arXiv subjects

Eduard Stefanescu

Publications and source records attributed to Eduard Stefanescu.

11 recordsLinked to original sources

arXiv2026

Resolvent bounds and eigenvalue estimates of generalized Schrödinger operators with complex potentials on compact manifolds

We extend Cuenin's compact-manifold spectral bounds for Schrödinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators $P+V$, where $P$ is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and $V$ is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for $P$ imply $L^p$-$L^{p'}$ resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

math.SP↗

arXiv2026

On large-scale wind-drift ocean currents: An asymptotic approach in spherical coordinates

Starting from the Navier--Stokes equations in rotating spherical coordinates with depth-varying density and eddy viscosity, we derive an asymptotic model describing non-equatorial wind-generated ocean drift currents. Our approach allows for large-scale flows that cannot be captured by classical tangent-plane approximations. The strategy is to perform a careful scaling and to perform a double asymptotic expansion with respect to two small parameters arising from the scaling: the Rossby number and the ratio between the Ekman depth and the Earth's radius. We obtain a system of linear ordinary differential equations with nonlinear boundary conditions governing the leading-order dynamics, highlighting that the dynamics is governed by the linear terms, whereas the nonlinear ones, related to the injection and dissipation of kinetic energy, appear only at higher order. We use the leading-order equations to compare our model with the simplest theory of ocean circulation due to Sverdrup and note that, even at this level of simplification, our equations have the potential to provide deeper insight. Subsequently, focusing on Ekman flows, we prove existence and uniqueness of the leading-order solution, which retains the classical Ekman spiral structure for arbitrary eddy viscosity profiles. Finally, we compute the surface deflection angle of the wind-driven current for three explicit eddy viscosity profiles, obtaining results consistent with observations. In addition, we derive the governing equations for the first-order correction with respect to the Rossby number and provide a priori bounds for its solution.

physics.flu-dyn↗

arXiv2026

Random Schrödinger operators on manifolds and abstract bounds for multiplier-type operators

We study random Schrödinger operators on closed Riemannian manifolds with Anderson-type potentials. We prove high-probability spectral inclusion bounds showing that eigenvalues remain close to those of the Laplacian, with deviations controlled by a norm of the potential coefficients. Compared with deterministic bounds, this yields a square-root cancellation gain. The proof is based on a general principle showing that randomisation improves operator norm bounds for multiplier-type operators, which we formulate in both discrete and continuous settings.

math.SP↗

arXiv2026

A Marcinkiewicz-Zygmund inequality and the Kadec Pełczynśki theorem in Orlicz spaces

In this paper, we extend the Marcinkiewicz--Zygmund inequality to the setting of Orlicz and Lorentz spaces. Furthermore, we generalize a Kadec--Pełczyński-type result -- originally established by the first and third authors for $L^p$ spaces with $1 \le p < 2$ -- to a broader class of Orlicz spaces defined via Young functions $ψ$ satisfying $x \le ψ(x) \le x^2$.

math.FA↗

arXiv2026

Lacunary Series, Nonlinear Functionals and Banach Space Structure

In a previous paper \cite{BT} we studied the asymptotic behavior of $\| \sum_{k=1}^N a_k X_{n_k}\|_p$ for lacunary sequences $(X_{n_k})$ of random variables in $L_p$ and used the result to give a necessary and sufficient condition for the first alternative in the Kadec-Pełczynski theorem in the case $1\le p<2$. In the present paper we extend this result for nonlinear functionals $f_k (a_1 X_{n_1}, \ldots, a_k X_{n_k})$, establishing a uniform version of the subsequence principle of Aldous \cite{ald}. Moreover, we prove Kadec-Pełczynski type theorems in Orlicz spaces $L_ψ$.

math.FA↗

arXiv2026

Eigenvalue bounds for non-self-adjoint Schrödinger operators and pseudodifferential generalizations

This is mostly a survey paper, where we collect results concerning the spectral bounds of deterministic and random Schrödinger operators with complex potentials, both on $\mathbb{R}^d$ and on compact manifolds. The survey part is complemented by a new theorem, where we extend the result on spectral bounds on compact manifolds to the case of fractional Laplacians, applying methods by Cuenin and Sogge. These bounds are formulated in terms of the $L^p$-norms of the corresponding potentials.

