128
D. Gómez-Ullate et al.
for some M ∈ M and constant λ. Since T M+n = T M + 2n, Theorem 1 implies that
for every rational extension T M , M ∈ M, and n ∈ Z, there exists a ladder operator
A M,K , where K = (M + n) M. By (14) no generality is lost if we index such
ladder operators in terms of unlabelled Maya diagrams [M] ∈ M/Z.
A recent result provides a characterization of translational multi-flips [5] in terms
of cyclic Maya diagrams. This characterization makes it possible to establish the
order of a ladder operator [6].
Theorem 2 Let M ∈ M and n = 1, 2, . . .. Then,
|(M + n) M| = n + 2
n−1
i=0
g i ,
(16)
where g i is the genus of the Maya diagram
M i = {m ∈ Z : mn + i ∈ M}, i = 0, 1, . . . , n − 1.
Proof Let B i ∈ Z 2g i +1 be the block coordinates of M i , and set
B =
n−1
i=0
(nB i + i) =
n−1
i=0
{nb + i : b ∈ B i }.
Since B i is the unique set such that f B i (M i ) = M i + 1, it follows that B is the
unique set such that f B (M) = M + n. Therefore B = (M + n) M.
Fix a Maya diagram M ∈ M. An immediate consequence of Theorem 2 is the
existence of a primitive ladder operator that intertwines T M and T M + 2n for every
n ∈ Z. The ladder operator in question is L n := A M,K , where K = (M + n) M.
The order of L n is given by (16). If n > 0, then both L n and L n
1 intertwine T M and
T M + 2n; it follows that there must be a syzygy of the form
L
n
1 = L n ◦ p(T M ),
where the roots of the polynomial p are determined by (15).
The action of ladder operators on states is that of a lowering or raising operator
according to
L n [ψ M,k ] = C M,n,k ψ M,k−n , k /
∈ M,
where C M,n,k is]zero if ψ M,k−n is not a bound state, i.e., if k − n ∈ M. Otherwise,
C M,n,k is a rational number whose explicit form can be derived on the basis of (7).
As a particular example, suppose that the index set of M consists of positive integers
0 < k 1 < · · · < k p , that n > 0, and that k /
∈ M. In this case,
C M,n,k =
i∈M\(M+n) (2i − 2j) × (k − n + 1) n 2 n if k − n /
∈ M
0
otherwise.
D. Gómez-Ullate et al.
for some M ∈ M and constant λ. Since T M+n = T M + 2n, Theorem 1 implies that
for every rational extension T M , M ∈ M, and n ∈ Z, there exists a ladder operator
A M,K , where K = (M + n) M. By (14) no generality is lost if we index such
ladder operators in terms of unlabelled Maya diagrams [M] ∈ M/Z.
A recent result provides a characterization of translational multi-flips [5] in terms
of cyclic Maya diagrams. This characterization makes it possible to establish the
order of a ladder operator [6].
Theorem 2 Let M ∈ M and n = 1, 2, . . .. Then,
|(M + n) M| = n + 2
n−1
i=0
g i ,
(16)
where g i is the genus of the Maya diagram
M i = {m ∈ Z : mn + i ∈ M}, i = 0, 1, . . . , n − 1.
Proof Let B i ∈ Z 2g i +1 be the block coordinates of M i , and set
B =
n−1
i=0
(nB i + i) =
n−1
i=0
{nb + i : b ∈ B i }.
Since B i is the unique set such that f B i (M i ) = M i + 1, it follows that B is the
unique set such that f B (M) = M + n. Therefore B = (M + n) M.
Fix a Maya diagram M ∈ M. An immediate consequence of Theorem 2 is the
existence of a primitive ladder operator that intertwines T M and T M + 2n for every
n ∈ Z. The ladder operator in question is L n := A M,K , where K = (M + n) M.
The order of L n is given by (16). If n > 0, then both L n and L n
1 intertwine T M and
T M + 2n; it follows that there must be a syzygy of the form
L
n
1 = L n ◦ p(T M ),
where the roots of the polynomial p are determined by (15).
The action of ladder operators on states is that of a lowering or raising operator
according to
L n [ψ M,k ] = C M,n,k ψ M,k−n , k /
∈ M,
where C M,n,k is]zero if ψ M,k−n is not a bound state, i.e., if k − n ∈ M. Otherwise,
C M,n,k is a rational number whose explicit form can be derived on the basis of (7).
As a particular example, suppose that the index set of M consists of positive integers
0 < k 1 < · · · < k p , that n > 0, and that k /
∈ M. In this case,
C M,n,k =
i∈M\(M+n) (2i − 2j) × (k − n + 1) n 2 n if k − n /
∈ M
0
otherwise.