arXiv subjects

A. Uribe

Publications and source records attributed to A. Uribe.

9 recordsLinked to original sources

arXiv2024

Spectral cluster asymptotics of the Dirichlet to Neumann operator on the two-sphere

We study the spectrum of the Dirichlet to Neumann operator of the two-sphere associated to a Schr\"odinger 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.

math.SP↗

arXiv2021

The semiclassical structure of the scattering matrix for a manifold with infinite cylindrical end

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.

math.SP↗

arXiv2019

Perturbations of the Landau Hamiltonian: Asymptotics of eigenvalue clusters

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.

math-ph↗

arXiv2011

Quantum cutting and a Szegö limit theorem

Given a representation of the circle group by semiclassical Fourier integral operators, we construct an algebra of semiclassical pseudodifferential operators that are a quantum analogue of the notion of symplectic cutting of Lerman, and we prove an associated Szegö limit theorem.

math.SP↗

arXiv2007

The spectral density function of a toric variety

For a Kahler manifold (X, ω) with a holomorphic line bundle L and metric h such that the Chern form of L is ω, the spectral measures are the measures μ_N = \sum |s_{N,i}|^2 ν, where \{s_{N,i}\}_i is an L^2-orthonormal basis for H^0(X, L^{\otimes N}), and νis Liouville measure. We study the asymptotics in N of μ_N for (X, L) a Hamiltonian toric manifold, and give a precise expansion in terms of powers 1/N^j and data on the moment polytope Δof the Hamiltonian torus K acting on X. In addition, for an infinitesimal character k of K and the unique unit eigensection s_{Nk} for the character Nk of the torus action on H^0(X, L^N), we give a similar expansion for the measures μ_{Nk} = |s_{Nk}|^2 ν. A final remark shows that the eigenbasis \{s_{k}, k \in Δ\cap \mathbb{Z}^{\dim K} \} is a Bohr-Sommerfeld basis in the sense of Tyurin. Some of the present results are related to work of Shiffman, Tate and Zelditch. The present paper uses no microlocal analysis, but rather an Euler-Maclaurin formula for Delzant polytopes.

math.SP↗

arXiv2006

"Bottom of the well" semi-classical trace invariants

Let $\hat H$ be an h-admissible pseudodifferential operator whose principal symbol, $H$, has a unique non-degenerate global minimum. We give a simple proof that the semi-classical asymptotics of the eigenvalues of $\hat H$ corresponding to the "bottom of the well" determine the Birkhoff normal form of $H$ at the minimum. We treat both the resonant and the non-resonant cases.

math.SP↗

arXiv2005

Some inverse spectral results for semi-classical Schrödinger operators

We consider a semi-classical Schrödinger operator, -h^2Δ+ V(x). Assuming that the potential admits a unique global minimum and that the eigenvalues of the Hessian are linearly independent over the rationals, we show that the low-lying eigenvalues of the operator determine the Taylor series of the potential at the minimum.

math.SP↗

arXiv2004

Remarks on the naturality of quantization

Hamiltonian quantization of an integral compact symplectic manifold M depends on a choice of compatible almost complex structure J. For open sets U in the set of compatible almost complex structures and small enough values of Planck's constant, the Hilbert spaces of the quantization form a bundle over U with a natural connection. In this paper we examine the dependence of the Hilbert spaces on the choice of J, by computing the semi-classical limit of the curvature of this connection. We also show that parallel transport provides a link between the action of the group Symp(M) of symplectomorphisms of M and the Schrodinger equation.

math.SG↗

arXiv1994

Legendrian Distributions with Applications to Poincaré Series

Let $X$ be a compact Kahler manifold and $L\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $Λ$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\{ |Λ, k\rangle \}_{k=1}^\infty$, where $\forall k$ $|Λ, k\rangle$ is a holomorphic section of $L^{\otimes k}$. The terms in each sequence concentrate on $Λ$, and a sequence itself has a symbol which is a half-form, $σ$, on $Λ$. We prove estimates, as $k\to\infty$, of the norm squares $\langle Λ, k|Λ, k\rangle$ in terms of $\int_Λσ\overlineσ$. More generally, we show that if $Λ_1$ and $Λ_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $\langleΛ_1, k|Λ_2, k\rangle$ have an asymptotic expansion as $k\to\infty$, the leading coefficient being an integral over the intersection $Λ_1\capΛ_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincaré series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products.

hep-th↗