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Marius Beceanu

Publications and source records attributed to Marius Beceanu.

At least 19 recordsLinked to original sources
arXiv2025

Dispersive estimates for Schr\"{o}dinger's and wave equations on Riemannian manifolds

This paper proves $L^p$ decay estimates for Schr\"{o}dinger's and wave equations with scalar potentials on three-dimensional Riemannian manifolds. The main result regards small perturbations of a metric with constant negative sectional curvature. We also prove estimates on $\mathbb S^3$, the three-dimensional sphere, and $\mathbb H^3$, the three-dimensional hyperbolic space. Most of the estimates hold for the perturbed Hamiltonian $H=H_0+V$, where $H_0$ is the shifted Laplacian $H_0=-\Delta+\kappa_0$, $\kappa_0$ is the constant (or asymptotic) sectional curvature, and $V$ is a small scalar potential. The results are based on direct estimates of the wave propagator. All results hold in three space dimensions. The metric is required to have four derivatives.

math.AP↗
arXiv2024

Decay estimates for Schr\"{o}dinger's equation with magnetic potentials in three dimensions

In this paper we prove that Schr\"{o}dinger's equation with a Hamiltonian of the form $H=-\Delta+i(A \nabla + \nabla A) + V$, which includes a magnetic potential $A$, has the same dispersive and solution decay properties as the free Schr\"{o}dinger equation. In particular, we prove $L^1 \to L^\infty$ decay and some related estimates for the wave equation. The potentials $A$ and $V$ are short-range and $A$ has four derivatives, but they can be arbitrarily large. All results hold in three space dimensions.

math.AP↗
arXiv2024

Local Exact Controllability to Stationary Solutions of a Semilinear Parabolic Equation

This paper establishes the local exact controllability of the quasilinear porous media equation with Dirichlet boundary condition.\\ Consider the equation $$\begin{aligned} &y_t - \Delta a(y) = mu+f \text{ on } Q\\ &y(0)=y_0,\ y \mid_{\Sigma} = 0 \end{aligned}$$ on the $n+1$-dimensional cylinder $Q = \Omega \times (0, T)$ with lateral boundary $\Sigma = \partial \Omega \times (0, T)$. The exact controllability in finite time is proved when $\|y_0 - y_s\|_{W^{1, n}_0(\Omega) \cap C(\overline \Omega)}$ is sufficiently small, $n > 1$, for every stationary solution $y_s$ such that $a(y_s) \in W^{2, q}(\Omega)$, where $q>n$. It is assumed that $\Omega$ is a bounded open set with $C^2$ boundary and that $a \in C^2(\mathbb R)$, $a'>0$.

math.AP↗
arXiv2024

Pluriharmonic solutions to Yang-Mills equations: a $C^*$-algebras approach

This partially expository paper provides a view of Yang-Mills equations from the perspective of complex variables, operator theory, and $C^{*}$-algebras. Through operator-valued pluriharmonic and skew-Hermitian differential forms, it constructs a new class of instanton solutions. Furthermore, it provides a complex variable version of the Yang-Mills Lagrangian and the Belavin-Polyakov-Schwartz-Tyupkin instanton.

math-ph↗
arXiv2023

Spectral multipliers III: Endpoint bounds, intertwining operators, and twisted Hardy spaces

We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on $\mathbb R^3$ to the case of perturbed Hamiltonians of the form $H=-\Delta+V$, where $V$ is a scalar real-valued potential. Results include sharp bounds for Mihlin multipliers, partial confirmation for a conjecture made in [BeGo3] about intertwining operators, a characterization of the twisted Hardy spaces that correspond to these perturbed Hamiltonians, Strichartz estimates, and maximum principles.

math.AP↗
arXiv2023

Spectral Multipliers II: Elliptic and Parabolic Operators and Bochner-Riesz Means

We establish estimates for the Poisson kernel, the heat kernel, and Bochner--Riesz means defined in terms of $H=-\Delta+V$, where $V$ is a possibly large rough real-valued scalar potential and $H$ can have negative eigenvalues. All results are in three space dimensions. We eliminate several unnecessary conditions on $V$, leaving just $V \in \mathcal K_0$, meaning that $V$ is locally integrable and $(-\Delta)^{-1}|V|$ is bounded. For the spectral multiplier bounds, we assume that $H$ has no zero or positive energy bound states. For $V \in \mathcal K_0$, we prove that $H$ has at most a finite number of negative bound states. If in addition $V \in \dot W^{-1/4, 4/3}$, then by [GoSc] and [KoTa] there are no positive energy bound states.

math.AP↗
arXiv2022

Spectral multipliers and wave propagation for Hamiltonians with a scalar potential

We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on $\mathbb R^3$ to the case of perturbed Hamiltonians of the form $-\Delta+V$, where $V$ is a scalar real-valued potential. In this paper, we prove resolvent estimates, a dispersive bound for the perturbed wave propagator, Mihlin multiplier and fractional integration bounds, and the full range of wave equation Strichartz estimates, under optimal or almost optimal scaling-invariant conditions on the potential and on the spectral multipliers themselves.