math.SP↗

arXiv2026

Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections

We study problems on covering $[0,1)$ by shrinking intervals centered at the points $\{q_n x\}$, where $(q_n)_{n\in \mathbb{N}}$ is a given real-valued sequence and $x \in [0,1)$ is random. For real-valued lacunary sequences $(q_n)_{n\in\mathbb{N}}$, we show that the covering radius $\frac{1}{n}$ is sharp up to a constant: there exist $C>c>0$ such that, for Lebesgue-almost all $x$, the intervals of length $\frac{C}{n}$ cover $[0,1)$ infinitely often, while this fails for intervals of length $\frac{c}{n}$. Moreover, the lower bound holds for certain sub-lacunary rates and the results partially extend to all probability measures with sufficiently fast Fourier decay. As an application, we obtain a new bound for a variant of the inhomogeneous Littlewood-Cassels problem: for any badly approximable $α$ and $γ\in\mathbb{R}$, there exists a set of badly approximable $β$ of full Hausdorff dimension such that $ \|nα-γ\| \|nβ-δ\|<C/(n\log n)$ for infinitely many $n\geqslant 1,$ uniformly in $δ\in\mathbb{R}$. This improves upon previous works of Haynes-Jensen-Kristensen, Chow-Technau, and the third author, and is best possible when one restricts to best approximations of the first factor. Second, under certain arithmetic restrictions on $(q_n)_{n\in\mathbb{N}}$, we compute the almost-sure Hausdorff dimension of limsup sets generated by intervals of size $\frac{1}{n^ν}$ for $ν\geqslant 1$, centered at $\{q_n x\}$, and intersected with Ahlfors regular compact sets such as the middle-third Cantor set. In particular, our results apply to all real-valued lacunary sequences, to integer-valued polynomials, and to powers of primes. This substantially extends the work of Bugeaud and Durand, which applies only to certain super-lacunary integer-valued sequences.

math.NT↗

arXiv2025

On the maximal volume of empty convex bodies amidst multivariate dilates of a lacunary integer sequence

Let $(a_n)_{n \in \mathbb{N}}$ be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension $d \geq 1$, we establish asymptotic upper bounds for the maximal gap in the set of dilates $\{\boldsymbolα a_n \}_{n \leq N}$ modulo 1 as $N \to \infty$, for Lebesgue--almost all dilation vectors $\boldsymbolα \in [0,1]^d$. More precisely, we prove that for any lacunary $(a_n)_{n \in \mathbb{N}}$ and Lebesgue--almost all $\boldsymbolα$, every convex set in $[0,1]^d$ of volume at least $(\log N)^{2+\varepsilon}/N$ must contain an element of the set $\{\boldsymbolα a_n \}_{n \leq N}$ mod 1, for all sufficiently large $N$. We also establish a generalized version of this result, where the $d$-dimensional Lebesgue measure is replaced by a general measure satisfying a certain Fourier decay condition. Our result is optimal up to logarithmic factors, and recovers as a special case a recent result for dimension $d=1$.

math.NT↗

arXiv2024

Global-in-time existence, uniqueness and stability of solutions to a model of the Antarctic Circumpolar Current

We consider a model for the Antarctic Circumpolar Current in rotating spherical coordinates. After establishing global-in-time existence and uniqueness of classical solutions, we turn our attention to the issue of stability of a class of steady zonal solutions (i.e., time-independent solutions that vary only with latitude). By identifying suitable conserved quantities and combining them to construct a Lyapunov function, we prove a stability result.

math.AP↗

arXiv2024

The atmospheric Ekman spiral for piecewise-uniform eddy viscosity

We investigate the boundary-value problem of atmospheric Ekman flows with piecewise-uniform eddy viscosity. In addition we present a method for finding more general solutions by considering eddy viscosity as an arbitrary step-function. We discuss the existence and uniqueness of the solutions obtained through this method, providing detailed proofs for cases with one and two "jumps" in eddy viscosity. For scenarios with more "jumps," we establish results inductively. Furthermore, we examine the angle between the bottom surface of the Ekman layer and geostrophic winds by extremizing variables such as the eddy viscosity and its point of change. These calculations reveal how the angle can differ from $45^\circ$, demonstrating that the extreme values of $0^\circ$ and $90^\circ$ are achievable, indicating the potential range of the deflection angle.

math-ph↗

arXiv2024

The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation

Let $(a_n)_{n \in \mathbb{N}}$ be a Hadamard lacunary sequence. We give upper bounds for the maximal gap of the set of dilates $\{a_n α\}_{n \leq N}$ modulo 1, in terms of $N$. For any lacunary sequence $(a_n)_{n \in \mathbb{N}}$ we prove the existence of a dilation factor $α$ such that the maximal gap is of order at most $(\log N)/N$, and we prove that for Lebesgue almost all $α$ the maximal gap is of order at most $(\log N)^{2+\varepsilon}/N$. The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation.

math.NT↗