Italo Guarneri

Band formalism.
The particle is moving in a circle S parametrized by x ∈ [−π, +π] with a distinguished
point O (x = 0).
H =
∫ ⊕
R
dq H α (q) + I ⊗ H osc ,
d 2
H α (q) = − 1 2
+ αq δ(x) .
dx 2
ψ(x, q) =
+∞∑
n=0
Q n (q) φ q,n (x) ,
H α (q)φ q,n = W n (q)φ q,n , φ q,n ∈ L 2 (S) . (3)