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

Abstract In this work, we introduce a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">PT</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> -symmetric infinite-dimensional representation of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>z</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">PT</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> -symmetric Hamiltonians can be determined. It is worth stressing that a relevant feature of some of the new <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>z</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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.