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

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 $\infty$. Moreover, in this situation, the point $\infty$ 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 $$(\int_{-1}^1\frac{1}{|r|}|f'|dx)^2 \le K^2 \int_{-1}^1|f|^2dx\int_{-1}^1\Big|\Big(\frac{1}{r}f'\Big)'\Big|^2dx$$ is satisfied with some positive constant $K>0$ for a certain class of functions $f$ on $[-1,1]$ if the eigenfunctions of the problem $$-y"=\lambda\, r(x)y,\quad y(-1)=y(1)=0$$ form a Riesz basis of the Hilbert space $L^2_{|r|}(-1,1)$. Here the weight $r\in L^1(-1,1)$ is assumed to satisfy $xr(x)>0$ a.e. on $[-1,1]$. We present two criteria in terms of Weyl-Titchmarsh $m$-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 $r$ is odd.Comment: 26 page