Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics

Ryu Sasaki; Satoru Odake

REVIEW

Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics

Satoru Odake, Ryu Sasaki

Quantum Physics · Submitted 25 May 2006 · 16 references

Pith's verdict: A careful, explicit unification of ladder operators via the Heisenberg-evolved sinusoidal coordinate; the discrete-QM half is the real contribution, and the scope word 'most' is the only thing I'd push back on.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDquant-ph/0605215 (version 1)

Pith IDGZQDK2NO, Pith's permanent record of this paper and its review

CitationRyu Sasaki, Satoru Odake. "Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics." arXiv:quant-ph/0605215 (2006). https://pith.science/paper/GZQDK2NO

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This is the Odake-Sasaki paper that ties ladder operators to the positive/negative frequency parts of the Heisenberg solution for the 'sinusoidal coordinate' η(x). The construction is elementary and it works: closed algebra at level two (2.12), exact Heisenberg solution (2.14), frequency split (2.28)-(2.29), and a clean proof that a^(+) and a^(-) are Hermitian conjugates (2.36)-(2.37). The relative weights in the operators come straight out of the energy spectrum, which is the nice conceptual point and the reason the method is uniform across systems that otherwise look unrelated. The action on eigenstates reproduces the three-term recurrences and gives shift-down/up operators for the orthogonal polynomials, which is the useful byproduct. Appendix B's X-compensation result (B.26)-(B.29) is genuinely new and worth citing on its own; it bridges the 'discrete' QM formalism to the Askey scheme in a way I haven't seen done this cleanly. The ordinary-QM cases overlap Nieto-Simmons [4], and the paper says so, so no complaint there. Appendix A is a real piece of work: it derives the necessary and sufficient condition for a sinusoidal coordinate in ordinary QM and shows the resulting class sits inside shape invariance, with Kepler and asymmetric Rosen-Morse explicitly excluded. That is the kind of negative result that makes a unification claim honest. Soft spots, in proportion. The abstract's 'most' (abstract:s2) is broader than what is proved. Appendix A restricts ordinary QM to a subgroup of shape-invariant potentials, and for the discrete systems the general closure condition is asserted, not derived; the authors say as much and leave the complete list as a challenge. That is a scope caveat, not an error, but a reader skimming the abstract will overestimate coverage. Second, the paper is formula-dense and several key actions (e.g. the Askey-Wilson (2.118)-(2.119), the Wilson (3.110)-(3.111)) are stated rather than derived line by line; I spot-checked the Meixner-Pollaczek and continuous dual Hahn cases and they hold, but a referee should verify the q-case independently. Third, several coherent-state sums (3.31), (3.73), (3.90), (3.113) are left as unsummed series; the authors say they don't know closed forms, which is fine, but it limits the practical value of that section. None of this threatens the central claim. Who should read it: people working on ladder operators, shape invariance, or the Askey scheme, and anyone who wants a dynamical (rather than algebraic) derivation of structure relations. It deserves a serious referee, mainly to check the q-formulas and to press on the scope of 'most'.

Referee report

Major objections (2)

[Abstract, sentence s2; Appendix A] The abstract states that the method applies to 'most of the solvable quantum mechanics of single degree of freedom', but the demonstrated class is narrower. Appendix A shows that the ordinary quantum-mechanical potentials admitting a sinusoidal coordinate are restricted to shape-invariant systems and explicitly excludes the Kepler problems and the asymmetric Rosen-Morse potential, Eqs. (A.16)-(A.19). For the 'discrete' systems the general closure condition (2.12) is asserted but not derived; Appendix A states that 'writing down corresponding conditions is easy' and leaves the complete list as a challenge, so the discrete case rests on the five worked examples. The authors acknowledge this restriction in Appendix A, but the abstract does not carry the qualification. The abstract should be revised to state the demonstrated scope (a broad but restricted class, with the ordinary-QM restriction and the open discrete classification noted), or the general discrete condition should be established.
[Abstract, sentence s1; Eq. (2.34); Eq. (3.10)] The abstract claims that 'the relative weights of various terms in them are solely determined by the energy spectrum'. This is not accurate as stated. The constant term in a^(±) involves R_{-1}(H)/R_0(H), which on eigenstates equals -B_n by Eq. (2.34), where B_n is the diagonal three-term recurrence coefficient. B_n is not fixed by the spectrum alone: for the x²+1/x² potential the spectrum is E_n=2n, independent of g, while Eq. (3.10) gives B_n=2n+β+1 with β=g-1/2, so two systems with different g share the same spectrum but have different B_n. The relative weight of the constant term is therefore determined by the spectrum together with the algebra coefficients R_i(H), equivalently the recurrence coefficients, not by {E_n} alone. The construction itself is unaffected, but this headline characterisation should be corrected.

