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O. Ragnisco
Hence it follows
Y (θ) ≡ Y
c N
c N−1 ,..,c 2
(θ 1 , θ 2 , . . . , θ N −1 ) ≡ Y
l
l N−2 ,..,l 1
(θ 1 , θ 2 , . . . , θ N −1 ),
(17)
Accordingly, the radial Schrödinger equation associated with ˆ
H reads
(ξ − r)r
−
¯
h 2
2ξ 2
d
2
dr 2 +
N − 1
r
d
dr
−
l(l + N − 2)
r 2
Φ(r)
+
k
ξ − r
Φ(r) = E Φ(r),
(18)
with k, ξ > 0 and 0 < r < ξ.
So we can formulate the following Proposition:
Proposition 1 The solution of the spectral problem associated with the radial
Schrödinger equation exists, and has the explicit closed form given by:
Φ n,l (r) = c n,l (ξ )
1 −
r
ξ
1
2
1+
8kξ
¯
h 2 +
1
2
r
ξ
l
P
1+
8kξ
¯
h 2 , 2l+N −2
n
2r
ξ
− 1
E n,l =
¯
h 2
8ξ 2
1+2n +
1 +
8kξ
¯
h 2
1 + 2n + 4
l +
N − 2
2
+
1 +
8kξ
¯
h 2
.
In the formula for Φ n,l (r), c n,l (ξ ) are normalization factors and P
(α,β)
n
(x) are the
Jacobi orthogonal polynomials with parameters α
.
=
1 +
8kξ
¯
h 2 and β
.
= 2l + N − 2.
The Jacobi orthogonal polynomials are defined for α, β > −1, meaning N +
2l > 1, i.e. N > 1. Moreover, the eigenfunctions Φ n,l (r) are square integrable in
r ∈ (0, ξ).
Finally, the (unnormalized) eigenfunctions of the confined Hamiltonian read:
Ψ n,l,l N−2 ,...,l 1 (r, θ ) ∝ Y
l
l N−2 ,..,l 1
(θ 1 , θ 2 , . . . , θ N −1 )Φ n,l (r) ,
(19)
and they are orthogonal with respect to the measure
dμ(r, θ )
.
=
r N −1
r(ξ − r)
sin
N −2 (θ 1 ) sin
N −3 (θ 2 ) . . . sin(θ N −2 )dr dθ 1 dθ 2 . . . dθ N −1 ,
with
θ 1 , . . . , θ N −2 ∈ [0, π),
θ N −1 ∈ [0, 2π), r ∈ (0, ξ).
In particular, the Jacobi orthogonal polynomials yield a δ nn at fixed l = l (for l = l
the orthogonality comes from the δ ll arising from the hyperspherical harmonics).