arXiv subjects

Antonio Arnal

Publications and source records attributed to Antonio Arnal.

6 recordsLinked to original sources

arXiv2026

Resolvent estimates for one-dimensional Dirac operators with imaginary potentials

We investigate massive one-dimensional Dirac operators perturbed by diagonal matrix potentials of the form $i V$ where the function $V$ is real-valued and unbounded at infinity. For such operators we find an $L^2$-realization with non-empty resolvent set using generalized coercivity and Schur complement dominance techniques. In the prototypical Airy-Dirac case $V(x)=x$, $x \in \mathbb{R}$, we derive the precise asymptotic behavior of the resolvent as the spectral parameter tends to infinity and also as the mass $m$ tends to $0$. Finally, we find the asymptotics of the resolvent norm for general potentials $V$ in terms of the Airy-Dirac resolvent, which in particular yields an asymptotic shape of $\varepsilon$-pseudospectral curves and establishes the optimality of the pseudospectral region found in [32].

math.SP↗

arXiv2026

Semigroup decay for the wave equation with unbounded damping

We study the damped wave equation with a damping coefficient which is possibly singular and unbounded at infinity. In general, zero belongs to the spectrum of the corresponding generator, which prevents a uniform (exponential) decay for the energy. However, for initial conditions in a suitable subspace, a detailed analysis of the resolvent norm for low frequencies leads to sharp polynomial time-decay rates for the solution and its energy.

math.AP↗

arXiv2025

Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators

We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex $L^p$-potentials on bounded and unbounded Lipschitz domains with compact boundaries.

math.SP↗

arXiv2025

Resolvent estimates for the one-dimensional damped wave equation with unbounded damping

We study the generator $G$ of the one-dimensional damped wave equation with unbounded damping. We show that the norm of the corresponding resolvent operator, $\| (G - λ)^{-1} \|$, is approximately constant as $|λ| \to +\infty$ on vertical strips of bounded width contained in the closure of the left-hand side complex semi-plane, $\overline{\mathbb{C}}_{-} := \{λ\in \mathbb{C}: \operatorname{Re} λ\le 0\}$. Our proof rests on a precise asymptotic analysis of the norm of the inverse of $T(λ)$, the quadratic operator associated with $G$.

math.SP↗

arXiv2022

Resolvent estimates for one-dimensional Schrödinger operators with complex potentials

We study one-dimensional Schrödinger operators $\operatorname{H} = -\partial_x^2 + V$ with unbounded complex potentials $V$ and derive asymptotic estimates for the norm of the resolvent, $Ψ(λ) := \| (\operatorname{H} - λ)^{-1} \|$, as $|λ| \to +\infty$, separately considering $λ\in \operatorname{Ran} V$ and $λ\in \mathbb{R}_+$. In each case, our analysis yields an exact leading order term and an explicit remainder for $Ψ(λ)$ and we show these estimates to be optimal. We also discuss several extensions of the main results, their interrelation with some aspects of semigroup theory and illustrate them with examples.

math.SP↗

arXiv2022

Generalised Airy Operators

We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form $A := -\partial_x + W(x)$, with $W(x) \ge 0$, defined in $L^2(\mathbb{R})$. We provide a sharp estimate for the norm of its resolvent operator, $\| (A - λ)^{-1} \|$, as the spectral parameter diverges $(λ\to +\infty)$. Furthermore, we describe the $C_0$-semigroup generated by $-A$ and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schrödinger and damped wave operators and also the optimality of abstract resolvent bounds based on Carleman-type estimates.

math.SP↗