Minor objections (7)

[Eq. (2.30)] The fraction R_{-1}(E_n)/(E_{n±1}-E_n) is typeset ambiguously as 'R_{-1}(En)En±1 − En'; please make the denominator explicit so that the two signs are unambiguous.
[Eqs. (2.22)-(2.23)] The text states that the overdetermined conditions (2.22), (2.23) and E_0=0 determine the entire spectrum completely for each Hamiltonian. This is demonstrated case by case but not proved in general; a sentence qualifying the claim as verified for the examples would be appropriate.
[Appendix A, Eq. (A.5)] The parametrisation R_0 linear in H, R_1 constant, and R_{-1} linear in H is justified by derivative counting in the paragraph preceding (A.5), but the argument is compressed. A short explicit statement of why R_1 cannot depend on H (otherwise [H,η]R_1(H) would produce a third derivative absent from the left-hand side) would make the derivation self-contained.
[Section 2, notation] The three normalisations a^(±), a'^(±) and a''^(±) are introduced in close succession, and the reader would benefit from a small summary table listing the definition, the Hermitian-conjugacy property, and the action on eigenstates for each.
[Reference [5]] Reference [5] is listed as an arXiv preprint (math.CA/0601303). Since the correspondence between the Heisenberg operator solution and the structure relation is one of the paper's stated connections, the citation should note the preprint status or point to the published version if available.
[Title and terminology] The word 'discrete' in the title and throughout refers to difference-equation quantum mechanics rather than to a discrete spectrum, which may confuse readers outside the subfield; a footnote at first use would remove the ambiguity.
[Sections 3.1.2, 3.2.1-3.2.3] The coherent states in Eqs. (3.31), (3.32), (3.73), (3.74), (3.90), (3.113) and (3.114) are left as formal sums. The authors state that they are not aware of concise summation formulas, which is acceptable, but a brief remark on the convergence of these sums (or the range of λ for which they define normalisable states) would help the reader.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

A single dynamical recipe is claimed to generate annihilation and creation operators for a wide class of solvable quantum systems, including discrete ones.

The reading

The paper asks whether the familiar harmonic-oscillator ladder operators are a special case of a more general dynamical construction. Its answer is yes for a broad family of one-dimensional solvable systems: identify a special coordinate $��(x)$ that executes sinusoidal motion under the Hamiltonian, solve the Heisenberg equation for that coordinate exactly, and split the solution into positive- and negative-frequency parts. Those two parts are taken to be the creation and annihilation operators. The payoff is that the relative coefficients inside the operators are not chosen by hand; they follow from the energy spectrum, and the same recipe is claimed to cover ordinary potentials and the difference-equation 'discrete' systems whose eigenfunctions are Askey-scheme orthogonal polynomials. The authors acknowledge that Appendix A derives the condition for ordinary quantum mechanics but leaves the complete discrete classification as a challenge, so the discrete side rests on five worked examples.

What carries the argument

The load-bearing object is the 'sinusoidal coordinate' $��(x)$—the argument of the orthogonal polynomial in the eigenfunction $��_n(x)=��_0(x)P_n(��(x))$—together with the closure relation (2.12). Solving the Heisenberg equation for $��$ gives the two frequencies $��_��(H)$ as roots of $��^2-R_1(H)��-R_0(H)=0$. The annihilation and creation operators are then $a^{(��)}=(��[H,��]��(��+R_{-1}/R_0)��_��)/(��_+-��_-)$. This is the same move Heisenberg used for the harmonic oscillator, but the paper applies it uniformly; by similarity transformation with $��_0$ it also yields the structure relation and shift-down/shift-up operators for the corresponding orthogonal polynomials.

