Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2602.18211

On the directional growth of the resolvent norm

Horia Cornean·Henrik Garde·Arne Jensen
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $A$ be a closed densely defined operator on a separable Hilbert space $\mathcal{H}$. Assume the resolvent set $ρ(A)$ is non-empty. For $z,z'\inρ(A)$ let $[z,z']$ denote the straight line segment from $z$ to $z'$. For each $z\inρ(A)$ we classify the behavior of the resolvent norm $ζ\mapsto\lVert R_A(ζ) \rVert$ near $z$. Either there are $z'\inρ(A)$, $z'\neq z$, $[z,z']\subsetρ(A)$, such that $\lVert R_A(ζ) \rVert \geq \lVert R_A(z) \rVert + C\lvert ζ-z \rvert^δ$ for $ζ\in[z,z']$ with $δ=1$ or $δ=2$, or the function $ζ\mapsto\lVert R_A(ζ) \rVert$ has a global minimum at $ζ=z$.

Explore related subjects

math.SPmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Horia Cornean, Henrik Garde, Arne Jensen. 2026-04-28. On the directional growth of the resolvent norm. https://arxiv.org/abs/2602.18211

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Schrödinger operators with accretive potentials in weighted spaces

We analyse Schrödinger operators with accretive potentials in weighted spaces. We find conditions on potentials and weights for which the Dirichlet realisation, introduced by generalised form methods, has non-empty resolvent set. We establish a domain and graph norm separation property, as well as sufficient conditions for the compactness and Schatten class of the resolvent. Moreover, we investigate the relation between discrete spectra and eigenfunctions of operators in standard and weighted spaces. As applications we extend results on the completeness of eigensystems of operators with accretive potentials from standard to weighted spaces and analyse operator matrices exhibiting a Schur dominance property, in particular, related to a wave equation with strong accretive damping.

math.SP↗
arXiv2026

Eisenstein scattering and Plancherel decomposition on cuspidal Bruhat-Tits quotients

For arithmetic quotients of Bruhat--Tits trees with finitely many cusps, we establish an explicit unitary correspondence between the spherical Eisenstein transform, with the Eisenstein series normalized by their constant terms, and the scattering transform of an associated Jacobi operator with finite core. Tracking the Haar measure, stabilizer weights, height coordinates, and cusp widths yields the Plancherel measure and shows that the absolutely continuous spectrum has multiplicity equal to the number of cusps. From a discrete Green identity we derive a matrix-valued Maass--Selberg formula for the Hermitian matrix $iS(θ)^*\partial_θS(θ)$, where $S(θ)$ is the scattering matrix. Its trace is determined by $\det S(θ)$, while the full matrix retains additional cusp-to-cusp information. After the corresponding change of normalization, the finite Schur complement obtained by eliminating the cusp rays agrees with the resonance matrix of Arends-Peterson-Weich. Using their resonance computations as input, we distinguish eigenvalues supported entirely in the finite core from poles of the scattering matrix. The Nagao and $Γ_0(T)$ quotients, together with a four-cusp quotient arising from an elliptic curve over $\mathbb F_3$, make the normalizations and matrix-valued conclusions explicit.

math.SP↗
arXiv2026

Dirichlet--Neumann bracketing for nonlocal operators

We establish Dirichlet--Neumann bracketing for the Dirichlet eigenvalues of $ψ(-Δ)$ on bounded Lipschitz domains, where $ψ$ is an arbitrary complete Bernstein function. The eigenvalues lie between $ψ$ applied to the corresponding Neumann and Dirichlet eigenvalues of the Laplacian. Both inequalities are strict whenever $ψ$ admits no meromorphic continuation to $\mathbb C \setminus \{0\}$. The proof uses quadratic forms, operator monotonicity, and an analysis of equality in resolvent comparisons. Applying the bracketing to intervals and balls gives a unified proof of simplicity of interval eigenvalues and antisymmetry of second eigenfunctions in balls under the same condition on $ψ$. For fractional powers, these recover results of Fall, Ghimenti, Micheletti and Pistoia for the interval, and of Fall, Feulefack, Temgoua and Weth and, independently, Benedikt, Bobkov, Dhara and Girg for the ball. The argument extends these conclusions to a broader class of nonlocal operators.

math.SP↗