86 research outputs found
Optimization of quasi-normal eigenvalues for 1-D wave equations in inhomogeneous media; description of optimal structures
The paper is devoted to optimization of resonances associated with 1-D wave
equations in inhomogeneous media. The medium's structure is represented by a
nonnegative function B. The problem is to design for a given a
medium that generates a resonance on the line \alpha + \i \R with a minimal
possible modulus of the imaginary part. We consider an admissible family of
mediums that arises in a problem of optimal design for photonic crystals. This
admissible family is defined by the constraints
with certain constants . The paper gives an accurate definition of
optimal structures that ensures their existence. We prove that optimal
structures are piecewise constant functions taking only two extreme possible
values and . This result explains an effect recently observed in
numerical experiments. Then we show that intervals of constancy of an optimal
structure are tied to the phase of the corresponding resonant mode and write
this connection as a nonlinear eigenvalue problem.Comment: Typos are correcte
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 a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems
In 1996, H. Volkmer observed that the inequality
is
satisfied with some positive constant for a certain class of functions
on if the eigenfunctions of the problem form a Riesz basis of the Hilbert space
. Here the weight is assumed to satisfy
a.e. on .
We present two criteria in terms of Weyl-Titchmarsh -functions for the
Volkmer inequality to be valid. Using these results we show that this
inequality is valid if the operator associated with the spectral problem
satisfies the linear resolvent growth condition. In particular, we show that
the Riesz basis property of eigenfunctions is equivalent to the linear
resolvent growth if is odd.Comment: 26 page
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