26 problems

Quantum-integrability necessity conjecture for exact solvability of time-dependent Hamiltonians
arXiv@arxiv
A time-dependent quantum Hamiltonian is called quantum integrable when it possesses nontrivial parameter-dependent commuting partners. Quantum-integrability necessity conjecture. Q…
Conjectureopenexact-solvability quantum-integrability time-dependent-quantum-systems
0
Tremblay–Turbiner–Winternitz conjecture on exact solvability
arXiv@arxiv
A quantum system with $n$ degrees of freedom is called maximally superintegrable when it possesses $2n-1$ integrals of motion, the maximum possible number. An exactly solvable quan…
Conjectureopenexact-solvability quantum-mechanics superintegrability
0
The conjecture that exact solvability is widespread among super-integrable systems
arXiv@arxiv
Exact-solvability conjecture. Exactly solvability should hold quite widely for super-integrable systems.
Conjectureopenexact-solvability quantum-integrable-systems super-integrability
1
The conjecture that all maximally superintegrable systems are exactly solvable
arXiv@arxiv
Superintegrability–exact-solvability conjecture. All maximally superintegrable systems are exactly solvable.
Conjectureopenexact-solvability mathematical-physics superintegrability
0
The Montreal conjecture for second-order superintegrable systems in Euclidean space
arXiv@arxiv
Let $E^n$ be $n$ -dimensional Euclidean space, and consider superintegrable systems whose integrals of motion are differential operators of order at most two. The Montreal conjectur…
Conjectureopeneuclidean-space exact-solvability superintegrability
1
The Montreal conjecture for second-order superintegrable systems in Euclidean space
arXiv@arxiv
Let $E^n$ denote $n$ -dimensional Euclidean space, and consider quantum superintegrable systems on $E^n$ whose integrals of motion have differential order at most two. The Montreal…
Conjectureopeneuclidean-space exact-solvability superintegrability
0
The Montreal conjecture on exact solvability of maximally superintegrable quantum systems
arXiv@arxiv
A maximally superintegrable quantum system in flat space is a quantum system whose number of functionally independent integrals of motion is maximal possible; an exactly-solvable s…
Conjectureopenexact-solvability quantum-mechanics superintegrability
2
Conjecture on new classical superintegrability and quantum exact solvability
arXiv@arxiv
Consider the classical Hamiltonian family $H^{(cl)}_n$ and its quantum version $H^{(q)}_n$ , with parameters as introduced in the paper. The authors distinguish known reductions fro…
Conjectureopenexact-solvability hamiltonian-systems superintegrability
0
Conjecture on exact solvability in the quantum Hamiltonian family
arXiv@arxiv
Let $H^{(q)}_n$ denote the quantum version of the Hamiltonian family introduced in the paper, with the same parameter space as the classical family. Quantum exact-solvability conje…
Conjectureopenexact-solvability quantum-mechanics
1
Patra's commutation-relations conjecture for multistate Landau–Zener models
arXiv@arxiv
A multistate Landau–Zener model is a quantum system whose Hamiltonian depends linearly on time, and a model is called solvable when its transition probabilities between asymptotic…
Conjectureopenexact-solvability landau-zener-models quantum-dynamics
1
SWKB exactness beyond conventional shape-invariant potentials
arXiv@arxiv
Let the SWKB condition be the supersymmetric WKB quantization condition for a potential, and let a conventional shape-invariant (SI) potential be one for which this condition is ex…
Conjectureopenexact-solvability semiclassical-quantization supersymmetric-quantum-mechanics
0
Nasuda's level-distribution conjecture for SWKB exactness
arXiv@arxiv
Let $I_{\mathrm{SWKB}}$ be the leading-order supersymmetric WKB quantization integral for a quantum system, and let the energy eigenvalues describe its level structure. Nasuda's co…
Conjectureopenexact-solvability semiclassical-quantization supersymmetric-quantum-mechanics
0
Shape invariance as a necessary condition for exactness of the SWKB condition
arXiv@arxiv
Let $V(x)$ be a supersymmetric quantum-mechanical potential with groundstate eigenfunction $\phi_0(x)$ , and let the SWKB quantization condition be … where $a$ and $b$ are the roots…
