The power of the disconjugacy properties of second-order differential equations of Schrödinger type to check the regularity of rationally extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re-examining the extensions of the isotonic oscillator (or radial oscillator) potential derived in kth-order supersymmetric quantum mechanics or multistep Darboux-Bäcklund transformation method. The function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term, whose value is calculated by induction over k. The sign of this term being the same as that of the already known highest degree term, the potential denominator has the same sign at both extremities of the definition interval, a property that is shared by the seed eigenfunction used in the potential construction. By virtue of disconjugacy, such a property implies the nodeless character of both the eigenfunction and the resulting potential.

1.
D.
Gómez-Ullate
,
N.
Kamran
, and
R.
Milson
, “
The Darboux transformation and algebraic deformations of shape-invariant potentials
,”
J. Phys. A
37
,
1789
(
2004
);
2.
D.
Gómez-Ullate
,
N.
Kamran
, and
R.
Milson
, “
Supersymmetry and algebraic Darboux transformations
,”
J. Phys. A
37
,
10065
(
2004
);
3.
D.
Gómez-Ullate
,
N.
Kamran
, and
R.
Milson
, “
An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
,”
J. Math. Anal. Appl.
359
,
352
(
2009
);
4.
D.
Gómez-Ullate
,
N.
Kamran
, and
R.
Milson
, “
An extension of Bochner's problem: Exceptional invariant subspaces
,”
J. Approx. Theory
162
,
987
(
2010
);
5.
C.
Quesne
, “
Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
,”
J. Phys. A
41
,
392001
(
2008
);
6.
L. E.
Gendenshtein
, “
Derivation of exact spectra of the Schrödinger equation by means of supersymmetry
,”
JETP Lett.
38
,
356
(
1983
).
7.
F.
Cooper
,
A.
Khare
, and
U.
Sukhatme
, “
Supersymmetry and quantum mechanics
,”
Phys. Rep.
251
,
267
(
1995
);
8.
J. F.
Cariñena
and
A.
Ramos
, “
Shape-invariant potentials depending onn parameters transformed by translation
,”
J. Phys. A
33
,
3467
(
2000
);
9.
B.
Bagchi
,
C.
Quesne
, and
R.
Roychoudhury
, “
Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of
$\mathcal {PT}$
PT
symmetry
,”
Pramana, J. Phys.
73
,
337
(
2009
);
10.
C.
Quesne
, “
Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics
,”
SIGMA
5
,
084
(
2009
);
11.
S.
Odake
and
R.
Sasaki
, “
Infinitely many shape invariant potentials and new orthogonal polynomials
,”
Phys. Lett. B
679
,
414
(
2009
);
12.
S.
Odake
and
R.
Sasaki
, “
Another set of infinitely many exceptional (X) Laguerre polynomials
,”
Phys. Lett. B
684
,
173
(
2010
);
13.
S.
Odake
and
R.
Sasaki
, “
Infinitely many shape-invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
,”
J. Math. Phys.
51
,
053513
(
2010
);
14.
C.-L.
Ho
,
S.
Odake
, and
R.
Sasaki
, “
Properties of the exceptional (X) Laguerre and Jacobi polynomials
,”
SIGMA
7
,
107
(
2011
);
15.
R.
Sasaki
,
S.
Tsujimoto
, and
A.
Zhedanov
, “
Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations
,”
J. Phys. A
43
,
315204
(
2010
);
16.
D.
Gómez-Ullate
,
N.
Kamran
, and
R.
Milson
, “
Exceptional orthogonal polynomials and the Darboux transformation
,”
J. Phys. A
43
,
434016
(
2010
);
17.
D.
Gómez-Ullate
,
N.
Kamran
, and
R.
Milson
, “
On orthogonal polynomials spanning a non-standard flag
,”
Contemp. Math.
563
,
51
(
2012
);
18.
Y.
Grandati
, “
Solvable rational extensions of the isotonic oscillator
,”
Ann. Phys. (N.Y.)
326
,
2074
(
2011
);
19.
C.-L.
Ho
, “
Prepotential approach to solvable rational potentials and exceptional orthogonal polynomials
,”
Prog. Theor. Phys.
126
,
185
(
2011
);
20.
S. Yu.
Dubov
,
V. M.
Eleonskii
, and
N. E.
Kulagin
, “
Equidistant spectra of anharmonic oscillators
,”
Sov. Phys. JETP
75
,
446
(
1992
).
