Alejandro Uribe

We study the spectrum of the Dirichlet to Neumann operator of the two-sphere associated to a Schrödinger operator in the unit ball. The spectrum forms clusters of size $O(1/k)$ around the sequence of natural numbers $k=1,2,\ldots$, and we compute the first three terms in the asymptotic distribution of the eigenvalues within the clusters, as $k\to\infty$ (band invariants). There are two independent aspects of the proof. The first is a study of the Berezin symbol of the Dirichlet to Neumann operator, which arises after one applies the averaging method. The second is the use of a symbolic calculus of Berezin-Toeplitz operators on the manifold of closed geodesics of the sphere.

We study the microlocal properties of the scattering matrix associated to the semiclassical Schr\"odinger operator $P=h^2\Delta_X+V$ on a Riemannian manifold with an infinite cylindrical end. The scattering matrix at $E=1$ is a linear operator $S=S_h$ defined on a Hilbert subspace of $L^2(Y)$ that parameterizes the continuous spectrum of $P$ at energy $1$. Here $Y$ is the cross section of the end of $X$, which is not necessarily connected. We show that, under certain assumptions, microlocally $S$ is a Fourier integral operator associated to the graph of the scattering map $\kappa:\mathcal{D}_{\kappa}\to T^*Y$, with $\mathcal{D}_\kappa\subset T^*Y$. The scattering map $\kappa$ and its domain $\mathcal{D}_\kappa$ are determined by the Hamilton flow of the principal symbol of $P$. As an application we prove that, under additional hypotheses on the scattering map, the eigenvalues of the associated unitary scattering matrix are equidistributed on the unit circle.

We consider the asymptotic behavior of the spectrum of the Landau Hamiltonian plus a rapidly decaying potential, as the magnetic field strength, $B$, tends to infinity. After a suitable rescaling, this becomes a semiclassical problem where the role of Planck's constant is played by $1/B$. The spectrum of the operator forms eigenvalue clusters. We obtain a Szeg\H{o} limit theorem for the eigenvalues in the clusters as a suitable cluster index and $B$ tend to infinity with a fixed ratio $\calE$. The answer involves the averages of the potential over circles of radius $\sqrt{\calE/2}$ (circular Radon transform). We also discuss related inverse spectral results.

We consider non-self-adjoint Schr\"odinger operators $\Delta+V$ where $\Delta$ is the Laplace-Beltrami operator on a Zoll manifold $X$ and $V\in C^\infty(X,\mathbb C)$. We obtain asymptotic results on the pseudo-spectrum and numerical range of such operators.

We consider Berezin-Toeplitz operators whose multipliers are compactly supported densities carried by a submanifold of ${\mathbb C}^N$ . We compute asymptotically the moments of their spectral measures, and we prove Szeg\"o limit theorems in cases when the submanifold is isotropic or co-isotropic, from which Weyl estimates follow. We also obtain asymptotics of the Schatten norms of such operators. Rescaled versions of these operators can be thought of as quantum mechanical mixed states, and our results give the semi-classical limit of their entropy.