Related papers: Extensions of natural Hamiltonians
The technique of Caffarelli and Silvestre, characterizing the fractional Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an elliptic equation in the upper half space with one extra spatial dimension, is shown to hold…
We discuss representation of certain functions of the Laplace operator $\Delta$ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $(-\Delta)^{1/2}$, the square root of the…
Superintegrable Hamiltonian systems in a two-dimensional Euclidean space are considered. We present all real standard potentials that allow separation of variables in polar coordinates and admit an independent fourth-order integral of…
Effective Hamiltonians are usually constructed by using canonical transformations or projection techniques. In contrast to this, we present a method for systems with arbitrary Hilbert space based on the introduction of cumulants. Cumulants…
The superintegrability of four Hamiltonians $\tilde{H_r} = \lambda\, H_r$, $r=a,b,c,d$, where $H_r$ are known Hamiltonians and $\lambda$ is a certain function defined on the configuration space and depending of a parameter $\kappa$, is…
While dealing in [1] with the supersymmetry of a tridiagonal Hamiltonian H, we have proved that its partner Hamiltonian H(+) also have a tridiagonal matrix representation in the same basis and that the polynomials associated with the…
$M$-dimensional extended objects $\Sigma$ can be described by projecting a Diff $\Sigma$ invariant Hamiltonian of time-independent Hamiltonian density {\cal H} onto the Diff $\Sigma$- singlet sector, which after Hamiltonian reduction, using…
It is shown that the standard formulation of quantum mechanics in terms of Hermitian Hamiltonians is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but…
We provide a natural generalization to submanifolds of the holographic method used to extract higher-order local invariants of both Riemannian and conformal embeddings, some of which depend on a choice of parallelization of the normal…
For a given boundary sequence $a=(a_n)_{n\in\mathbb{Z}}$, we construct harmonic extensions $U,V:\mathbb{Z}\times\ \mathbb{N}\to \mathbb{R}$ that serve as discrete analogs of the Poisson and conjugate-Poisson integrals. The construction is…
We discuss four off-shell N=4 D=1 supersymmetry transformations, their associated one-dimensional sigma-models and their mutual relations. They are given by I) the (4,4)_{lin} linear supermultiplet (supersymmetric extension of R^4), II) the…
In these three lectures we will discuss some fundamental aspects of the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on compact Riemannian manifolds with smooth boundary emphasizing the relation…
Described is n-level quantum system realized in the n-dimensional ''Hilbert'' space H with the scalar product G taken as a dynamical variable. The most general Lagrangian for the wave function and G is considered. Equations of motion and…
Recently, there has been an increasing interest in modelling and computation of physical systems with neural networks. Hamiltonian systems are an elegant and compact formalism in classical mechanics, where the dynamics is fully determined…
An N-dimensional position-dependent mass Hamiltonian (depending on a parameter \lambda) formed by a curved kinetic term and an intrinsic oscillator potential is considered. It is shown that such a Hamiltonian is exactly solvable for any…
In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincar\'e algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the…
In two recent papers [N. Aizawa, Y. Kimura, J. Segar, J. Phys. A 46 (2013) 405204] and [N. Aizawa, Z. Kuznetsova, F. Toppan, J. Math. Phys. 56 (2015) 031701], representation theory of the centrally extended l-conformal Galilei algebra with…
The two-dimensional extension of the one-dimensional PDM-Lagrangians and their nonlocal point transformation mappings into constant unit-mass exactly solvable Lagrangians is introduced. The conditions on the related two-dimensional…
Motivated mainly by certain interesting recent extensions of the Gamma, Beta and hypergeometric functions, we introduce here new extensions of the Beta function, hypergeometric and confluent hypergeometric functions. We systematically…
We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…