Abstract:
Precise asymptotics known for the Green's function of the Laplace operator have found their analogs for periodic elliptic operators of the second order at and below the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. As the result of this work shows, this is possible at a spectral edge in dimensions d>2.

Abstract:
The paper contains the Weyl formula for the counting function of the interior transmission problem when the latter is parameter-elliptic. Branching billiard trajectories are constructed, and the second term of the Weyl asymptotics is estimated from above under some conditions on the set of periodic billiard trajectories.

Abstract:
At the example of two coupled waveguides we construct a periodic second order differential operator acting in a Euclidean domain and having spectral gaps whose edges are attained strictly inside the Brillouin zone. The waveguides are modeled by the Laplacian in two infinite strips of different width that have a common interior boundary. On this common boundary we impose the Neumann boundary condition but cut out a periodic system of small holes, while on the remaining exterior boundary we impose the Dirichlet boundary condition. It is shown that, by varying the widths of the strips and the distance between the holes, one can control the location of the extrema of the band functions as well as the number of the open gaps. We calculate the leading terms in the asymptotics for the gap lengths and the location of the extrema.

Abstract:
The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for "generic" periodic second-order elliptic operators on noncompact Riemannian co-compact coverings with abelian deck groups. Previously, analogous results have been known for the case of $\mathbb{R}^n$ only. One of the interesting features discovered is that the rank of the deck group plays more important role than the dimension of the manifold.

Abstract:
We prove a Weyl asymptotics $N(r) = c r^d + {\mathcal O}_{\epsilon}(r^{d - \kappa + \epsilon})$, $\forall\, 0< \epsilon \ll 1$, for the counting function $N(r) = \sharp\{\lambda_j \in {\mathbb C} \setminus \{0\}:\: |\lambda_j| \leq r^2\}$, $r>1$, of the interior transmission eigenvalues (ITE), $\lambda_j$. Here $0<\kappa\leq 1$ is such that there are no (ITE) in the region $\{\lambda\in {\mathbb C}:\: |{\rm Im}\:\lambda|\geq C(| {\rm Re}\:\lambda|+1)^{1-\frac{\kappa}{2}}\}$ for some $C>0$.

Abstract:
We propose a new state of matter in which the pairing interactions carve out a gap within the interior of a large Fermi ball, while the exterior surface remains gapless. This defines a system which contains both a superfluid and a normal Fermi liquid simultaneously, with both gapped and gapless quasiparticle excitations. This state can be realized at weak coupling. We predict that a cold mixture of two species of fermionic atoms with different mass will exhibit this state. For electrons in appropriate solids, it would define a material that is simultaneously superconducting and metallic.

Abstract:
Canopy gaps, i.e. openings in the forest cover caused by the fall of structural elements, are considered to be important for the maintenance of diversity and for the forest cycle. A gap can be considered as a young forest patch in the forest matrix, composed of interior surrounded by an edge, both enclosed by a perimeter. Much of the attention has been focused on the gap interior. However, at gap edges the spectrum of regeneration opportunities for plants may be larger than in the interior. Although definitions of gap are still discussed, any definition can describe it in an acceptable way, if justified, but defining edges is complicated and appropriate descriptors should be used. A method to determine gap interior and edge, using light as a descriptor, is presented with an example of gaps from a beech forest (Fagus sylvatica) in Belgium. Also, the relevance and implications of gap edges for plant diversity and calculation of forest turnover is discussed.

Abstract:
We investigate superfluidity in the interior-gap states proposed by Liu and Wilczek. At weak coupling, we find the {\em gapless} interior-gap state unstable in physically accessible regimes of the parameter space, where the superfluid density is shown to be always negative. We therefore conclude that the spatially-uniform interior-gap phase is extremely unstable unless it is fully gapped; in this case, however, the state is rather similar to conventional BCS states.

Abstract:
We outline an approach recently used to prove formulae for the multiplicative constants in the asymptotics for the sine-kernel and Airy-kernel determinants appearing in random matrix theory and related areas.

Abstract:
We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided by the method of signaling solutions by Mooers, though crucial arguments in the conical case obviously do not carry over to the setup of edges. We then establish the heat kernel asymptotics for the algebraic extensions of the Hodge operator on edges, and elaborate on the exotic phenomena in the heat trace asymptotics which appear in case of a non-Friedrichs extension.