Related papers: Square-integrable eigenfunctions in quantizing the…
It is known that a self-adjoint, time-independent hamiltonian can be defined for the quantum damped harmonic oscillator. We show here that the two vacua naturally associated to this operator, when expressed in terms of pseudo-bosonic…
We carry out an exact quantization of a PT symmetric (reversible) Li\'{e}nard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard…
New exact vacuum solutions with various singularities in the plane-symmetric spacetime are shown, and they are applied to the analysis of inhomogeneous cosmological models and colliding gravitational waves. One of the singularities can be…
The superintegrability of a rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) with non-linear "centrifugal" terms is studied. In the first part, the system is directly studied in the Euclidean…
Several completely integrable, indeed solvable, Hamiltonian many-body problems are exhibited, characterized by Newtonian equations of motion ("acceleration equal force"), with linear and cubic forces, in N-dimensional space (N being an…
A countable class of integrable dynamical systems, with four dimensional phase space and conserved quantities in involution (H\_n,I\_n) are exhibited. For $n=1$ we recover Neumann sytem on T*S^2. All these systems are also integrable at the…
Non-trivial outer algebra automorphisms may be utilized in $\lambda$-deformations of (gauged) WZW models thus providing an efficient way to construct new integrable models. We provide two such integrable deformations of the exact coset CFT…
We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal…
Spherically symmetric models of loop quantum gravity have been studied recently by different methods that aim to deal with structure functions in the usual constraint algebra of gravitational systems. As noticed by Gambini and Pullin, a…
We consider a general non-linear sigma model coupled to Einstein gravity and show that in spherical symmetry and for a simple realization of self-similarity, the spacetime can be completely determined. We also examine some more specific…
Spherical Bessel functions appear commonly in many areas of physics wherein there is both translation and rotation invariance, and often integrals over products of several arise. Thus, analytic evaluation of such integrals with different…
We formulate a scalar realization of Sciama's Machian programme within the general Bergmann-Wagoner class of scalar--tensor gravity. Starting from a universally conformally coupled matter sector, we rewrite the field equations in terms of…
Harmonic oscillator in noncommutative two dimensional lattice are investigated. Using the properties of non-differential calculus and its applications to quantum mechanics, we provide the eigenvalues and eigenfunctions of the corresponding…
According to Belinsky, Khalatnikov and Lifshitz, gravity near a space-like singularity reduces to a set of decoupled one-dimensional mechanical models at each point in space. We point out that these models fall into a class of conformal…
The classical and the quantal problem of a particle interacting in one-dimension with an external time-dependent quadratic potential and a constant inverse square potential is studied from the Lie-algebraic point of view. The integrability…
A class of three-dimensional models which satisfy supersymmetric intertwining relations with the simplest - oscillator-like - variant of shape invariance is constructed. It is proved that the models are not amenable to conventional…
Quantum superintegrable systems in two dimensions are obtained from their classical counterparts, the quantum integrals of motion being obtained from the corresponding classical integrals by a symmetrization procedure. For each quantum…
For fields that vary slowly on the scale of the lightest mass the logarithm of the vacuum functional can be expanded as a sum of local functionals, however this does not satisfy the obvious form of the Schr\"odinger equation. For…
A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the…
We show that quadratic Hamiltonians in involution coming from a St\"ackel system are quantizable, in the sense that one can construct commutative self-adjoint operators whose symbols are the quadratic Hamiltonians. Moreover, they allow…