Asymptotics of random resonances generated by a point process of delta-interactions

We introduce and study the following model for random resonances: we take a collection of point interactions $Upsilon_j$ generated by a simple finite point process in the 3-D space and consider the resonances of associated random Schrodinger Hamiltonians $H_Upsilon = -Delta + ``sum mathfrak{m}(alpha) delta (x - Upsilon_j)``$. These resonances are zeroes of a random exponential polynomial, and so form a point process $Sigma (H_Upsilon)$ in the complex plane $mathbb{C}$. We show that the counting function for the set of random resonances $Sigma (H_Upsilon)$ in $mathbb{C}$-discs with growing radii possesses Weyl-type asymptotics almost surely for a uniform binomial process $Upsilon$, and obtain an explicit formula for the limiting distribution as $m to infty$ of the leading parameter of the asymptotic chain of `most narrow resonances generated by a sequence of uniform binomial processes $Upsilon^m$ with $m$ points. We also pose a general question about the limiting behavior of the point process formed by leading parameters of asymptotic sequences of resonances. Our study leads to questions about metric characteristics for the combinatorial geometry of $m$ samples of a random point in the 3-D space and related statistics of extreme values.

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.