Extended reading notes

Core claim

The central claim is that for a solvable Hamiltonian $H$ with a coordinate $��(x)$ satisfying the level-two closure $[H,[H,��]]=��R_0(H)+[H,��]R_1(H)+R_{-1}(H)$, the exact Heisenberg operator $e^{itH}��e^{-itH}$ can be written as $a^{(+)}e^{i��_+t}+a^{(-)}e^{i��_-t}-R_{-1}/R_0$, where $��_��$ are fixed by $R_0$ and $R_1$. The coefficient of the positive-frequency term is the creation operator and the coefficient of the negative-frequency term is the annihilation operator; they are Hermitian conjugates, and their action on the $n$-th eigenstate is fixed by neighbouring energy differences. The paper works this out for ordinary potentials ($1/��sin^2 x$, $x^2+1/x^2$, Pöschl-Teller, soliton, Morse) and for five discrete systems (Meixner-Pollaczek, continuous Hahn, continuous dual Hahn, Wilson, Askey-Wilson), and argues that the construction applies to most single-degree-of-freedom solvable quantum mechanics.

Load-bearing premise

The load-bearing premise is that every system claimed to be covered possesses a coordinate $��(x)$—the same argument that appears in the orthogonal-polynomial factor of the eigenfunction—satisfying the level-two closure $[H,[H,��]]=��R_0(H)+[H,��]R_1(H)+R_{-1}(H)$; for the discrete systems this is shown on five worked examples rather than proved as a general condition, and Appendix A leaves the discrete classification as a challenge.

Editorial extensions

If this is right

For any system in the covered class, the discrete spectrum can be recovered from the requirement that the Heisenberg solution have only two frequencies plus a constant, following Heisenberg and Pauli's oscillator argument.
The three-term recurrence coefficients $A_n$, $B_n$, $C_n$ of the orthogonal polynomials are tied directly to matrix elements of $��$, so the polynomial recurrence relation and the operator ladder become two views of one calculation.
Coherent states defined as eigenstates of the new annihilation operator can be written down from the recurrence coefficients; in equi-spaced cases they retain simple time evolution.
The construction supplies shift-down and shift-up operators for the Askey-scheme polynomials from the Hamiltonian rather than from a separately postulated structure relation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's 'most' claim would become a classification theorem if the difference-equation analogue of Appendix A were solved; until then, discrete applicability is an extrapolation from five worked examples.
Because the normalisation is fixed only up to an overall function of $H$, different choices of that function produce different coherent-state families; a reader could test whether one choice is preferred by temporal stability or by a classical limit.
The same level-two closure could be used as a search filter for new solvable difference equations: impose the closure with low-degree $R_0$, $R_1$, $R_{-1}$ and ask which potentials admit a sinusoidal coordinate.
If the construction is right, known structure relations for Askey-Wilson and Wilson polynomials should be re-derivable mechanically from the Heisenberg solution; a mismatch in any coefficient would falsify the unified claim for that family.

Original abstract

The annihilation-creation operators $a^{(\pm)}$ are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the `sinusoidal coordinate'. Thus $a^{(\pm)}$ are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the `discrete' quantum mechanics.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,979 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.44 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.10 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The annihilation-creation operators $a^{(\pm)}$ are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the `sinusoidal coordinate'. Thus $a^{(\pm)}$ are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the `discrete' quantum mechanics.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The annihilation-creation operators $a^{(\pm)}$ are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the `sinusoidal coordinate'.	method 0.61	0.75	0.14	asserted 1.00
s1	Thus $a^{(\pm)}$ are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum.	result 0.59	0.71	0.16	asserted 1.00
s2	This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the `discrete' quantum mechanics.	contribution 0.72	0.73	0.32	asserted 0.99

What kind of paper the classifier reports this is: method or tool 0.93 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (16)

[4]

M. M. Nieto and L. M. Simmons, Jr., “Coherent States For Genera l Potentials”, Phys. Rev. Lett. 41 (1978) 207-210; “Coherent States For General Potentials”. 1. Formalism, 41 Phys. Rev. D 20 (1979) 1321-1331; 2. Conﬁning One-Dimensional Examples, Phys. Rev. D 20 (1979) 1332-1341; 3. Nonconﬁning One-Dimensional Examples, Ph ys. Rev. D 20 (1979) 1342-1350

View on Pith 1978
[1]

See, for example: L. Infeld and T. E. Hull, “The factorization met hod”, Rev. Mod. Phys. 23 (1951) 21-68. F. Cooper, A. Khare and U. Sukhatme, “Supersym metry and quantum mechanics”, Phys. Rept. 251 (1995) 267-385

View on Pith 1951
[2]

