Two-dimensional integrable systems with position-dependent mass via complex holomorphic functions

We study the non-relativistic charge-monopole system when the charged particle has a position-dependent mass written as M(r) = m0rw. The angular wave functions are the well-known monopole harmonics, and the radial ones are ordinary Bessel functions which depend on the magnetic and electric charge product as well as on the w parameter. We investigate mappings—approximate and exact—between the charge-monopole system with constant mass and the charge with a position-dependent mass solving the position-dependent mass Schrödinger equation for the mass distribution.

This paper is a continuation of our investigations of the dynamics and integrability of a relativistic particle moving under the influence of potential forces. Here, we examine particle motion in a general two-dimensional Riemannian manifold subjected to an external potential. In the limit of small momentum this system transforms into the corresponding non-relativistic system in a curved space extensively investigated in recent years. Our integrability results are formulated for relativistic systems with the metric and potential which are homogeneous functions of degrees n and k, respectively. First, we show that the integrability of a non-relativistic Hamiltonian is a necessary condition for the integrability of its relativistic version. Next, we derive the necessary integrability conditions from the analysis of the differential Galois groups of variational equations along two different particular solutions. These conditions, together with those for non-relativistic counterparts, give the final integrability obstructions. Obtained obstructions are used to explain integrability results for relativistic versions of well-known classical systems: the Elmandouh system with variable Gaussian curvature, the Hénon–Heiles-type systems, the position-dependent mass systems and the swinging Atwood machine. We compare the dynamics of the classical and corresponding relativistic versions systems in curved space using Poincaré sections and demonstrate that relativistic corrections frequently induce chaos. Moreover, we prove the non-integrability of these systems using the obtained necessary integrability conditions. Unexpectedly, we have also identified a new family of relativistic integrable Hamiltonian systems with position-dependent mass, which extends both classical integrable potentials and conditionally integrable systems defined at specific energy levels. Our results provide a deeper understanding of integrability and chaos in relativistic mechanics, which is currently of great importance for mathematical physics and astrophysics.

This study examines the phenomenon of vibrational resonance (VR) in a classical positiondependent mass (PDM) system characterized by three types of single-well potentials. These potentials are influenced by an amplitude-modulated (AM) signal with $\Omega\gg\omega$. Our analysis is limited to the following parametric choices: <br> (i) $\omega_0^2$, $\beta$, $m_0$, $\lambda>0$ (type-1 single-well), <br> (ii) $\omega_0^2>0$, $\beta <0$, $2< m_0 <3$, $1< \lambda <2$ (type-2 single-well), <br> (iii) $\omega_0^2>0$, $\beta <0$, $0< m_0 <2$, $0<\lambda<1$ (type-3 single-well). <br> The system presents an intriguing scenario in which the PDM function significantly contributes to the occurrence of VR. In addition to the analytical derivation of the equation for slow motions of the system based on the high-frequency signal’s parameters using the method of direct separation of motion, numerical evidence is presented for VR and its basic dynamical behaviors are investigated. Based on the findings presented in this paper, the weak low-frequency signal within the single-well PDM system can be either attenuated or amplified by manipulating PDM parameters, such as mass amplitude ($m_0$) and mass spatial nonlinearity $\lambda$. The outcomes of the analytical investigations are validated and further supported through numerical simulations.

PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative PDM-Hamiltonian

Complex holomorphic functions are defined using a complex derivative. In higher dimensions the meaningful generalization of complex derivative is not straight forward. Sudbery defined a derivative for quaternion regular functions using differential forms. Gurlebeck and Malonek generalized that for monogenic functions. In this paper we find similar characterizations for k-hypermonogenic functions which are holomorphic functions based on the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}}{x_{2}^{2k}}. \end{aligned}$$ When \(k=0\), we obtain the hypercomplex derivative by Gurlebeck and Malonek. Just like in the complex case derivative of k-hypermonogenic is the usual partial derivative with respect to the first coordinate.

We focus on integrable systems with two degrees of freedom that are integrable in the Liouville sense and are obtained as real and imaginary parts of a polynomial (or entire) complex function in two complex variables. We propose definitions of the actions for such systems (which are not of the Arnol'd-Liouville type). We show how to compute the actions from a complex Hamilton-Jacobi equation and apply these techniques to several examples including those recently considered in relation to perturbations of the Ruijsenaars-Schneider system. These examples introduce the crucial problem of the semiclassical approach to the corresponding quantum systems.

We examine the problem of integrability of two-dimensional Hamiltonian systems by means of separation of variables. The systematic approach to construction of the special non-pure coordinate separation of variables for certain natural two-dimensional Hamiltonians is presented. The relations with SUSY quantum mechanics are discussed.

We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schrödinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering, we show that the class of systems can also be exactly solvable and is also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Liénard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

PDM-charged particles in PD-magnetic plus Aharonov–Bohm flux fields: Unconfined “almost-quasi-free” and confined in a Yukawa plus Kratzer exact solvability

Analysis of vibrational resonance in an oscillator with exponential mass variation

The dynamics of position-dependent mass systems is considered from both, classical and quantum mechanical points of view, by means of the factorization method. Some examples are presented, with particular choices of the mass function, for the harmonic oscillator in order to illustrate our results. In the quantum regime, new isospectral position-dependent mass potentials are also constructed by the intertwiningte chnique.

Using an adaptation of the Gelfand-Dickey analysis of two-dimensional integrable Hamiltonian systems to the harmonic superspace (HS), we derive a class of completely solvable four-dimensional N = 2 hypermultiplet self-couplings. The known harmonic non-localities are shown to factor out in the HS integrable models. As an illustration, we rederive the Taub-Nut result and obtain a new hyper-Kähler metric associated with a HS superpotential having a pure imaginary coupling constant. A comparison with classes of metrics in the literature is given.

We consider a two-dimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence of a second order integral of motion does not guarantee the separation of variables in the Schrödinger equation. We introduce the concept of “quasiseparation of variables” and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra.

Recent results on two-dimensional integrable Hamiltonian systems

A second-order supersymmetric approach is taken to the system describing motion of a quantum particle in a potential endowed with position-dependent effective mass. It is shown that the intertwining relations between second-order partner Hamiltonians may be exploited to obtain a simple shape-invariant condition. Indeed, a novel relation between potential and mass functions is derived, which leads to a class of exactly solvable models. As an illustration of our procedure, two examples are given for which one obtains whole spectra algebraically. Both shape-invariant potentials exhibit harmonic-oscillator-like or singular-oscillator-like spectra depending on the values of the shape-invariant parameter.