math.AP↗
arXiv2021

The Eigenvalue Problem for the Laplacian via Conformal Mapping and the Gohberg--Sigal Theory

We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a planar, simply connected domain. The eigenvalues admit a characterization in terms of a layer potential of the Helmholtz equation. Using the exterior conformal mapping associated with the given domain, we reformulate the layer potential as an infinite-dimensional matrix. Based on this matrix representation, we develop a finite section approach for approximating the Laplacian eigenvalues and provide a convergence analysis by applying the Gohberg--Sigal theory for operator-valued functions. Moreover, we derive an asymptotic formula for the Laplacian eigenvalues on deformed domains that results from the changes in the conformal mapping coefficients.

math.NA↗
arXiv2021

Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$

In this paper we prove standard and reversed Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$. Instead of the Fourier theory, our analysis is based on fundamental solutions of the free equations and fractional integrations. In the final part of this paper, we apply Strichartz estimates in the study of a semilinear Klein--Gordon equation.

math.AP↗
arXiv2019

A semilinear Schroedinger equation with random potential

We study a non-linear Schroedinger equation with a Hartree-type nonlinearity and a localized random time-dependent external potential. Sharp dispersive estimates for the linear Schroedinger equation with a random time-dependent potential enable us to also treat the case of small semi-linear perturbations. In both the linear and the nonlinear instances, we prove that, on average, energy remains bounded and solutions scatter.

math.AP↗
arXiv2019

Large global solutions for nonlinear Schrödinger equations III, energy-supercritical cases

In this work, we mainly focus on the energy-supercritical nonlinear Schrödinger equation, $$ i\partial_{t}u+Δu= μ|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $μ=\pm1$ and $p>\frac4{d-2}$. %In this work, we consider the energy-supercritical cases, that is, $p\in (\frac4{d-2},+\infty)$. We prove that for radial initial data with high frequency, if it is outgoing (or incoming) and in rough space $H^{s_1}(\mathbb{R}^d)$ $(s_1<s_c)$ or its Fourier transform belongs to $W^{s_2,1}(\mathbb{R}^d)$ $(s_2<s_c)$, the corresponding solution is global and scatters forward (or backward) in time. We also construct a class of large global and scattering solutions starting with many bubbles, which are mingled with in the physical space and separate in the frequency space. The analogous results are also valid for the energy-subcritical cases.

math.AP↗
arXiv2018

Large global solutions for nonlinear Schr\"odinger equations II, mass-supercritical, energy-subcritical cases

In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= |u|^p u, \quad (t,x)\in \mathbb R^{d+1}, $$ with $p\in (\frac4d,\frac4{d-2})$. We prove that under some restrictions on $d,p$, any radial function in the rough space $H^{s_0}(\mathbb R^d),\textit{for some } s_0<s_c$ with the support away from the origin, there exists an incoming/outgoing decomposition, such that the initial data in the outgoing part leads to the global well-posedness and scattering forward in time; while the initial data in the incoming part leads to the global well-posedness and scattering backward in time. The proof is based on Phase-Space analysis of the nonlinear dynamics.

math.AP↗
arXiv2018

Large global solutions for nonlinear Schr\"odinger equations I, mass-subcritical cases

In this paper, we consider the nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= \mu|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $\mu=\pm1, p>0$. In this work, we consider the mass-subcritical cases, that is, $p\in (0,\frac4d)$. We prove that under some restrictions on $d,p$, any radial initial data in the critical space $\dot H^{s_c}(\mathbb{R}^d)$ with compact support, implies global well-posedness.

math.AP↗
arXiv2017

Large Outgoing Solutions to Supercritical Wave Equations

We prove the existence of global solutions to the energy-supercritical wave equation in R^{3+1} u_{tt}-Δu + |u|^N u = 0, u(0) = u_0, u_t(0) = u_1, 4<N<\infty, for a large class of radially symmetric finite-energy initial data. Functions in this class are characterized as being outgoing under the linear flow --- for a specific meaning of "outgoing" defined below. In particular, we construct global solutions for initial data with large (even infinite) critical Sobolev, Besov, Lebesgue, and Lorentz norms and several other large critical norms.

math.AP↗
arXiv2017

Structure formulas for wave operators

We establish quantitative estimates on the structure function arising in the representation of the intertwining wave operators of a Schroedinger operator in three dimensions. Regularity of zero energy is assumed throughout. This paper is related to, and corrects some inaccuracies in, the first author's work https://arxiv.org/abs/1101.0502

math.AP↗
arXiv2016

A Positivity Criterion for the Wave Equation and Global Existence of Large Solutions

In dimensions one to three, the fundamental solution to the free wave equation is positive. Therefore, there exists a simple positivity criterion for solutions. We use this to obtain large global solutions to two well-studied energy-supercritical semilinear wave equations, as well as some new results in the subcritical and critical cases.

math.AP↗