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 $\alpha \in \R$ 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 $0\leq b_1 \leq B (x) \leq b_2$ with certain constants $b_{1,2}$. 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 $b_1$ and $b_2$. 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

Indefinite Sturm-Liouville operators with the singular critical point zero

We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh $m$-functions. Also we obtain necessary conditions for regularity of the critical points 0 and $\infty$ of $J$-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular critical point zero. In particular, it is shown that 0 is a singular critical point of the operator -\frac{(\sgn x)}{(3|x|+1)^{-4/3}} \frac{d^2}{dx^2} acting in the Hilbert space $L^2(\R, (3|x|+1)^{-4/3}dx)$ and therefore this operator is not similar to a self-adjoint one. Also we construct a J-nonnegative Sturm-Liouville operator of type (\sgn x)(-d^2/dx^2+q(x)) with the same properties.Comment: 24 pages, LaTeX2e <2003/12/01

On the nature of ill-posedness of the forward-backward heat equation

We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by the J-self-adjoint linear operator L depending on a small parameter. The problem has been originated from the lubrication approximation of a viscous fluid film on the inner surface of the rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on the numerical evidence, that the set of eigenvectors of the operator $L$ does not form a Riesz basis in \L^2 (-\pi,\pi). Our method can be applied to a wide range of the evolutional problems given by $PT-$symmetric operators.Comment: 21 pages; Remark 5.2 added, acknowledgements added, several typos fixe