(A) closed form of g_jl on (0,pi) agrees with quadrature
  j=1: sum (E_l-E_j) g^2 = 1.000000000000 (target 1; raw partial sum 1.00000000),  sum (E_l-E_j)^2 g^2 = 4.0000000000 (target 4)
  j=2: sum (E_l-E_j) g^2 = 1.000000000000 (target 1; raw partial sum 1.00000000),  sum (E_l-E_j)^2 g^2 = 16.0000000000 (target 16)
  j=3: sum (E_l-E_j) g^2 = 1.000000000000 (target 1; raw partial sum 1.00000000),  sum (E_l-E_j)^2 g^2 = 36.0000000000 (target 36)
  j=4: sum (E_l-E_j) g^2 = 1.000000000000 (target 1; raw partial sum 1.00000000),  sum (E_l-E_j)^2 g^2 = 64.0000000000 (target 64)
  j=5: sum (E_l-E_j) g^2 = 1.000000000000 (target 1; raw partial sum 1.00000000),  sum (E_l-E_j)^2 g^2 = 100.0000000000 (target 100)
  closed form: sum_{l-j odd} l^2/(l^2-j^2)^2 = pi^2/16 for j < 30, hence the second sum rule = 4 j^2 exactly

(B) Proposition 3.2 on (0,pi), n = 1
  J=1 z=   1.00: -n Q_J(z) =    -0.000000   sum_(j<=J<l) c_jl g_jl^2 =     0.000000
  J=1 z=   2.50: -n Q_J(z) =     3.750000   sum_(j<=J<l) c_jl g_jl^2 =     3.750000
  J=1 z=   4.00: -n Q_J(z) =     3.000000   sum_(j<=J<l) c_jl g_jl^2 =     3.000000
  J=2 z=   4.00: -n Q_J(z) =     3.000000   sum_(j<=J<l) c_jl g_jl^2 =     3.000000
  J=2 z=   6.50: -n Q_J(z) =    25.500000   sum_(j<=J<l) c_jl g_jl^2 =    25.500000
  J=2 z=   9.00: -n Q_J(z) =    23.000000   sum_(j<=J<l) c_jl g_jl^2 =    23.000000
  J=3 z=   9.00: -n Q_J(z) =    23.000000   sum_(j<=J<l) c_jl g_jl^2 =    23.000000
  J=3 z=  12.50: -n Q_J(z) =    91.250000   sum_(j<=J<l) c_jl g_jl^2 =    91.250000
  J=3 z=  16.00: -n Q_J(z) =    86.000000   sum_(j<=J<l) c_jl g_jl^2 =    86.000000
  J=4 z=  16.00: -n Q_J(z) =    86.000000   sum_(j<=J<l) c_jl g_jl^2 =    86.000000
  J=4 z=  20.50: -n Q_J(z) =   239.000000   sum_(j<=J<l) c_jl g_jl^2 =   239.000000
  J=4 z=  25.00: -n Q_J(z) =   230.000000   sum_(j<=J<l) c_jl g_jl^2 =   230.000000
  J=5 z=  25.00: -n Q_J(z) =   230.000000   sum_(j<=J<l) c_jl g_jl^2 =   230.000000
  J=5 z=  30.50: -n Q_J(z) =   518.750000   sum_(j<=J<l) c_jl g_jl^2 =   518.750000
  J=5 z=  36.00: -n Q_J(z) =   505.000000   sum_(j<=J<l) c_jl g_jl^2 =   505.000000
  J=6 z=  36.00: -n Q_J(z) =   505.000000   sum_(j<=J<l) c_jl g_jl^2 =   505.000000
  J=6 z=  42.50: -n Q_J(z) =   992.500000   sum_(j<=J<l) c_jl g_jl^2 =   992.500000
  J=6 z=  49.00: -n Q_J(z) =   973.000000   sum_(j<=J<l) c_jl g_jl^2 =   973.000000

(C) Proposition 3.2 on the square (0,pi)^2, n = 2, degenerate eigenvalues
  J=1 E_J=2 E_J+1=5 z=  2.00: -n Q_J(z) =  -0.000000   sum_(j<=J<l) =   0.000000
  J=1 E_J=2 E_J+1=5 z=  3.50: -n Q_J(z) =   7.500000   sum_(j<=J<l) =   7.500000
  J=1 E_J=2 E_J+1=5 z=  5.00: -n Q_J(z) =   6.000000   sum_(j<=J<l) =   6.000000
  J=2 E_J=5 E_J+1=5 z=  5.00: -n Q_J(z) =   6.000000   sum_(j<=J<l) =   6.000000
  J=3 E_J=5 E_J+1=8 z=  5.00: -n Q_J(z) =   6.000000   sum_(j<=J<l) =   6.000000
  J=3 E_J=5 E_J+1=8 z=  6.50: -n Q_J(z) =  46.500000   sum_(j<=J<l) =  46.500000
  J=3 E_J=5 E_J+1=8 z=  8.00: -n Q_J(z) =  60.000000   sum_(j<=J<l) =  60.000000
  J=4 E_J=8 E_J+1=10 z=  8.00: -n Q_J(z) =  60.000000   sum_(j<=J<l) =  60.000000
  J=4 E_J=8 E_J+1=10 z=  9.00: -n Q_J(z) =  84.000000   sum_(j<=J<l) =  84.000000
  J=4 E_J=8 E_J+1=10 z= 10.00: -n Q_J(z) =  92.000000   sum_(j<=J<l) =  92.000000
  J=5 E_J=10 E_J+1=10 z= 10.00: -n Q_J(z) =  92.000000   sum_(j<=J<l) =  92.000000
  J=6 E_J=10 E_J+1=13 z= 10.00: -n Q_J(z) =  92.000000   sum_(j<=J<l) =  92.000000
  J=6 E_J=10 E_J+1=13 z= 11.50: -n Q_J(z) = 185.000000   sum_(j<=J<l) = 185.000000
  J=6 E_J=10 E_J+1=13 z= 13.00: -n Q_J(z) = 224.000000   sum_(j<=J<l) = 224.000000

(D) harmonic oscillator on R (exact rationals)
  sum rules with ||u'||^2 = E/2 and -Q~_J(z) = J(z-E_J)(E_(J+1)-z) hold exactly for J < 200; the analogue of (HS) is an equality for every J

(E) unit disk: overlaps <x u_1, u_(1,k)> with u_1 ~ J0(j01 r), u_(1,k) ~ J1(j1k r) cos(theta)
  3.291e-01, -2.529e-02, 7.247e-03, -3.073e-03, 1.589e-03, -9.289e-04, 5.900e-04, -3.982e-04, 2.814e-04, -2.062e-04
  all nonzero: x u_1 has components in infinitely many eigenspaces (consistent with Theorem 1.2)

ALL COMMUTATOR CHECKS PASSED (18.0 s)
