906 research outputs found
The five-dimensional Kepler Problem as an SU(2) Gauge System: Algebraic Constraint Quantization
Starting from the structural similarity between the quantum theory of gauge
systems and that of the Kepler problem, an SU(2) gauge description of the
five-dimensional Kepler problem is given. This non-abelian gauge system is used
as a testing ground for the application of an algebraic constraint quantization
scheme which can be formulated entirely in terms of observable quantities. For
the quantum mechanical reduction only the quadratic Casimir of the constraint
algebra, interpreted as an observable, is needed.Comment: 29 pages, Latex, no figure
Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x
We consider a singular Sturm-Liouville expression with the indefinite weight
sgn x. To this expression there is naturally a self-adjoint operator in some
Krein space associated. We characterize the local definitizability of this
operator in a neighbourhood of . Moreover, in this situation, the point
is a regular critical point. We construct an operator A=(\sgn
x)(-d^2/dx^2+q) with non-real spectrum accumulating to a real point. The
obtained results are applied to several classes of Sturm-Liouville operators.Comment: 21 pages, LaTe
On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)
In the recent years a generalization of Hermiticity was investigated using a
complex deformation H=p^2 +x^2(ix)^\epsilon of the harmonic oscillator
Hamiltonian, where \epsilon is a real parameter. These complex Hamiltonians,
possessing PT symmetry (the product of parity and time reversal), can have real
spectrum. We will consider the most simple case: \epsilon even. In this paper
we describe all self-adjoint (Hermitian) and at the same time PT symmetric
operators associated to H=p^2 +x^2(ix)^\epsilon. Surprisingly it turns out that
there are a large class of self-adjoint operators associated to H=p^2
+x^2(ix)^\epsilon which are not PT symmetric
Eigenvalue estimates for singular left-definite Sturm-Liouville operators
The spectral properties of a singular left-definite Sturm-Liouville operator
are investigated and described via the properties of the corresponding
right-definite selfadjoint counterpart which is obtained by substituting
the indefinite weight function by its absolute value. The spectrum of the
-selfadjoint operator is real and it follows that an interval
is a gap in the essential spectrum of if and only
if both intervals and are gaps in the essential spectrum of
the -selfadjoint operator . As one of the main results it is shown that
the number of eigenvalues of in differs at most by
three of the number of eigenvalues of in the gap ; as a byproduct
results on the accumulation of eigenvalues of singular left-definite
Sturm-Liouville operators are obtained. Furthermore, left-definite problems
with symmetric and periodic coefficients are treated, and several examples are
included to illustrate the general results.Comment: to appear in J. Spectral Theor
Variational principles for self-adjoint operator functions arising from second-order systems
Variational principles are proved for self-adjoint operator functions arising
from variational evolution equations of the form Here and are densely defined,
symmetric and positive sesquilinear forms on a Hilbert space . We associate
with the variational evolution equation an equivalent Cauchy problem
corresponding to a block operator matrix , the forms where
and are in the domain of the form , and a corresponding
operator family . Using form methods we define a generalized
Rayleigh functional and characterize the eigenvalues above the essential
spectrum of by a min-max and a max-min variational principle. The
obtained results are illustrated with a damped beam equation.Comment: to appear in Operators and Matrice
Spectral bounds for singular indefinite Sturm-Liouville operators with --potentials
The spectrum of the singular indefinite Sturm-Liouville operator
with a real
potential covers the whole real line and, in addition,
non-real eigenvalues may appear if the potential assumes negative values. A
quantitative analysis of the non-real eigenvalues is a challenging problem, and
so far only partial results in this direction were obtained. In this paper the
bound on the absolute values of the non-real
eigenvalues of is obtained. Furthermore, separate bounds on the
imaginary parts and absolute values of these eigenvalues are proved in terms of
the -norm of the negative part of .Comment: to appear in Proc. Amer. Math. So
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