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Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the spectral parameter $k$. For potentials \( q\in \langle \cdot \rangle^{-2} H^{s}(\mathbb{C}) \) for some $s \in]1,2]$, it is shown that the solution converges as the geometric series in $1/|k|^{s-1}$. For potentials $q$ being the characteristic function of a strictly convex open set with smooth boundary, this still holds with $s=3/2$ i.e., with $1/\sqrt{|k|}$ instead of $1/|k|^{s-1}$. The leading order controbutions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk., Comment: references updated

In this article, we investigate a density problem coming from the linearization of Calder\'on's problem with partial data. More precisely, we prove that the set of products of harmonic functions on a bounded smooth domain $\Omega$ vanishing on any fixed closed proper subset of the boundary are dense in $L^{1}(\Omega)$ in all dimensions $n \geq 2$. This is proved using ideas coming from the proof of Kashiwara's Watermelon theorem.

Consider in $L^2(R^d)$, $d\geq 1$, the operator family $H(g):=H_0+igW$. $\ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2$ is the quantum harmonic oscillator with rational frequencies, $W$ a $P$ symmetric bounded potential, and $g$ a real coupling constant. We show that if $|g|<\rho$, $\rho$ being an explicitly determined constant, the spectrum of $H(g)$ is real and discrete. Moreover we show that the operator $\ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1$ has real discrete spectrum but is not diagonalizable., Comment: 20 pages

For semiclassical problems we establish upper bounds on the number of resonances in boxes of size $h$ along the real axis, in terms of the dimension of the set of trapped trajectories. The proof uses second microlocalization.

This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2. We assume that the unperturbed operator has a periodic Hamilton flow, and study the remaining generic case; when the flow average of the perturbation has a saddle point

In this paper we improve an earlier result by Bukhgeim and Uhlmann, by showing that in dimension larger than or equal to three, the knowledge of the Cauchy data for the Schr\"odinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of Bukhgeim and Uhlmann but use a richer set of solutions to the Dirichlet problem., Comment: Revised version (Nov 5, 2004) fixing a mistake in section 6 and adding an application to the original problem for the conductivity. Revised version (Sept 14, 2005) modifying a few lines in the introduction and correcting the formula (1.8)

We discuss the general method of Grushin problems, closely related to Shur complements, Feshbach projections and effective Hamiltonians, and describe various appearances in spectral theory, pdes, mathematical physics and numerical problems., Comment: 2 figures

We apply recent results on semi-classical trace formulae and on Birkhoff normal forms for semi-classical Fourier integral operators to a wide range of semi-classical and high energy spectral inverse problems.

Consider an h-pseudodifferential operator P, whose symbol extends holomorphically to a tubular neighborhood of the real phase space and converges sufficiently fast to 1, so that the determinant of P is well-defined. We show that the modulus of this determinant is asymptotically bounded by an exponential of the integral of the logarithm of the modulus of the symbol along a certain complex deformation of the real phase space. Since there are many possible such deformations, we get a variational problem. The paper is devoted to the corresponding variational calculus.

For a class of non-selfadjoint h-pseudodifferential operators in dimension 2, we determine all eigenvalues in an h-independent domain in the complex plane and show that they are given by a Bohr-Sommerfeld quantization condition. No complete integrability is assumed, and as a geometrical step in our proof, we get a KAM-type theorem (without small divisors) in the complex domain.

In this work we construct logarithms and Birkhoff normal forms for elliptic Fourier integral operators in the semi-classical limit under more general assumptions than in aprevious work by the first author. The methods are similar but slightly different., Comment: 30 pages