Ladder Operators and Rational Extensions
129
6 Examples
The articles [9, 10] considered a particular class of ladder operators corresponding
to Maya diagrams obtained by a single state-adding transformation. Fix some n =
1, 2, . . ., and let ˜
M n be the Maya diagram with index set {−n}, i.e., let ˜
M n = Z − \
{−n}. We set
ˆ
M n = ˜
M n + n = Z − ∪ {1, . . . , n − 1},
and observe that ˆ
M n has index set {1, . . . , n − 1}. Hence,
L n := A ˜
M n ,{−n,1,...,n−1} ,
is an nth order ladder operator that intertwines T ˜
M n
and T ˆ
M n
. Ordering the flips in
ascending order, we obtain the following factorization into first-order intertwiners:
L n = A ˆ
M n−1 ,{n−1} · · · A ˆ
M 2 ,{2} A ˆ
M 1 ,{1} A ˜
M n ,{−n} ;
each flip corresponds to a state-deleting transformation.
Let us also observe that ˜
M n is a genus 1 Maya diagram. It follows that
L 1 := A ˜
M n ,{−n,−n+1,0}
is a third-order ladder operator that intertwines ˜
M n and ˜
M n + 1.
The composition L n
1 is represented by the multi-set
n−1
j =0
{−n + j, −n + j + 1, j} = {−n, 1, . . . , n − 1} ∪ {(−n + 1)
2 , . . . , (−1)
2 , (0)
2
},
where the superscripts indicate repetition (and not a square). The syzygy between
L n and L 1 is therefore
L
n
1 = L n
0
j =−n+1
(2j + 1 − T ˜
M n
).
References
1. V.E. Adler, A modification of Crum’s method. Theoret. Math. Phys. 101, 1381–1386 (1994)
2. M.A. García-Ferrero, D. Gómez-Ullate, R. Milson, A Bochner type characterization theorem
for exceptional orthogonal polynomials. J. Math. Anal. Appl. 472, 584–626 (2019)
3. D. Gómez-Ullate, Y. Grandati, R. Milson, Rational extensions of the quantum harmonic
oscillator and exceptional Hermite polynomials. J. Phys. A 47, 015203 (2013)