S. Odake and R. Sasaki, “Shape Invariant Potentials in ‘Discrete’ Quantum Mechanics”, J. Nonlinear Math. Phys. 12 Suppl. 1 (2005) 507-521; “Calogero-Sutherland-Moser Sys- tems, Ruijsenaars-Schneider-van Diejen Systems and Orthogon al Polynomials”, Prog. Theor. Phys. 114 (2005) 1245-1260

View on Pith 2005
[3]

S. Odake and R. Sasaki, “Equilibria of ’discrete’ integrable system s and deformations of classical orthogonal polynomials”, J. Phys. A37 (2004) 11841-11876; “Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials”, J. Math . Phys. 46 (2005) 063513, 10 pages

View on Pith 2004
[5]

T. Koornwinder, “The structure relation for Askey-Wilson polyn omials”, arXiv: math.CA/0601303

View on Pith arXiv
[6]

R. Koekoek and R. F. Swarttouw, “The Askey-scheme of hyper geometric orthogonal polynomials and its q-analogue”, math.CA/9602214; G. E. Andrews, R. Askey and R. Roy, “ Special Functions ”, Encyclopedia of mathematics and its applications, Cam- bridge, (1999)

View on Pith arXiv 1999
[7]

G. Szeg¨ o, “Orthogonal polynomials ”, Amer. Math. Soc. New York (1939)

View on Pith 1939
[8]

L. E. Gendenshtein, “Derivation of exact spectra of the Schro dinger equation by means of supersymmetry”, JETP Lett. 38 (1983) 356-359; G. Levai, “A Search For Shape Invariant Solvable Potentials”, J. Phys. A22 (1989) 689-702

View on Pith 1983
[9]

T. Fukui and N. Aizawa, “Shape-invariant potentials and an asso ciated coherent state”, Phys. Lett. A180 (1993) 308–313; J.-P. Gazeau and J. R. Klauder, “Coherent stat es for systems with discrete and continuous spectrum”, J. Phys. A32 (1999) 123–132; J.-P. An- toine, J.-P. Gazeau, P. Monceau, J. R. Klauder and K. A. Penson, “ Temporally stable coherent s...

View on Pith Pith review arXiv 1993
[10]

F. Calogero, “Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials”, J. Math. Phys. 12 (1971) 419-436; B. Sutherland, “Exact results for a quantum many-body problem in one-dimension. II”, Phys. Rev. A5 (1972) 1372-1376

View on Pith 1971
[11]

S. P. Khastgir, A. J. Pocklington and R. Sasaki, “Quantum Calog ero-Moser Models: Integrability for all Root Systems”, J. Phys. A33 (2000) 9033-9064. 42

View on Pith 2000
[12]

A. Degasperis and S. N. M. Ruijsenaars, “Newton-Equivalent H amiltonians for the Harmonic Oscillator”, Ann. of Phys. 293 (2001) 92–109

View on Pith 2001
[13]

S. N. M Ruijsenaars and H. Schneider, “A New Class Of Integrab le Systems And Its Relation To Solitons”, Ann. of Phys. 170 (1986) 370-405; J. F. van Diejen, “Integrability of diﬀerence Calogero-Moser systems”, J. Math. Phys. 35 (1994) 2983-3004

View on Pith 1986
[14]

G. S. Agarwal and S. Chaturvedi, “Calogero-Sutherland oscilla tor: classical behaviour and coherent states”, J. Phys. A28 (1995) 5747-5755

View on Pith 1995
[15]

P. J. Gambardella, “Exact results in quantum many-body syste ms of interacting par- ticles in many dimensions with SU (1, 1) as the dynamical group”, J. Math. Phys. 16 (1975) 1172-1187

View on Pith 1975
[16]

A. J. Bordner, N. S. Manton and R. Sasaki, “Calogero-Moser M odels V: Supersymmetry, and Quantum Lax Pair”, Prog. Theor. Phys. 103 (2000) 463-487. 43

View on Pith 2000

How the literature uses it

43 arXiv papers cite this in 173 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 139 sentences without a confident label

Builds on it 3 papers

Following our embryonic work [ref] on the construction of exact Heisenberg operator solutions for various degree one quantum mechanics, we present in this paper a modest first step in the quest of deriving exact Heisenberg operator solutions for a family of interacting multi-particle dynamics.

