179 research outputs found
Sharp eigenvalue enclosures for the perturbed angular Kerr-Newman Dirac operator
A certified strategy for determining sharp intervals of enclosure for the
eigenvalues of matrix differential operators with singular coefficients is
examined. The strategy relies on computing the second order spectrum relative
to subspaces of continuous piecewise linear functions. For smooth perturbations
of the angular Kerr-Newman Dirac operator, explicit rates of convergence due to
regularity of the eigenfunctions are established. Existing benchmarks are
validated and sharpened by several orders of magnitude in the unperturbed
setting.Comment: 27 pages, 2 figures, 5 tables. Some errors fixe
Spectral theory of some non-selfadjoint linear differential operators
We give a characterisation of the spectral properties of linear differential
operators with constant coefficients, acting on functions defined on a bounded
interval, and determined by general linear boundary conditions. The boundary
conditions may be such that the resulting operator is not selfadjoint.
We associate the spectral properties of such an operator with the
properties of the solution of a corresponding boundary value problem for the
partial differential equation . Namely, we are able to
establish an explicit correspondence between the properties of the family of
eigenfunctions of the operator, and in particular whether this family is a
basis, and the existence and properties of the unique solution of the
associated boundary value problem. When such a unique solution exists, we
consider its representation as a complex contour integral that is obtained
using a transform method recently proposed by Fokas and one of the authors. The
analyticity properties of the integrand in this representation are crucial for
studying the spectral theory of the associated operator.Comment: 1 figur
On Unbounded Composition Operators in -Spaces
Fundamental properties of unbounded composition operators in -spaces are
studied. Characterizations of normal and quasinormal composition operators are
provided. Formally normal composition operators are shown to be normal.
Composition operators generating Stieltjes moment sequences are completely
characterized. The unbounded counterparts of the celebrated Lambert's
characterizations of subnormality of bounded composition operators are shown to
be false. Various illustrative examples are supplied
On local linearization of control systems
We consider the problem of topological linearization of smooth (C infinity or
real analytic) control systems, i.e. of their local equivalence to a linear
controllable system via point-wise transformations on the state and the control
(static feedback transformations) that are topological but not necessarily
differentiable. We prove that local topological linearization implies local
smooth linearization, at generic points. At arbitrary points, it implies local
conjugation to a linear system via a homeomorphism that induces a smooth
diffeomorphism on the state variables, and, except at "strongly" singular
points, this homeomorphism can be chosen to be a smooth mapping (the inverse
map needs not be smooth). Deciding whether the same is true at "strongly"
singular points is tantamount to solve an intriguing open question in
differential topology
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