Conjectureopenexact-solvability semiclassical-quantization supersymmetric-quantum-mechanics
0
Non-orthogonal constant-coefficient, closure, and operator conjecture
arXiv@arxiv
Let $P_{\mathcal{D},n}$ be the multi-indexed Meixner–Pollaczek or continuous Hahn polynomial for an arbitrary index set $\mathcal{D}$ , including cases in which it is not orthogonal…
Conjectureopenexact-solvability orthogonal-polynomials recurrence-relations
0
Creation and annihilation operator conjecture for arbitrary-index multi-indexed eigenpolynomials
arXiv@arxiv
Let $\check{P}_{\mathcal{D},n}(x)$ be the eigenpolynomials of the multi-indexed system, let $\widetilde{\mathcal{H}}_{\mathcal{D}}$ be its similarity-transformed Hamiltonian, and l…
Conjectureopenexact-solvability orthogonal-polynomials special-functions
0
Order- $2L$ closure relation conjecture for multi-indexed systems
arXiv@arxiv
Let $X(\eta)$ be the polynomial used in the constant-coefficient recurrence theorem, with degree $L$ , and let $\alpha_j(z)$ ( $1\leq j\leq 2L$ ) be the corresponding spectral factors…
Conjectureopenexact-solvability orthogonal-polynomials special-functions
0
The exact-solvability conjecture for superintegrable systems
arXiv@arxiv
Superintegrable systems are dynamical systems possessing sufficiently many integrals of motion; in the context of this paper, the claim concerns such systems and their associated s…
Conjectureopenexact-solvability orthogonal-polynomials superintegrable-systems
5
Conjecture that solved multistate Landau–Zener models are limits of more complex solvable models
arXiv@arxiv
A multistate Landau–Zener model is a quantum system with diabatic levels whose energies depend linearly on time and whose couplings are time-independent. An already solved model is…
Conjectureopenexact-solvability landau-zener-models quantum-dynamics
1
Exact two-Schur correlator formula for the r-fold q-deformed logarithmic matrix model
arXiv@arxiv
The exact two-Schur correlator conjecture. The average of the two Schur-polynomial insertions satisfies
Conjectureopenexact-solvability matrix-models schur-functions
0
The conjecture that maximally superintegrable systems in Euclidean spaces are exactly solvable
arXiv@arxiv
Exact-solvability conjecture. All maximally superintegrable quantum systems in Euclidean spaces are exactly solvable.
Conjectureopenexact-solvability quantum-mechanics superintegrability
0
The hypergeometric reduction conjecture for exact solvability of Schrödinger equations
arXiv@arxiv
Hypergeometric reduction conjecture. Exact solvability of the Schrödinger equation should depend on whether the equation can be suitably reduced to the hypergeometric equation or t…
Conjectureopenexact-solvability schrodinger-equations special-functions
0
Tempesta–Turbiner–Winternitz conjecture on solvability of superintegrable systems
arXiv@arxiv
A two-dimensional system is maximally superintegrable if it has the maximal number of functionally independent integrals of motion compatible with superintegrability. Tempesta–Turb…
Conjectureopenexact-solvability mathematical-physics superintegrability
0
The exact-solvability conjecture for superintegrable systems
arXiv@arxiv
A superintegrable system is a Hamiltonian system that admits more integrals of motion than degrees of freedom; here the Hamiltonian is … with two additional integrals polynomial in…
Conjectureopenexact-solvability orthogonal-polynomials superintegrable-systems
4
The conjecture that all maximally superintegrable systems are exactly solvable
arXiv@arxiv
A quantum-mechanical system is maximally superintegrable if it possesses the maximal number of functionally independent integrals of motion compatible with its phase-space dimensio…
Conjectureopenexact-solvability superintegrability supersymmetry
0
The conjecture that maximally superintegrable scalar systems are exactly solvable
arXiv@arxiv
A scalar system is maximally superintegrable if it possesses the maximal number of functionally independent integrals of motion compatible with its degrees of freedom. The scalar s…
Conjectureopenexact-solvability integrable-systems superintegrability
0