arXiv subjects

Borbala Gerhat

Publications and source records attributed to Borbala Gerhat.

5 recordsLinked to original sources

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↗

arXiv2023

Criticality transition for positive powers of the discrete Laplacian on the half line

We study the criticality and subcriticality of powers $(-\Delta)^\alpha$ with $\alpha>0$ of the discrete Laplacian $-\Delta$ acting on $\ell^2(\mathbb{N})$. We prove that these positive powers of the Laplacian are critical if and only if $\alpha \ge 3/2$. We complement our analysis with Hardy type inequalities for $(-\Delta)^\alpha$ in the subcritical regimes $\alpha \in (0,3/2)$. As an illustration of the critical case, we describe the negative eigenvalues emerging by coupling the discrete bilaplacian with an arbitrarily small localized potential.

math.SP↗

arXiv2022

An improved discrete Rellich inequality on the half-line

Based on a new idea of factorization, we prove an improved discrete Rellich inequality and discuss its optimality. We also give a conjecture on improved higher order discrete Hardy-like inequalities and formulate an open problem for the discrete Rellich inequality on the full space.

math.SP↗

arXiv2022

Schur complement dominant operator matrices

We propose a method for the spectral analysis of unbounded operator matrices in a general setting which fully abstains from standard perturbative arguments. Rather than requiring the matrix to act in a Hilbert space $\mathcal{H}$, we extend its action to a suitable distributional triple $\mathcal{D} \subset \mathcal{H} \subset \mathcal{D}_-$ and restrict it to its maximal domain in $\mathcal{H}$. The crucial point in our approach is the choice of the spaces $\mathcal{D}$ and $\mathcal{D}_-$ which are essentially determined by the Schur complement of the matrix. We show spectral equivalence between the resulting operator matrix in $\mathcal{H}$ and its Schur complement, which allows to pass from a suitable representation of the Schur complement (e.g. by generalised form methods) to a representation of the operator matrix. We thereby generalise classical spectral equivalence results imposing standard dominance patterns. The abstract results are applied to damped wave equations with possibly unbounded and/or singular damping, to Dirac operators with Coulomb-type potentials, as well as to generic second order matrix differential operators. By means of our methods, previous regularity assumptions can be weakened substantially.

math.SP↗

arXiv2022

Pseudo Numerical Ranges and Spectral Enclosures

We introduce the new concepts of pseu\-do numerical range for operator functions and families of sesquilinear forms as well as the pseu\-do block numerical range for $n \times n$ operator matrix functions. While these notions are new even in the bounded case, we cover operator polynomials with unbounded coefficients, unbounded holomorphic form families of type (a) and associated operator families of type (B). Our main results include spectral inclusion properties of pseudo numerical ranges and pseudo block numerical ranges. For diagonally dominant and off-diagonally dominant operator matrices they allow us to prove spectral enclosures in terms of the pseudo numerical ranges of Schur complements that no longer require dominance order $0$ and not even $<1$. As an application, we establish a new type of spectral bounds for linearly damped wave equations with possibly unbounded and/or singular damping.

math.SP↗