Third-order ladder operators, generalized Okamoto and exceptional
orthogonal polynomials
- URL: http://arxiv.org/abs/2101.12313v2
- Date: Mon, 15 Feb 2021 09:16:31 GMT
- Title: Third-order ladder operators, generalized Okamoto and exceptional
orthogonal polynomials
- Authors: V\'eronique Hussin, Ian Marquette, and Kevin Zelaya
- Abstract summary: We extend and generalize the construction of Sturm-Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-in condition.
We identify a link between the eigenfunctions of the Hamiltonian operator and a special family of exceptional Hermite.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We extend and generalize the construction of Sturm-Liouville problems for a
family of Hamiltonians constrained to fulfill a third-order shape-invariance
condition and focusing on the "$-2x/3$" hierarchy of solutions to the fourth
Painlev\'e transcendent. Such a construction has been previously addressed in
the literature for some particular cases but we realize it here in the most
general case. The corresponding potential in the Hamiltonian operator is a
rationally extended oscillator defined in terms of the conventional Okamoto
polynomials, from which we identify three different zero-modes constructed in
terms of the generalized Okamoto polynomials. The third-order ladder operators
of the system reveal that the complete set of eigenfunctions is decomposed as a
union of three disjoint sequences of solutions, generated from a set of
three-term recurrence relations. We also identify a link between the
eigenfunctions of the Hamiltonian operator and a special family of exceptional
Hermite polynomial.
Related papers
- Butson Hadamard matrices, bent sequences, and spherical codes [15.98720468046758]
We explore a notion of bent sequence attached to the data consisting of an Hadamard matrix of order $n$ defined over the complex $qth$ roots of unity.
In particular we construct self-dual bent sequences for various $qle 60$ and lengths $nle 21.$ construction methods comprise the resolution of systems by Groebner bases and eigenspace computations.
arXiv Detail & Related papers (2023-11-01T08:03:11Z) - Generalized $ \left\{ h (1) \oplus h(1) \right\} \uplus u(2) $
commensurate anisotropic Hamiltoninan and ladder operators; energy spectrum,
eigenstates and associated coherent and squeezed states [0.0]
Several families of generalized Hamiltonian systems are found.
Explicit expressions for the normalized eigenstates of the Hamiltonian and its associated lowering operator are given.
arXiv Detail & Related papers (2023-06-13T16:30:56Z) - Rationally-extended Dunkl oscillator on the line [0.0]
It is shown that the extensions of exactly-solvable quantum mechanical problems connected with the replacement of ordinary derivatives by Dunkl ones can be easily combined.
arXiv Detail & Related papers (2023-04-12T13:20:37Z) - Some Remarks on the Regularized Hamiltonian for Three Bosons with
Contact Interactions [77.34726150561087]
We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions.
In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $mathcal H$ can be constructed.
We show that the threshold value $gamma_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $gammagamma_c$.
arXiv Detail & Related papers (2022-07-01T10:01:14Z) - Krylov complexity and orthogonal polynomials [30.445201832698192]
Krylov complexity measures operator growth with respect to a basis, which is adapted to the Heisenberg time evolution.
The construction of that basis relies on the Lanczos method of recursion.
arXiv Detail & Related papers (2022-05-25T14:40:54Z) - On the general family of third-order shape-invariant Hamiltonians
related to generalized Hermite polynomials [0.0]
This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermites.
It is achieved by exploiting the intrinsic relation between third-order shape-invariant Hamiltonians and the fourth Painlev'e equation.
arXiv Detail & Related papers (2022-03-10T20:45:37Z) - Polynomial algebras of superintegrable systems separating in Cartesian
coordinates from higher order ladder operators [0.618778092044887]
We introduce the general algebras characterizing a class of higher order superintegrable systems that separate in coordinates.
The construction relies on underlying Heisenberg algebras and their defining higher order ladder operators.
arXiv Detail & Related papers (2022-02-27T03:33:26Z) - Progressive approximation of bound states by finite series of
square-integrable functions [0.0]
We use the "tridiagonal representation approach" to solve the time-independent Schr"odinger equation for bound states in a basis set of finite size.
arXiv Detail & Related papers (2022-02-20T00:25:35Z) - Intrinsic decoherence dynamics in the three-coupled harmonic oscillators
interaction [77.34726150561087]
We give an explicit solution for the complete equation, i.e., beyond the usual second order approximation used to arrive to the Lindblad form.
arXiv Detail & Related papers (2021-08-01T02:36:23Z) - Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range.
For a large part of the presentation, infinite scattering length will be considered.
arXiv Detail & Related papers (2020-06-03T17:54:43Z) - A refinement of Reznick's Positivstellensatz with applications to
quantum information theory [72.8349503901712]
In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares.
Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
arXiv Detail & Related papers (2019-09-04T11:46:26Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.