9,050 research outputs found
Quantum Monodromy in the Isotropic 3-Dimensional Harmonic Oscillator
The isotropic harmonic oscillator in dimension 3 separates in several
different coordinate systems. Separating in a particular coordinate system
defines a system of three commuting operators, one of which is the Hamiltonian.
We show that the joint spectrum of the Hamilton operator, the component of
the angular momentum, and a quartic integral obtained from separation in
prolate spheroidal coordinates has quantum monodromy for sufficiently large
energies. This means that one cannot globally assign quantum numbers to the
joint spectrum. The effect can be classically explained by showing that the
corresponding Liouville integrable system has a non-degenerate focus-focus
point, and hence Hamiltonian monodromy.Comment: 15 pages, 8 figure
On occurrence of spectral edges for periodic operators inside the Brillouin zone
The article discusses the following frequently arising question on the
spectral structure of periodic operators of mathematical physics (e.g.,
Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can
obtain the correct spectrum by using the values of the quasimomentum running
over the boundary of the (reduced) Brillouin zone only, rather than the whole
zone? Or, do the edges of the spectrum occur necessarily at the set of
``corner'' high symmetry points? This is known to be true in 1D, while no
apparent reasons exist for this to be happening in higher dimensions. In many
practical cases, though, this appears to be correct, which sometimes leads to
the claims that this is always true. There seems to be no definite answer in
the literature, and one encounters different opinions about this problem in the
community.
In this paper, starting with simple discrete graph operators, we construct a
variety of convincing multiply-periodic examples showing that the spectral
edges might occur deeply inside the Brillouin zone. On the other hand, it is
also shown that in a ``generic'' case, the situation of spectral edges
appearing at high symmetry points is stable under small perturbations. This
explains to some degree why in many (maybe even most) practical cases the
statement still holds.Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in
the new versio
Modification of the standard model for the lanthanides
We show that incorporation of strong electron correlations into the Kohn-Sham
scheme of band structure calculations leads to a modification of the standard
model of the lanthanides and that this procedure removes the existing
discrepancy between theory and experiment concerning the ground state
properties. Within the picture suggested, part of the upper Hubbard -band is
occupied due to conduction band--mixing interaction (that is renormalized
due to correlations) and this contributes to the cohesive energy of the
crystal. The lower Hubbard band has zero width and describes fermionic
excitations in the shell of localized -s. Fully self-consistent calculations
(with respect to both charge density and many-electron population numbers of
the -shell) of the equilibrium volume and the bulk modulus of selected
lanthanides have been performed and a good agreement is obtained.Comment: 1 fi
New Concept of Solvability in Quantum Mechanics
In a pre-selected Hilbert space of quantum states the unitarity of the
evolution is usually guaranteed via a pre-selection of the generator (i.e., of
the Hamiltonian operator) in self-adjoint form. In fact, the simultaneous use
of both of these pre-selections is overrestrictive. One should be allowed to
make a given Hamiltonian self-adjoint only after an {\em ad hoc} generalization
of Hermitian conjugation. We argue that in the generalized, hidden-Hermiticity
scenario with nontrivial metric, the current concept of solvability (meaning,
most often, the feasibility of a non-numerical diagonalization of Hamiltonian)
requires a generalization allowing for a non-numerical form of metric. A few
illustrative solvable quantum models of this type are presented.Comment: 24 pages, 8 figure
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