Related papers: Shape Invariance and Its Connection to Potential A…
Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized…
A non-associative algebra of observables cannot be represented as operators on a Hilbert space, but it may appear in certain physical situations. This article employs algebraic methods in order to derive uncertainty relations and…
We show that the conditional shape invariance symmetry can be used as a very powerful tool to calculate the eigenvalues of the mixed potential V (r) = ar + br^2 +c/r + l(l+1)/r^2 for a restricted set of potential parameters. The energy for…
General non-commutative supersymmetric quantum mechanics models in two and three dimensions are constructed and some two and three dimensional examples are explicitly studied. The structure of the theory studied suggest other possible…
A number of physical systems exhibit a particular form of asymptotic conformal invariance: within a particular range of distances, they are characterized by a long-range conformal interaction (inverse square potential), the absence of…
The procedure commonly used in textbooks for determining the eigenvalues and eigenstates for a particle in an attractive Coulomb potential is not symmetric in the way the boundary conditions at $r=0$ and $r \rightarrow \infty$ are…
In this work we use the deformation procedure and explore the route to obtain distinct field theory models that present similar stability potentials. Starting from systems that interact polynomially or hyperbolically, we use a deformation…
Affine transformations (dilatations and translations) are used to define a deformation of one-dimensional $N=2$ supersymmetric quantum mechanics. Resulting physical systems do not have conserved charges and degeneracies in the spectra.…
A nonrelativistic approach to quantum gravity is studied. At least for weak gravitational fields it should be a valid approximation. Such an approach can be used to point out problems and prospects inherent in a more exact theory of quantum…
The study of the convergence of power series expansions of energy eigenvalues for anharmonic oscillators in quantum mechanics differs from general understanding, in the case of quasi-exactly solvable potentials. They provide examples of…
Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…
Most approaches towards a quantum theory of gravitation indicate the existence of a minimal length scale of the order of the Planck length. Quantum mechanical models incorporating such an intrinsic length scale call for a deformation of…
The use of geometric invariants has recently played an important role in the solution of classification problems in non-commutative ring theory. We construct geometric invariants of non-commutative projectivizations, a significant class of…
We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically…
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…
In this work, we delve into the theory of sheared potentials in non-relativistic quantum mechanics. After defining what we mean by a family of sheared potentials, we consider these families in two particular but emblematic cases, the…
With the present trend in experimental particle physics of probing yet shorter distances and with the requirement on the theoretical side of renormalizability, conformal invariance becomes an attractive symmetry for particle interactions.…
Second order integrals of motion for 3d quantum mechanical systems with position dependent masses (PDM) are classified. Namely, all PDM systems are specified which, in addition to their rotation invariance, admit at least one second order…
We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…