相关论文: Two exactly-solvable problems in one-dimensional q…

The q-deformed coherent states for a quantum particle on a circle are introduced and their properties investigated.

A procedure for constructing bound state potentials is given. We show that, under the natural conditions imposed on a radial eigenvalue problem, the only special cases of the general central potential, which are exactly solvable and have…

Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved…

Besides the standard quantum version of the Coulomb/Kepler problem, an alternative quantum model with not too dissimilar phenomenological (i.e., spectral and scattering) as well as mathematical (i.e., exact-solvability) properties may be…

Motivated by a recent analysis which presents explicitly the general solution, we consider the eigenvalue problem of the spinless Salpeter equation with a ("hard-core amended") Coulomb potential in one dimension. We prove the existence of a…

We reexamine the solvable model problem of two static, fundamental quarks interacting with a SU(2) Yang-Mills field on a spatial circle, introduced by Engelhardt and Schreiber. If the quarks are at the same point, the model exhibits a…

A correspondence between the sextic anharmonic oscillator and a pair of third-order ordinary differential equations is used to investigate the phenomenon of quasi-exact solvability for eigenvalue problems involving differential operators…

The cosmological Friedmann equation for the universe filled with a scalar field is reduced to a system of two equations of the first order, one of which is an equation with separable variables. For the second equation the exact solutions…

Solvable loop quantum cosmology provides a simple model of spatially flat, homogeneous, and isotropic quantum cosmology where the initial singularity is resolved. A close inspection of the literature reveals that there exist two different…

We investigate the spectral and symmetry properties of a quantum particle moving on a circle with a pointlike singularity (or point interaction). We find that, within the U(2) family of the quantum mechanically allowed distinct…

We construct a quantum mechanical model of the Calogero type for the icosahedral group as the structural group. Exact solvability is proved and the spectrum is derived explicitly.

Exact solutions of Dirac equation in two spatial dimensions in the Coulomb field are obtained. Equation which determines the so-called critical charge of the Coulomb field is derived and solved for a simple model.

New inverse and half-inverse problems: {\it sliding problems} are introduced. In this way several physically important equations are recovered from the quantum defect. In particular, sliding problems are solved for radial Schr\"odinger…

A nonlinear model of the quantum harmonic oscillator on two-dimensional spaces of constant curvature is exactly solved. This model depends of a parameter $\la$ that is related with the curvature of the space. Firstly the relation with other…

It is demonstrated that quasi-exactly solvable models of quantum mechanics admit an interesting duality transformation which changes the form of their potentials and inverts the sign of all the exactly calculable energy levels. This…

In this paper, we investigate the two-point boundary value problems for linear wave equation defined on a circle and prove that the equation possesses the exact controllability. We also investigate the two-point boundary value problems for…

The nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem. Any classical supersymmetric system characterized by the nonlinear in the Hamiltonian superalgebra is symplectomorphic to…

The Levi-Civita transformation is applied in the two-dimensional (2D) Dirac and Klein-Gordon (KG) equations with equal external scalar and vector potentials. The Coulomb and harmonic oscillator problems are connected via the Levi-Civita…

This review is devoted to the problem of Coulomb (dyon)-oscillator duality in non-relativistic quantum mechanics, which is based on the so-called non-bijective quadratic transformations, i.e. Levi-Civita transformation,…

We give a self-contained presentation and comparison of two different algorithms to explicitly solve quantum many body models of indistinguishable particles moving on a circle and interacting with two-body potentials of $1/\sin^2$-type. The…