The semi-classical Weyl law on complete manifolds

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.

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arXiv2026

New matrix perturbation bounds with relative strength: Perturbation of eigenspaces

Matrix perturbation bounds (such as Weyl and Davis--Kahan) are used abundantly in many areas of mathematics and data science. Many bounds (such as the above two) involve the spectral norm of the noise matrix and are sharp in worst-case analysis. In order to refine these classical bounds, we introduce a new parameter, which we refer to as the relative strength. This parameter measures the strength of the action of the noise matrix on the relevant eigenvectors of the ground matrix. It has turned out that in a number of situations, we can use the relative strength as a replacement for the spectral norm (which can be seen as the absolute strength). This has led to a number of notable improvements under certain sets of assumptions, which are frequently met in practice. A representative example is the case when the noise matrix is random. For the purpose of our study, we introduce a new method of analysis, which combines the classical contour integral argument with new (combinatorial) ideas. This method is robust and of independent interest. In the current paper, we focus on the perturbation of eigenspaces (Davis--Kahan type results). Perturbation bounds for eigenspaces are essential in statistics and theoretical computer science, and thus deserve a special treatment. Furthermore, this will lay the ground for the more technical treatment of general matrix functionals, which appears in a future paper.

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arXiv2026

One-dimensional optimisation of indefinite-weight principal eigenvalues with asymmetric Robin parameters and a Schrödinger-type perturbation

We study the minimisation of the positive principal eigenvalue for an indefinite-weight problem with asymmetric Robin parameters. The model is motivated by diffusive logistic equations in spatially heterogeneous environments, where the weight describes allocatable favourable resources and the Robin parameters measure boundary loss. After recalling the variational setting and the bang--bang reduction, we analyse the one-dimensional optimisation problem: the optimal favourable set is an interval, and the placement problem is reduced to a branchwise criterion. The key analytical tool is a shape-derivative formula for $a\mapstoλ(a)$, which shows that interior candidates are characterised by equality of the endpoint values of the positive eigenfunction, equivalently by the coupled transfer-matrix equations $f=0$ and $g=0$. We also introduce a Schrödinger-type extension with a fixed nonnegative background potential. In the coercive case we establish the corresponding principal-eigenvalue and bang--bang results, and in one dimension with constant potential we prove a compactness-type stability result showing that minimisers for small background potential converge, along subsequences, to minimisers of the unperturbed problem. No placement classification is claimed for general positive background potential. The computations are presented as numerical illustrations generated with an adaptive root-search protocol.

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