4,116 research outputs found
Pseudodifferential operators on manifolds with foliated boundaries
Let X be a smooth compact manifold with boundary. For smooth foliations on
the boundary of X admitting a `resolution' in terms of a fibration, we
construct a pseudodifferential calculus generalizing the fibred cusp calculus
of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a
simple description of the Fredholm operators inside the calculus. When the
leaves of the fibration `resolving' the foliation are compact, we also obtain
an index formula for Fredholm perturbations of Dirac-type operators. Along the
way, we obtain a formula for the adiabatic limit of the eta invariant for
invertible perturbations of Dirac-type operators, a result of independent
interest generalizing the well-known formula of Bismut and Cheeger.Comment: 49 pages, added references, strengthened the results, added an index
calculation for some quotients of gravitational instantons. To appear in the
Journal of Functional Analysi
Families Index for Pseudodifferential Operators on Manifolds with Boundary
An analytic index is defined for a family of cusp pseudodifferential
operators, on a fibration with fibres which are compact manifolds with
boundaries, provided the family is elliptic and has invertible indicial family
at the boundary. In fact there is always a perturbation by a family of
cusp operators of order such that each is invertible. Thus
any elliptic family of symbols has a realization as an invertible family of
cusp pseudodifferential operators, which is a form of the cobordism invariance
of the index. A crucial role is played by the weak contractibility of the group
of cusp smoothing operators on a compact manifold with non-trivial boundary and
the associated exact sequence of classifying spaces of odd and even K-theory.Comment: 21 pages; corrected typos, changed the abstract, added a paragraph in
the introductio
Bicomplex Quantum Mechanics: II. The Hilbert Space
Using the bicomplex numbers which is a commutative ring with
zero divisors defined by where , we construct hyperbolic and bicomplex Hilbert spaces.
Linear functionals and dual spaces are considered and properties of linear
operators are obtained; in particular it is established that the eigenvalues of
a bicomplex self-adjoint operator are in the set of hyperbolic numbers.Comment: 25 pages, no figur
Bicomplex quantum mechanics: I. The generalized Schr\"odinger equation
We introduce the set of bicomplex numbers which is a commutative
ring with zero divisors defined by where
$\bold{i^{\text 2}_1}=-1, \bold{i^{\text 2}_2}=-1, \bold{j}^2=1,\
\bold{i_1}\bold{i_2}=\bold{j}=\bold{i_2}\bold{i_1}$. We present the conjugates
and the moduli associated with the bicomplex numbers. Then we study the
bicomplex Schr\"odinger equation and found the continuity equations. The
discrete symmetries of the system of equations describing the bicomplex
Schr\"odinger equation are obtained. Finally, we study the bicomplex Born
formulas under the discrete symetries. We obtain the standard Born's formula
for the class of bicomplex wave functions having a null hyperbolic angle
The Inverse Iteration Method for Julia Sets in the 3-Dimensional Space
In this article, we introduce the adapted inverse iteration method to
generate bicomplex Julia sets associated to the polynomial map . The
result is based on a full characterization of bicomplex Julia sets as the
boundary of a particular bicomplex cartesian set and the study of the fixed
points of . The inverse iteration method is used in particular to
generate and display in the usual 3-dimensional space bicomplex Julia sets that
are dendrites.Comment: 16 pages, 4 figure
Finite-Dimensional Bicomplex Hilbert Spaces
This paper is a detailed study of finite-dimensional modules defined on
bicomplex numbers. A number of results are proved on bicomplex square matrices,
linear operators, orthogonal bases, self-adjoint operators and Hilbert spaces,
including the spectral decomposition theorem. Applications to concepts relevant
to quantum mechanics, like the evolution operator, are pointed out.Comment: 21 page
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