21.
S. Yu.
Dubov
,
V. M.
Eleonskii
, and
N. E.
Kulagin
, “
Equidistant spectra of anharmonic oscillators
,”
Chaos
4
,
47
(
1994
).
22.
V. G.
Bagrov
and
B. F.
Samsonov
, “
Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics
,”
Theor. Math. Phys.
104
,
1051
(
1995
).
23.
B. F.
Samsonov
, “
New features in supersymmetry breakdown in quantum mechanics
,”
Mod. Phys. Lett. A
11
,
1563
(
1996
);
24.
G.
Junker
and
P.
Roy
, “
Conditionally exactly solvable problems and non-linear algebras
,”
Phys. Lett. A
232
,
155
(
1997
).
25.
G.
Junker
and
P.
Roy
, “
Conditionally exactly solvable potentials: A supersymmetric construction method
,”
Ann. Phys. (N.Y.)
270
,
155
(
1998
);
26.
J. F.
Cariñena
,
A. M.
Perelomov
,
M. F.
Rañada
, and
M.
Santander
, “
A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator
,”
J. Phys. A
41
,
085301
(
2008
);
27.
J. M.
Fellows
and
R. A.
Smith
, “
Factorization solution of a family of quantum nonlinear oscillators
,”
J. Phys. A
42
,
335303
(
2009
).
28.
D.
Gómez-Ullate
,
N.
Kamran
, and
R.
Milson
, “
Two-step Darboux transformations and exceptional Laguerre polynomials
,”
J. Math. Anal. Appl.
387
,
410
(
2012
);
29.
S.
Odake
and
R.
Sasaki
, “
Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials
,”
Phys. Lett. B
702
,
164
(
2011
);
30.
C.
Quesne
, “
Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials
,”
Mod. Phys. Lett. A
26
,
1843
(
2011
);
31.
C.
Quesne
, “
Rationally-extended radial oscillators and Laguerre exceptional orthogonal polynomials inkth-order SUSYQM
,”
Int. J. Mod. Phys. A
26
,
5337
(
2011
);
32.
Y.
Grandati
, “
Multistep DBT and regular rational extensions of the isotonic oscillator
,”
Ann. Phys. (N.Y.)
327
,
2411
(
2012
);
33.
D.
Gómez-Ullate
,
N.
Kamran
, and
R.
Milson
, “
A conjecture on exceptional orthogonal polynomials
,”
Found. Comput. Math.
(
2012
);
34.
A.
Erdélyi
,
W.
Magnus
,
F.
Oberhettinger
, and
F. G.
Tricomi
,
Higher Transcendental Functions
(
McGraw-Hill
,
New York
,
1953
).
35.
G.
Szegö
,
Orthogonal Polynomials
(
American Mathematical Society
,
Providence, RI
,
1939
).
36.
P.
Hartman
,
Ordinary Differential Equations
(
John Wiley
,
New York
,
1964
).
37.
W. A.
Coppel
,
Disconjugacy
(
Springer
,
Berlin
,
1971
).
38.
M.
Böcher
,
Leçons sur les méthodes de Sturm
(
Gauthier-Villars
,
Paris
,
1917
).
39.
A. A.
Andrianov
,
M. V.
Ioffe
, and
V.
Spiridonov
, “
Higher-derivative supersymmetry and the Witten index
,”
Phys. Lett. A
174
,
273
(
1993
);
40.
A. A.
Andrianov
,
M. V.
Ioffe
,
F.
Cannata
, and
J.-P.
Dedonder
, “
Second order derivative supersymmetry,q deformations and the scattering problem
,”
Int. J. Mod. Phys. A
10
,
2683
(
1995
);
41.
D. J.
Fernández C.
and
N.
Fernández-García
, “
Higher-order supersymmetric quantum mechanics
,”
AIP Conf. Proc.
744
,
236
(
2004
);
42.
M. M.
Crum
, “
Associated Sturm-Liouville systems
,”
Quart. J. Math. Oxford Ser. (2)
6
,
121
(
1955
).
43.
T.
Muir
,
A Treatise on the Theory of Determinants
(
Dover
,
New York
,
1960
) (revised and enlarged by
W. H.
Metzler
).
44.
A.
Messiah
,
Mécanique Quantique
(
Dunod
,
Paris
,
1969
), Vol.
1
.
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