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D. Gómez-Ullate et al.
There has been some recent interest in rational extensions possessing ladder
operators, which may be thought of as higher order analogues of the classical
creation and annihilation operators. There are applications of such ladder operators
to superintegrable systems [9, 10], rational solutions of Painlevé equations [11], and
coherent states [7].
In this note we classify the ladder operators corresponding to the class of rational
extensions of the harmonic oscillator. Rational extensions are naturally associated
with combinatorial objects called Maya diagrams. We show that any two rational
extensions are related by an intertwining relation. It therefore makes sense to endow
both Maya diagrams and rational extensions with the structure of a category, and
to interpret the relation Maya diagram → rational extension as a functor between
these categories. This approach allows us to classify ladder operators and syzygies
of ladder operators, and thereby to generalize the results of [9, 10].
2 Maya Diagrams
A Maya diagram is a set of integers M ⊂ Z containing a finite number of positive
integers, and excluding a finite number of negative integers. We visualize a Maya
diagram as a horizontally extended sequence of •
and symbols, with the filled
symbol •
in position m indicating membership m ∈ M. The defining assumption
now manifests as the condition that a Maya diagram begins with an infinite filled •
segment and terminates with an infinite empty segment.
A Maya diagram may also be regarded as a strictly decreasing sequence of
integers m 1 > m 2 > · · · , subject to the constraint that m i+1 = m i − 1 for i
sufficiently large. It follows that there exists a unique integer σ , called the index of
M, such that m i = −i + σ for i sufficiently large.
Let M denote the set of all Maya diagrams. The flip at position k ∈ Z is the
involution f k : M → M defined by
f k : M →
M ∪ {k}, if k /
∈ M,
M \ {k}, if k ∈ M.
M ∈ M.
(1)
In the first case, we say that the flip acts on M by a state-deleting transformation
( → •
), and in the second case, by a state-adding transformation ( •
→ ).
Let Z p denote the set of subsets of Z having cardinality p, and Z =
p Z p the
set of all finite subsets of Z. For K ∈ Z p consisting of distinct k 1 , . . . , k p ∈ Z we
define the multi-flip f K : M → M by
f K (M) = (f k 1 ◦ · · · ◦ f k p )(M), M ∈ M.
(2)
Since flips commute, the action of f K does not depend upon the order of k 1 , . . . , k p .
D. Gómez-Ullate et al.
There has been some recent interest in rational extensions possessing ladder
operators, which may be thought of as higher order analogues of the classical
creation and annihilation operators. There are applications of such ladder operators
to superintegrable systems [9, 10], rational solutions of Painlevé equations [11], and
coherent states [7].
In this note we classify the ladder operators corresponding to the class of rational
extensions of the harmonic oscillator. Rational extensions are naturally associated
with combinatorial objects called Maya diagrams. We show that any two rational
extensions are related by an intertwining relation. It therefore makes sense to endow
both Maya diagrams and rational extensions with the structure of a category, and
to interpret the relation Maya diagram → rational extension as a functor between
these categories. This approach allows us to classify ladder operators and syzygies
of ladder operators, and thereby to generalize the results of [9, 10].
2 Maya Diagrams
A Maya diagram is a set of integers M ⊂ Z containing a finite number of positive
integers, and excluding a finite number of negative integers. We visualize a Maya
diagram as a horizontally extended sequence of •
and symbols, with the filled
symbol •
in position m indicating membership m ∈ M. The defining assumption
now manifests as the condition that a Maya diagram begins with an infinite filled •
segment and terminates with an infinite empty segment.
A Maya diagram may also be regarded as a strictly decreasing sequence of
integers m 1 > m 2 > · · · , subject to the constraint that m i+1 = m i − 1 for i
sufficiently large. It follows that there exists a unique integer σ , called the index of
M, such that m i = −i + σ for i sufficiently large.
Let M denote the set of all Maya diagrams. The flip at position k ∈ Z is the
involution f k : M → M defined by
f k : M →
M ∪ {k}, if k /
∈ M,
M \ {k}, if k ∈ M.
M ∈ M.
(1)
In the first case, we say that the flip acts on M by a state-deleting transformation
( → •
), and in the second case, by a state-adding transformation ( •
→ ).
Let Z p denote the set of subsets of Z having cardinality p, and Z =
p Z p the
set of all finite subsets of Z. For K ∈ Z p consisting of distinct k 1 , . . . , k p ∈ Z we
define the multi-flip f K : M → M by
f K (M) = (f k 1 ◦ · · · ◦ f k p )(M), M ∈ M.
(2)
Since flips commute, the action of f K does not depend upon the order of k 1 , . . . , k p .