Exact Heisenberg operator solutions for multi-particle quantum mechanics · Introduction · 0.96
In our previous work on discrete quantum mechanics [ref] , only the special case of the Meixner-Pollaczek polynomial with the phase angle $ϕ=π/2$ and the special case of the continuous Hahn polynomial with two real parameters $a_1$ and $a_2$ are discussed, partly because these special cases of the two polynomials appear in several other dynamical contexts [ref] and, in particular, they appear as describing the equilibrium positions [ref] of the classical Ruijsenaars-Schneider van Diejen systems [ref] .

New Quasi Exactly Solvable Difference Equation · Hamiltonian Formulation for Dynamics of Hypergeometric Orthogonal Polynomials · 0.88
The present paper supplements the earlier results [ref] .

Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states · Summary and Comments · 0.94

Uses its method or tool 7 papers

A complete set of exact Heisenberg solutions for `sinusoidal coordinates' [ref] , as many as the degree of freedom, is derived elementarily in terms of the universal Lax pair [ref] , which is a well-established solution mechanism for classical and quantum Calogero-Sutherland-Moser systems [ref] based on any root system [ref] .

Exact Heisenberg operator solutions for multi-particle quantum mechanics · Introduction · 0.80
From the closure relation [ref], the exact Heisenberg operator solution of the sinusoidal coordinate $ℰ$ can be obtained [ref] , [equation] where $a^(±)=a^(±)(ℋ^X dual_𝒟, ℰ )$ are [equation] Note that square roots in $α_±(ℋ^X dual_𝒟)$ are well-defined, because the matrix $R_1(ℋ^X dual_𝒟)^2 +4R_0(ℋ^X dual_𝒟)$ is positive semi-definite, see [ref].

Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials · Closure relation · 0.88
The closure relation [ref] allows us to obtain the exact Heisenberg operator solution for $η(x)$, and the annihilation and creation operators $a^(±)$ are extracted from this exact Heisenberg operator solution [ref] .

Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm · Closure relation · 0.90

Compares with it 1 paper

The two types of creation-annihilation operators are closely related with each other [ref] [equation] with [equation] and the operators $X$ and $X^†$ map the eigenfunction $ϕ_n$ to itself: [equation] The structure of these operators is better understood by the similarity transformation in terms of the groundstate wavefunction $ϕ_0$ [equation] In fact, their actions on polynomials ${H_n(cosx|q)}$ are essentially identical: [equation] The main part of $X$ and $X^†$, defined by [equation] was also introduced by Atakishiyev-Klimyk [ref] eq(9).

q-oscillator from the q-Hermite Polynomial · Heisenberg operator solution · 0.87

Cites it as support 2 papers

As shown in previous work [ref] and in [ref], the essential information on exact solvability is contained only in the closure relation [ref].

Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism · Askey-Wilson algebra · 0.98
As mentioned repeatedly, these standard properties are well satisfied in explicit examples of discrete QM in [ref] .

Crum's Theorem for `Discrete' Quantum Mechanics · Summary and Comments · 0.97

Cites it as background 16 papers

These orthogonal polynomials can be studied in the framework of idQM, in which they appear as part of the eigenfunction as follows: [equation] Here $η(x)$ is a sinusoidal coordinate [ref] and $P_n(η)$ is a orthogonal polynomial of degree $n$ in $η$.

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types · Discrete Quantum Mechanics With Pure Imaginary Shifts and Deformed Systems · 0.81
The exactly solvable quantum mechanical systems described by the classical orthogonal polynomials in the Askey scheme have two properties, shape invariance and closure relation [ref] .

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators · Introduction · 0.89
The eigenfunctions of the Hamiltonian $ϕ_n(x)=ϕ_n(x ;λ)$ have the following general structure: [equation] Here $P_n$ is a polynomial in $η(x)$, which is called a sinusoidal coordinate [ref] .

