The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian α < 1/2, and global existence as well as analyticity of solution for α ≥ 1/2. We also prove the existence of solutions with very rough initial data u 0 ∈ L p , 1 < p < ∞. Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.

We prove the complete asymptotic expansion of the integrated density of states of a Schrödinger operator H = −∆ + b acting in R d when the potential b is either smooth periodic, or generic quasi-periodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions.

We prove the unitary equivalence of the inverse of the Krein-von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S ≥ εIH for some ε > 0 in a Hilbert space H to an abstract buckling problem operator.In the concrete case wheren an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein-von Neumann extension of S),is in one-to-one correspondence with the problem of the buckling of a clamped plate,where u and v are related via the pair of formulaswith SF the Friedrichs extension of S. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.).

Abstract. In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein-von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, ≥ ℋ for some > 0 in a Hilbert space ℋ to an abstract buckling problem operator.In the concrete case where

We consider Schrödinger operators on [0, ∞) with compactly supported, possibly complex-valued potentials in L 1 ([0, ∞)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a byproduct we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.