491 research outputs found
Infinite matrices may violate the associative law
The momentum operator for a particle in a box is represented by an infinite
order Hermitian matrix . Its square is well defined (and diagonal),
but its cube is ill defined, because . Truncating these
matrices to a finite order restores the associative law, but leads to other
curious results.Comment: final version in J. Phys. A28 (1995) 1765-177
One-dimensional quasi-relativistic particle in the box
Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional
quasi-relativistic Hamiltonian (-h^2 c^2 d^2/dx^2 + m^2 c^4)^(1/2) + V_well(x)
(the Klein-Gordon square-root operator with electrostatic potential) with the
infinite square well potential V_well(x) is given: the n-th eigenvalue is equal
to (n pi/2 - pi/8) h c/a + O(1/n), where 2a is the width of the potential well.
Simplicity of eigenvalues is proved. Some L^2 and L^infinity properties of
eigenfunctions are also studied. Eigenvalues represent energies of a `massive
particle in the box' quasi-relativistic model.Comment: 40 pages, 4 figures; minor correction
Positivity and conservation of superenergy tensors
Two essential properties of energy-momentum tensors T_{\mu\nu} are their
positivity and conservation. This is mathematically formalized by,
respectively, an energy condition, as the dominant energy condition, and the
vanishing of their divergence \nabla^\mu T_{\mu\nu}=0. The classical Bel and
Bel-Robinson superenergy tensors, generated from the Riemann and Weyl tensors,
respectively, are rank-4 tensors. But they share these two properties with
energy momentum tensors: the Dominant Property (DP) and the divergence-free
property in the absence of sources (vacuum). Senovilla defined a universal
algebraic construction which generates a basic superenergy tensor T{A} from any
arbitrary tensor A. In this construction the seed tensor A is structured as an
r-fold multivector, which can always be done. The most important feature of the
basic superenergy tensors is that they satisfy automatically the DP,
independently of the generating tensor A. In a previous paper we presented a
more compact definition of T{A} using the r-fold Clifford algebra. This form
for the superenergy tensors allowed to obtain an easy proof of the DP valid for
any dimension. In this paper we include this proof. We explain which new
elements appear when we consider the tensor T{A} generated by a
non-degree-defined r-fold multivector A and how orthogonal Lorentz
transformations and bilinear observables of spinor fields are included as
particular cases of superenergy tensors. We find some sufficient conditions for
the seed tensor A, which guarantee that the generated tensor T{A} is
divergence-free. These sufficient conditions are satisfied by some physical
fields, which are presented as examples.Comment: 19 pages, no figures. Language and minor changes. Published versio
Improved quantum algorithms for the ordered search problem via semidefinite programming
One of the most basic computational problems is the task of finding a desired
item in an ordered list of N items. While the best classical algorithm for this
problem uses log_2 N queries to the list, a quantum computer can solve the
problem using a constant factor fewer queries. However, the precise value of
this constant is unknown. By characterizing a class of quantum query algorithms
for ordered search in terms of a semidefinite program, we find new quantum
algorithms for small instances of the ordered search problem. Extending these
algorithms to arbitrarily large instances using recursion, we show that there
is an exact quantum ordered search algorithm using 4 log_{605} N \approx 0.433
log_2 N queries, which improves upon the previously best known exact algorithm.Comment: 8 pages, 4 figure
Green's function for a Schroedinger operator and some related summation formulas
Summation formulas are obtained for products of associated Lagurre
polynomials by means of the Green's function K for the Hamiltonian H =
-{d^2\over dx^2} + x^2 + Ax^{-2}, A > 0. K is constructed by an application of
a Mercer type theorem that arises in connection with integral equations. The
new approach introduced in this paper may be useful for the construction of
wider classes of generating function.Comment: 14 page
Classical 5D fields generated by a uniformly accelerated point source
Gauge fields associated with the manifestly covariant dynamics of particles
in spacetime are five-dimensional. In this paper we explore the old
problem of fields generated by a source undergoing hyperbolic motion in this
framework. The 5D fields are computed numerically using absolute time
-retarded Green-functions, and qualitatively compared with Maxwell fields
generated by the same motion. We find that although the zero mode of all fields
coincides with the corresponding Maxwell problem, the non-zero mode should
affect, through the Lorentz force, the observed motion of test particles.Comment: 36 pages, 8 figure
Critical strength of attractive central potentials
We obtain several sequences of necessary and sufficient conditions for the
existence of bound states applicable to attractive (purely negative) central
potentials. These conditions yields several sequences of upper and lower limits
on the critical value, , of the coupling constant
(strength), , of the potential, , for which a first
-wave bound state appears, which converges to the exact critical value.Comment: 18 page
Green's function for the Hodge Laplacian on some classes of Riemannian and Lorentzian symmetric spaces
We compute the Green's function for the Hodge Laplacian on the symmetric
spaces M\times\Sigma, where M is a simply connected n-dimensional Riemannian or
Lorentzian manifold of constant curvature and \Sigma is a simply connected
Riemannian surface of constant curvature. Our approach is based on a
generalization to the case of differential forms of the method of spherical
means and on the use of Riesz distributions on manifolds. The radial part of
the Green's function is governed by a fourth order analogue of the Heun
equation.Comment: 18 page
The optimized Rayleigh-Ritz scheme for determining the quantum-mechanical spectrum
The convergence of the Rayleigh-Ritz method with nonlinear parameters
optimized through minimization of the trace of the truncated matrix is
demonstrated by a comparison with analytically known eigenstates of various
quasi-solvable systems. We show that the basis of the harmonic oscillator
eigenfunctions with optimized frequency ? enables determination of boundstate
energies of one-dimensional oscillators to an arbitrary accuracy, even in the
case of highly anharmonic multi-well potentials. The same is true in the
spherically symmetric case of V (r) = {\omega}2r2 2 + {\lambda}rk, if k > 0.
For spiked oscillators with k < -1, the basis of the pseudoharmonic oscillator
eigenfunctions with two parameters ? and {\gamma} is more suitable, and
optimization of the latter appears crucial for a precise determination of the
spectrum.Comment: 22 pages,8 figure
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