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Non-standard quantum algebras and infinite-dimensional PT-symmetric systems

Ángel Ballesteros, Romina Ramírez, Marta Reboiro

TL;DR

This work constructs a PT-symmetric, infinite-dimensional representation of the non-standard Uz(sl(2,R)) Hopf algebra and builds a multiparametric family of Hamiltonians from its generators. These Hamiltonians are systematically mapped to position-dependent-mass (PDM) systems via a similarity transformation and then recast, through point canonical transformations, as constant-mass Schrödinger problems with solvable potentials. The deformation parameter z is shown to tune the effective potentials from infinite barriers to double-well structures and, in the large-z limit, to harmonic-oscillator behavior, with analytical results in special cases and numerical/perturbative treatments otherwise. The framework is connected to physical models such as double-well and DWT potentials, including NH3 inversion and KHCO3 hydrogen bonding, highlighting the method’s relevance for molecular physics and its potential to illuminate exceptional points and non-PT phases.

Abstract

In this work, we introduce a PT-symmetric infinite-dimensional representation of the Uz(sl(2,R)) Hopf algebra, and we analyse a multiparametric family of Hamiltonians constructed from such representation of the generators of this non-standard quantum algebra. It is shown that all these Hamiltonians can be mapped to equivalent systems endowed with a position-dependent mass. From the latter presentation, it is shown how appropriate point canonical transformations can be further defined in order to transform them into Hamiltonians with constant mass over suitable domains. By following this approach, the bound-state spectrum and the corresponding eigenfunctions of the initial PT-symmetric Hamiltonians can be determined. It is worth stressing that a relevant feature of some of the new Uz(sl(2,R)) systems here presented is found to be their connection with double-well and Pöschl-Teller potentials. In fact, as an application we present a particular Hamiltonian that can be expressed as an effective double-well trigonometric potential, which is commonly used to model several relevant systems in molecular physics.

Non-standard quantum algebras and infinite-dimensional PT-symmetric systems

TL;DR

This work constructs a PT-symmetric, infinite-dimensional representation of the non-standard Uz(sl(2,R)) Hopf algebra and builds a multiparametric family of Hamiltonians from its generators. These Hamiltonians are systematically mapped to position-dependent-mass (PDM) systems via a similarity transformation and then recast, through point canonical transformations, as constant-mass Schrödinger problems with solvable potentials. The deformation parameter z is shown to tune the effective potentials from infinite barriers to double-well structures and, in the large-z limit, to harmonic-oscillator behavior, with analytical results in special cases and numerical/perturbative treatments otherwise. The framework is connected to physical models such as double-well and DWT potentials, including NH3 inversion and KHCO3 hydrogen bonding, highlighting the method’s relevance for molecular physics and its potential to illuminate exceptional points and non-PT phases.

Abstract

In this work, we introduce a PT-symmetric infinite-dimensional representation of the Uz(sl(2,R)) Hopf algebra, and we analyse a multiparametric family of Hamiltonians constructed from such representation of the generators of this non-standard quantum algebra. It is shown that all these Hamiltonians can be mapped to equivalent systems endowed with a position-dependent mass. From the latter presentation, it is shown how appropriate point canonical transformations can be further defined in order to transform them into Hamiltonians with constant mass over suitable domains. By following this approach, the bound-state spectrum and the corresponding eigenfunctions of the initial PT-symmetric Hamiltonians can be determined. It is worth stressing that a relevant feature of some of the new Uz(sl(2,R)) systems here presented is found to be their connection with double-well and Pöschl-Teller potentials. In fact, as an application we present a particular Hamiltonian that can be expressed as an effective double-well trigonometric potential, which is commonly used to model several relevant systems in molecular physics.
Paper Structure (14 sections, 96 equations, 7 figures)

This paper contains 14 sections, 96 equations, 7 figures.

Table of Contents

  1. Introduction.
  2. Formalism
  3. Non-standard $U_z(sl(2, \mathbb R))$ algebra
  4. PDM Hamiltonians and the PCT method
  5. The $sl(2,\mathbb{R})$ case
  6. Results and discussion
  7. Example 1. The case with $\mu_\pm$ and $\mu_0$ being constant functions
  8. Case 1: $\mu_+=0$.
  9. Case 2: $\mu_+ \neq 0$
  10. Example 2
  11. Example 3
  12. Transformations for a generic Hamiltonian
  13. Physical applications.
  14. Conclusions and outlook

Figures (7)

  • Figure 1: The figure displays the behaviour of the Potential,$\frac{2}{ \mu_-z}U(y,z)$ of \ref{['hex2']}, the first eigenvalues in units of $[\mu_- z]$ and the corresponding eigenfunctions, as a function of the scaled parameter $y=x \sqrt{z}$. The value of $\lambda$ and $\mu_0$ were fixed to $\lambda=2$ and $\mu_0=2$, respectively. We have taken $\mu_+=0$ for the left panel, $\mu_+=-26$ for the central panel and $\mu_+=26$ for the right panel.
  • Figure 2: The same as Figure \ref{['fig:f1']} for $\mu_0=3/2$.
  • Figure 3: The same as Figure \ref{['fig:f1']} for $\mu_0=1$
  • Figure 4: Absolute value of the difference between $E_n$ of \ref{['en']}$E_k^{FD}$ of \ref{['ekfd1']} as a function of $z$. We have taken $\lambda=2$ and $\mu_-=1$. For the upper Panels, (a) and (b), $\mu_0=0$, while for the lower Panels,(c) and (d), $\mu_0=2$. Left Panels, (a) and (c), correspond to values of $\mu_+=0.5$ and the right Panels, (b) and (d), to values of $\mu_+=10$, respectively. We have plotted dotted, dashed, dashed-dotted, and long-dashed lines for $n=0,~1,~2,~3$, respectively.
  • Figure 5: Potential $U(y,z)$ of \ref{['potej3']} as a function of $y$. We have considered the coupling constants values $\mu_0=3/2,~\mu_-=\mu_+=1$. Dashed-black, blue-, red- and solid-black curves correspond to values of $z=0, ~0.4,~0.6$, and the limit $z \rightarrow \infty$, respectively.
  • ...and 2 more figures