arXiv subjects

Andrés Larraín-Hubach

Publications and source records attributed to Andrés Larraín-Hubach.

7 recordsLinked to original sources

arXiv2026

Exponential local energy decay of solutions to the wave equation with $L^\infty$ electric and magnetic potentials

In this paper we prove sharp resolvent estimates for the magnetic Schrödinger operator in $\mathbb{R}^d$, $d\ge 3$, with $L^\infty$ short-range electric and magnetic potentials. We also show that these resolvent estimates still hold for the Dirichlet self-adjoint realization of the Schrödinger operator in the exterior of a non-trapping obstacle in $\mathbb{R}^d$, $d\ge 2$, provided the magnetic potential is supposed identically zero. As an application of the resolvent estimates, we obtain an exponential decay of the local energy of solutions to the wave equation with $L^\infty$ electric and magnetic potentials which decay exponentially at infinity, in all odd and even dimensions, provided the low frequencies are cut off in a suitable way. We also show that in odd dimensions there is no need to cut off the low frequencies in order to get an exponential local energy decay, provided we assume that zero is neither an eigenvalue nor a resonance.

math.AP↗

arXiv2026

Resolvent bounds for repulsive potentials

We prove limiting absorption resolvent bounds for the semiclassical Schrödinger operator with a repulsive potential in dimension $n\ge 3$, which may have a singularity at the origin. As an application, we obtain time decay for the weighted energy of the solution to the associated wave equation with a short range repulsive potential and compactly supported initial data.

math.AP↗

arXiv2026

Resolvent estimates for the Schrödinger operator with $L^\infty$ electric and magnetic potentials and applications to the local energy decay

We establish resolvent estimates that extend earlier results to a larger class of electric potentials $V\in L^\infty(\mathbb{R}^d;\mathbb{R})$, $d\ge 3$, and magnetic potentials $b\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$ such that $V(x), b(x)=O_k\left(|x|^{-k}\right)$, $|x|\gg 1$, for every integer $k$. More precisely, we prove estimates for the derivatives of the weighted resolvent of the corresponding magnetic Schrödinger operator, which are uniform with respect to both the spectral parameter and the order of derivation. We also show that these resolvent estimates still hold for the Dirichlet self-adjoint realization of the Schrödinger operator in the exterior of a non-trapping obstacle in $\mathbb{R}^d$, $d\ge 2$, provided the magnetic potential is supposed identically zero. As an application of these resolvent estimates, we obtain the rate of decay of the local energy of solutions to the corresponding wave equation. In particular, we show that for potentials satisfying $|V(x)|+|b(x)|\le Ce^{-c|x|^s}$, $c,C>0$, $0 0$, where $t\gg 1$ is the time variable.

math.AP↗

arXiv2025

Semiclassical estimates for the magnetic Schrödinger operator on the line

We prove a weighted Carleman estimate for a class of one-dimensional, self-adjoint Schrödinger operators $P(h)$ with low regularity electric and magnetic potentials, where $h > 0$ is a semiclassical parameter. The long range part of either potential has bounded variation. The short range part of the magnetic potential belongs to $L^1(\mathbb{R}) \cap L^2(\mathbb{R})$, while the short range part of the electric potential is a finite signed measure. The proof is a one dimensional instance of the energy method, which is used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result lies in the weak regularity assumptions on the coefficients. As a consequence of the Carleman estimate, we establish an optimal limiting absorption resolvent estimates for $P(h)$. We also present standard applications to the distribution of resonances for $P(1)$ and to associated evolution equations.

math.AP↗

arXiv2025

Instantons on multi-Taub-NUT Spaces III: Down Transform, Completeness, and Isometry

The index bundle of a family of Dirac operators associated to an instanton on a multi-Taub-NUT space forms a bow representation. We prove that the gauge equivalence classes of solutions of this bow representation are in one-to-one correspondence with the instantons. We also prove that this correspondence establishes an isometry of the bow and instanton moduli spaces.

math.DG↗

arXiv2024

Semiclassical estimates for measure potentials on the real line

We prove an explicit weighted estimate for the semiclassical Schrödinger operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$ a finite signed measure, and where $h >0$ is the semiclassical parameter. The proof is a one dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues for $P$, and a limiting absorption resolvent estimate with sharp $h$-dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.

math.AP↗

arXiv2021

Instantons on multi-Taub-NUT Spaces II: Bow Construction

Unitary anti-self-dual connections on Asymptotically Locally Flat (ALF) hyperkähler spaces are constructed in terms of data organized in a bow. Bows generalize quivers, and the relevant bow gives rise to the underlying ALF space as the moduli space of its particular representation -- the small representation. Any other representation of that bow gives rise to anti-self-dual connections on that ALF space. We prove that each resulting connection has finite action, i.e. it is an instanton. Moreover, we derive the asymptotic form of such a connection and compute its topological class.

math.DG↗