Darboux Transformation of the Laguerre Operator

• Published in: Complex analysis and operator theory Vol. 12; no. 3; pp. 787 - 809
• Main Author: Kovalyov, Ivan
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-03-2018; Springer Nature B.V
• Subjects: Analysis; Eigenvectors; Mathematics; Mathematics and Statistics; Operator Theory; Operators; Transformations
• ISSN: 1661-8254; 1661-8262
• DOI: 10.1007/s11785-018-0769-6

Let
L
α
be a Laguerre operator with
α
∉
Z
-
=
{
-
1
,
-
2
,
…
}
. Constructing the pairs of the factorization operators
L
,
R
or
L
,
R
such that
L
α
=
LR
or
L
α
=
L
R
. In the first case, the Darboux transformation
D
α
+
of
L
α
is the Laguerre operator
L
α
+
1
=
RL
and the eigenfunctions transform by a Christoffel formula. In the second case, the Darboux transformation
D
α
-
of
L
α
is the Laguerre operator
L
α
-
1
=
R
L
and the eigenfunctions transform by a Geronimus formula. The Darboux transformations
D
α
+
and
D
α
-
establish the relations between classical and non-classical Laguerre operators.