SciPost Submission Page
Linking ladder operators for the Rosen-Morse and Pöschl-Teller systems
by Simon Garneau-Desroches, Véronique Hussin
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Simon Garneau-Desroches |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202212_00014v1 (pdf) |
| Date submitted: | 2022-12-06 03:09 |
| Submitted by: | Garneau-Desroches, Simon |
| Submitted to: | SciPost Physics Proceedings |
| Proceedings issue: | 34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
An analysis of the realizations of the ladder operators for the Rosen-Morse and Pöschl- Teller quantum systems is carried out. The failure of the algebraic method of construction in the general Rosen-Morse case is exposed and explained. We present the reduction of a recently obtained set of (2n ± 1)-th-order Rosen-Morse ladder operators to the usual first-order realization for the Pöschl-Teller case known in the literature.
Current status:
Reports on this Submission
Anonymous Report 1 on 2023-3-20 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202212_00014v1, delivered 2023-03-20, doi: 10.21468/SciPost.Report.6933
Strengths
The presented results are interesting not only from the point of view of the considered quantum systems, but also due to their close connection with completely integrable systems (KdV hierarchy) and solitons.
Weaknesses
Two important points indicated in the report, unfortunately, escaped the attention of the authors.
Report
In the work, the construction of ladder operators for the Rosen-Morse and Pöschl-Teller systems is carried out. The presented results are interesting not only from the point of view of these quantum systems, but also due to their close connection with completely integrable systems (KdV hierarchy) and solitons. Two important points, unfortunately, escaped the attention of the authors.
1) It is known that there are broad classes of exactly solvable quantum systems characterized by a discrete spectrum resembling the spectrum of finite-gap quantum systems (DOI: 10.1088/1751-8121/aa739b , 10.1103/PhysRevD.98.026017 , 10.1103/PhysRevD.106.089901 ). As a consequence, they are characterized by the presence of a triple of basic (primary) pairs of the lowering and raising ladder operators, which form complete sets of spectrum generating operators. Such systems belong to the family of systems mentioned in the Introduction, and therefore this important feature from the point of view of physics should be pointed out.
2) The origin of such operators in rational deformations of a quantum harmonic oscillator and conformal mechanics bears some resemblance to a peculiar property of reflectionless hyperbolic Pöschl-Teller systems characterized by the presence of non-trivial Lax-Novikov integrals, which are the Darboux-dressed momentum operator of a free quantum particle (see DOI: 10.1016/j.aop.2006.12.002 , 10.1103/PhysRevLett.101.030403 , 10.1016/j.aop.2009.01.009 , 10.1007/JHEP12(2017)061 ). It is this nontrivial differential operator of higher odd order that underlies the exotic non-linear supersymmetry associated with (multi-)soliton potentials and their reflectionless nature, detects all bound states of the corresponding quantum systems, and distinguishes doubly degenerate states in the continuous parts of their spectrum. A special case of such systems corresponds to the Pöschl-Teller systems considered in the article (s=1,2,…). I suggest adding an appropriate comment related to the indicated feature of the Pöschl-Teller systems.
After appropriately taking into account the remarks, the article can be recommended for publication.
Requested changes
1) It is known that there are broad classes of exactly solvable quantum systems characterized by a discrete spectrum resembling the spectrum of finite-gap quantum systems (DOI: 10.1088/1751-8121/aa739b , 10.1103/PhysRevD.98.026017 , 10.1103/PhysRevD.106.089901 ). As a consequence, they are characterized by the presence of a triple of basic (primary) pairs of the lowering and raising ladder operators, which form complete sets of spectrum generating operators. Such systems belong to the family of systems mentioned in the Introduction, and therefore this important feature from the point of view of physics should be pointed out.
2) The origin of such operators in rational deformations of a quantum harmonic oscillator and conformal mechanics bears some resemblance to a peculiar property of reflectionless hyperbolic Pöschl-Teller systems characterized by the presence of non-trivial Lax-Novikov integrals, which are the Darboux-dressed momentum operator of a free quantum particle (see DOI: 10.1016/j.aop.2006.12.002 , 10.1103/PhysRevLett.101.030403 , 10.1016/j.aop.2009.01.009 , 10.1007/JHEP12(2017)061 ). It is this nontrivial differential operator of higher odd order that underlies the exotic non-linear supersymmetry associated with (multi-)soliton potentials and their reflectionless nature, detects all bound states of the corresponding quantum systems, and distinguishes doubly degenerate states in the continuous parts of their spectrum. A special case of such systems corresponds to the Pöschl-Teller systems considered in the article (s=1,2,…). I suggest adding an appropriate comment related to the indicated feature of the Pöschl-Teller systems.