojected law.
The background galaxian profile need not have finite mass and may be
cuspy. For example, a basis set tailored to the singular isothermal
sphere only requires one to specify appropriate boundary conditions.
Boundary conditions corresponding to a disturbance not felt by in the
singular core and at large radii are:
and
where 
. These same boundary conditions
apply to the profile above. The l=0 boundary conditions
ensure that the potential-density pairs are asymptotic to the
spherical background at small and large radii. The 
boundary
condition at small radius is the standard zero potential that ensures
a single valued function. At large radius, we choose the condition
obtained for an outer multipole. The four lowest-order l=0 pairs are
shown in Figure 1. The density functions are
multiplied by 
and, again, the lowest order
relative density function is constant as expected.
In addition, the background galaxian profile need not have an analytic
form. For example, the spherically symmetric profile that results in
the empirical surface density law may be numerically
deprojected, tabulated and used as input to the SLE routines described
above. A few of the lowest order potential-density pairs are shown in
Figure 2. The density functions are shown relative to
the background density. Notice that the lowest order relative density
function is constant as expected.