New Quasi Exactly Solvable Difference Equation · General Setting · 0.89

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [4]

M. M. Nieto and L. M. Simmons, Jr., “Coherent States For Genera l Potentials”, Phys. Rev. Lett. 41 (1978) 207-210; “Coherent States For General Potentials”. 1. Formalism, 41 Phys. Rev. D 20 (1979) 1321-1331; 2. Conﬁning One-Dimensional Examples, Phys. Rev. D 20 (1979) 1332-1341; 3. Nonconﬁning One-Dimensional Examples, Ph ys. Rev. D 20 (1979) 1342-1350

View on Pith 1978

[2] [1]

See, for example: L. Infeld and T. E. Hull, “The factorization met hod”, Rev. Mod. Phys. 23 (1951) 21-68. F. Cooper, A. Khare and U. Sukhatme, “Supersym metry and quantum mechanics”, Phys. Rept. 251 (1995) 267-385

View on Pith 1951

[3] [2]

S. Odake and R. Sasaki, “Shape Invariant Potentials in ‘Discrete’ Quantum Mechanics”, J. Nonlinear Math. Phys. 12 Suppl. 1 (2005) 507-521; “Calogero-Sutherland-Moser Sys- tems, Ruijsenaars-Schneider-van Diejen Systems and Orthogon al Polynomials”, Prog. Theor. Phys. 114 (2005) 1245-1260

View on Pith 2005

[4] [3]

S. Odake and R. Sasaki, “Equilibria of ’discrete’ integrable system s and deformations of classical orthogonal polynomials”, J. Phys. A37 (2004) 11841-11876; “Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials”, J. Math . Phys. 46 (2005) 063513, 10 pages

View on Pith 2004

[5] [5]

T. Koornwinder, “The structure relation for Askey-Wilson polyn omials”, arXiv: math.CA/0601303

View on Pith arXiv

[6] [6]

R. Koekoek and R. F. Swarttouw, “The Askey-scheme of hyper geometric orthogonal polynomials and its q-analogue”, math.CA/9602214; G. E. Andrews, R. Askey and R. Roy, “ Special Functions ”, Encyclopedia of mathematics and its applications, Cam- bridge, (1999)

View on Pith arXiv 1999

[7] [7]

G. Szeg¨ o, “Orthogonal polynomials ”, Amer. Math. Soc. New York (1939)

View on Pith 1939

[8] [8]

L. E. Gendenshtein, “Derivation of exact spectra of the Schro dinger equation by means of supersymmetry”, JETP Lett. 38 (1983) 356-359; G. Levai, “A Search For Shape Invariant Solvable Potentials”, J. Phys. A22 (1989) 689-702

View on Pith 1983

[9] [9]

T. Fukui and N. Aizawa, “Shape-invariant potentials and an asso ciated coherent state”, Phys. Lett. A180 (1993) 308–313; J.-P. Gazeau and J. R. Klauder, “Coherent stat es for systems with discrete and continuous spectrum”, J. Phys. A32 (1999) 123–132; J.-P. An- toine, J.-P. Gazeau, P. Monceau, J. R. Klauder and K. A. Penson, “ Temporally stable coherent s...

View on Pith Pith review arXiv 1993

[10] [10]

F. Calogero, “Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials”, J. Math. Phys. 12 (1971) 419-436; B. Sutherland, “Exact results for a quantum many-body problem in one-dimension. II”, Phys. Rev. A5 (1972) 1372-1376

View on Pith 1971

[11] [11]

S. P. Khastgir, A. J. Pocklington and R. Sasaki, “Quantum Calog ero-Moser Models: Integrability for all Root Systems”, J. Phys. A33 (2000) 9033-9064. 42

View on Pith 2000

[12] [12]

A. Degasperis and S. N. M. Ruijsenaars, “Newton-Equivalent H amiltonians for the Harmonic Oscillator”, Ann. of Phys. 293 (2001) 92–109

View on Pith 2001

[13] [13]

S. N. M Ruijsenaars and H. Schneider, “A New Class Of Integrab le Systems And Its Relation To Solitons”, Ann. of Phys. 170 (1986) 370-405; J. F. van Diejen, “Integrability of diﬀerence Calogero-Moser systems”, J. Math. Phys. 35 (1994) 2983-3004

View on Pith 1986

[14] [14]

G. S. Agarwal and S. Chaturvedi, “Calogero-Sutherland oscilla tor: classical behaviour and coherent states”, J. Phys. A28 (1995) 5747-5755

View on Pith 1995

[15] [15]

P. J. Gambardella, “Exact results in quantum many-body syste ms of interacting par- ticles in many dimensions with SU (1, 1) as the dynamical group”, J. Math. Phys. 16 (1975) 1172-1187

View on Pith 1975

[16] [16]

A. J. Bordner, N. S. Manton and R. Sasaki, “Calogero-Moser M odels V: Supersymmetry, and Quantum Lax Pair”, Prog. Theor. Phys. 103 (2000) 463-487. 43

View on Pith 2000