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Joaquim Duran

Publications and source records attributed to Joaquim Duran.

7 recordsLinked to original sources
arXiv2026

A Payne-Weinberger inequality for quantum dot Dirac operators

In this work we prove a Payne-Weinberger type inequality for quantum dot Dirac operators defined on bounded and simply connected planar domains. This is a sharp upper bound for their first positive eigenvalue depending only on the isoperimetric deficit of the domain. To this end, we use a recently studied connection with the so-called $\overline\partial$-Robin Laplacian and we establish an analogous inequality for its first eigenvalue, relying on the corresponding inequality for the Robin Laplacian.

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arXiv2026

Eigenvalue bounds for quantum dot Dirac operators

We exploit the connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians. First, we find a graphical relation between their smallest positive eigenvalues, which allows us to deduce a recipe for translating bounds (from above and below) from one to the other. As an application, we provide new upper and lower bounds for the eigenvalues of the quantum dot Dirac operators, which depend only on geometric quantities of the underlying domain. In particular, we obtain some Faber-Krahn type inequalities for convex thin domains.

math.AP↗
arXiv2026

A connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians in the context of shape optimization problems

This work addresses Faber-Krahn-type inequalities for quantum dot Dirac operators with nonnegative mass on bounded domains in $\mathbb{R}^2$. We show that this family of inequalities is equivalent to a family of Faber-Krahn-type inequalities for $\overline\partial$-Robin Laplacians. Thanks to this, we prove them in the case of simply connected domains for quantum dot boundary conditions asymptotically close to zigzag boundary conditions. Finally, we also study the case of negative mass.

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arXiv2026

The $\overline\partial$-Robin Laplacian

We study the family of operators $\{\mathcal{R}_a\}_{a\in [0,+\infty)}$ associated to the Robin-type problems in a bounded domain $Ω\subset\mathbb{R}^2$ $$ \begin{cases} -Δu = f & \text{in } Ω, \\ 2 \bar ν\partial_{\bar z} u + au = 0 & \text{on } \partialΩ, \end{cases} $$ and their dependency on the boundary parameter $a$ as it moves along $[0,+\infty)$. In this regard, we study the convergence of such operators in a resolvent sense. We also describe the eigenvalues of such operators and show some of their properties, both for all fixed $a$ and as functions of the parameter $a$. As shall be seen in more detail in arXiv:2507.18698, the eigenvalues of these operators characterize the positive eigenvalues of quantum dot Dirac operators.

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arXiv2025

A survey on the resolvent convergence

This chapter deals with the notion of the resolvent of a self-adjoint operator. We pay special attention to the convergence of unbounded self-adjoint operators in several resolvent senses, and how they are related to the convergence of their spectra. We also explore the relations that these notions of convergence have with the so-called strong graph limit, $G$-convergence, and $Γ$-convergence.

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arXiv2024

Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions

This work addresses the resolvent convergence of generalized MIT bag operators to Dirac operators with zigzag type boundary conditions. We prove that the convergence holds in strong but not in norm resolvent sense. Moreover, we show that the only obstruction for having norm resolvent convergence is the existence of an eigenvalue of infinite multiplicity for the limiting operator. More precisely, we prove the convergence of the resolvents in operator norm once projected into the orthogonal of the corresponding eigenspace.

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