$m$-step rational extensions of the trigonometric Darboux-Pöschl-Teller potential based on para-Jacobi polynomials

• URL: http://arxiv.org/abs/2410.05003v1
• Date: Mon, 7 Oct 2024 13:03:02 GMT
• Title: $m$-step rational extensions of the trigonometric Darboux-Pöschl-Teller potential based on para-Jacobi polynomials
• Authors: Yves Grandati, Christiane Quesne,
• Abstract summary: A previous construction of regular rational extensions of the trigonometric Darboux-P"oschler potential, obtained by one-step Darboux, is studied. Some novel families of exceptional transformations depending on $m$ discrete parameters are studied. The restrictions imposed on these parameters by the rational extensions regularity conditions are studied in detail.
• Score: 0.20482269513546453
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A previous construction of regular rational extensions of the trigonometric Darboux-P\"oschl-Teller potential, obtained by one-step Darboux transformations using seed functions associated with the para-Jacobi polynomials of Calogero and Yi, is generalized by considering $m$-step Darboux transformations. As a result, some novel families of exceptional orthogonal polynomials depending on $m$ discrete parameters, as well as $m$ continuous real ones $\lambda_1$, $\lambda_2$, \ldots, $\lambda_m$, are obtained. The restrictions imposed on these parameters by the rational extensions regularity conditions are studied in detail.

Related papers

• Efficient unitary designs and pseudorandom unitaries from permutations [35.66857288673615]
  We show that products exponentiated sums of $S(N)$ permutations with random phases match the first $2Omega(n)$ moments of the Haar measure. The heart of our proof is a conceptual connection between the large dimension (large-$N$) expansion in random matrix theory and the method.
  arXiv  Detail & Related papers  (2024-04-25T17:08:34Z)
• One continuous parameter family of Dirac Lorentz scalar potentials associated with exceptional orthogonal polynomials [3.8415024264641624]
  We get one $(lambda)$ family of rationally extended Dirac Lorentz scalar potentials with their explicit solutions in terms of $X_$ exceptional.
  arXiv  Detail & Related papers  (2023-09-22T16:02:35Z)
• Rational extensions of the Dunkl oscillator in the plane and exceptional orthogonal polynomials [0.0]
  It is shown that rational extensions of the isotropic Dunkl oscillator in the plane can be obtained by adding some terms. In the latter, it becomes an anisotropic potential, whose explicit form has been found in the simplest case.
  arXiv  Detail & Related papers  (2023-05-09T14:23:14Z)
• 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)
• $O(N^2)$ Universal Antisymmetry in Fermionic Neural Networks [107.86545461433616]
  We propose permutation-equivariant architectures, on which a determinant Slater is applied to induce antisymmetry. FermiNet is proved to have universal approximation capability with a single determinant, namely, it suffices to represent any antisymmetric function. We substitute the Slater with a pairwise antisymmetry construction, which is easy to implement and can reduce the computational cost to $O(N2)$.
  arXiv  Detail & Related papers  (2022-05-26T07:44:54Z)
• Conditions for realizing one-point interactions from a multi-layer structure model [77.34726150561087]
  A heterostructure composed of $N$ parallel homogeneous layers is studied in the limit as their widths shrink to zero. The problem is investigated in one dimension and the piecewise constant potential in the Schr"odinger equation is given.
  arXiv  Detail & Related papers  (2021-12-15T22:30:39Z)
• Tight High Probability Bounds for Linear Stochastic Approximation with Fixed Stepsize [41.38162218102825]
  This paper provides a non-asymptotic analysis of linear approximation (LSA) algorithms with fixed stepsize. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence $(bf A_n, bf b_n): n in mathbbN*$. We show that our conclusions cannot be improved without additional assumptions on the sequence $bf A_n: n in mathbbN*$.
  arXiv  Detail & Related papers  (2021-06-02T16:10:37Z)
• Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials [0.0]
  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.
  arXiv  Detail & Related papers  (2021-01-28T22:54:56Z)
• On the Rademacher Complexity of Linear Hypothesis Sets [45.06091849856641]
  We present a tight analysis of the empirical Rademacher complexity of the family of linear hypothesis classes with weight vectors bounded in $ell_p$-norm for any $p geq 1$. This provides a tight analysis of generalization using these hypothesis sets and helps derive sharp data-dependent learning guarantees.
  arXiv  Detail & Related papers  (2020-07-21T19:08:21Z)
• One parameter family of rationally extended isospectral potentials [7.343280016515051]
  We obtain one continuous $lambda$ family of rationally extended strictly isospectral potentials. In the special case of $lambda = 0$ and $-1$, we obtain two new exactly solvable rationally extended potentials.
  arXiv  Detail & Related papers  (2020-04-28T13:15:12Z)
• 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.