19 research outputs found
Spectrality of ordinary differential operators
We prove the long standing conjecture in the theory of two-point boundary
value problems that completeness and Dunford's spectrality imply Birkhoff
regularity. In addition we establish the even order part of S.G.Krein's
conjecture that dissipative differential operators are Birkhoff-regular and
give sharp estimate of the norms of spectral projectors in the odd case.
Considerations are based on a new direct method, exploiting \textit{almost
orthogonality} of Birkhoff's solutions of the equation , which
was discovered earlier by the author.Comment: AmsLaTeX, 26 pages, added section on dissipative operators and
reference
Nonlinear eigenvalue problem for optimal resonances in optical cavities
The paper is devoted to optimization of resonances in a 1-D open optical
cavity. The cavity's structure is represented by its dielectric permittivity
function e(s). It is assumed that e(s) takes values in the range 1 <= e_1 <=
e(s) <= e_2. The problem is to design, for a given (real) frequency, a cavity
having a resonance with the minimal possible decay rate. Restricting ourselves
to resonances of a given frequency, we define cavities and resonant modes with
locally extremal decay rate, and then study their properties. We show that such
locally extremal cavities are 1-D photonic crystals consisting of alternating
layers of two materials with extreme allowed dielectric permittivities e_1 and
e_2. To find thicknesses of these layers, a nonlinear eigenvalue problem for
locally extremal resonant modes is derived. It occurs that coordinates of
interface planes between the layers can be expressed via arg-function of
corresponding modes. As a result, the question of minimization of the decay
rate is reduced to a four-dimensional problem of finding the zeroes of a
function of two variables.Comment: 16 page