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Konstantin Merz

Publications and source records attributed to Konstantin Merz.

At least 19 recordsLinked to original sources
arXiv2026

Random Schr\"odinger operators on manifolds and abstract bounds for multiplier-type operators

We study random Schr\"odinger operators on closed Riemannian manifolds with Anderson-type potentials. We prove high-probability spectral inclusion bounds showing that eigenvalues remain close to those of the Laplacian, with deviations controlled by a norm of the potential coefficients. Compared with deterministic bounds, this yields a square-root cancellation gain. The proof is based on a general principle showing that randomisation improves operator norm bounds for multiplier-type operators, which we formulate in both discrete and continuous settings.

math.SP↗
arXiv2024

Equivalence of Sobolev norms for Kolmogorov operators with scaling-critical drift

We consider the ordinary or fractional Laplacian plus a homogeneous, scaling-critical drift term. This operator is non-symmetric but homogeneous, and generates scales of $L^p$-Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. Unlike in previous studies concerning Hardy operators, i.e., ordinary or fractional Laplacians plus scaling-critical scalar perturbations, handling the drift term requires an additional, possibly technical, restriction on the range of comparable Sobolev spaces, which is related to the unavailability of gradient bounds for the associated semigroup.

math.AP↗
arXiv2024

Hardy perturbations of subordinated Bessel heat kernels

Motivated by the spectral theory of relativistic atoms, we prove matching upper and lower bounds for the transition density of Hardy perturbations of subordinated Bessel heat kernels. The analysis is based on suitable supermedian functions, in particular invariant functions.

math.AP↗
arXiv2023

Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space

We consider Hardy operators, i.e., homogeneous Schr\"odinger operators consisting of the ordinary or fractional Laplacian in a half-space plus a potential, which only depends on the appropriate power of the distance to the boundary of the half-space. We compare the scales of homogeneous $L^p$-Sobolev spaces generated by these Hardy operators with and without potential with each other. To that end, we prove and use new square function estimates for operators with slowly decaying heat kernels. Our results hold for all admissible coupling constants in the local case and for repulsive potentials in the fractional case, and extend those obtained recently in $L^2$. They also cover attractive potentials in the fractional case, once expected heat kernel estimates are available.

math.AP↗
arXiv2023

Subordinated Bessel heat kernels

We prove new bounds for Bessel heat kernels and Bessel heat kernels subordinated by stable subordinators. In particular, we provide a 3G inequality in the subordinated case.

math.AP↗
arXiv2023

Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels

Motivated by the study of relativistic atoms, we consider the Hardy operator $(-\Delta)^{\alpha/2}-\kappa|x|^{-\alpha}$ acting on functions of the form $u(|x|) |x|^{\ell} Y_{\ell,m}(x/|x|)$ in $L^2(\mathbb{R}^d)$, when $\kappa\geq0$ and $\alpha\in(0,2]\cap(0,d+2\ell)$. We give a ground state representation of the corresponding form on the half-line (Theorem 1.5). For the proof we use subordinated Bessel heat kernels.

math.AP↗
arXiv2023

On Sobolev norms involving Hardy operators in a half-space

We consider Hardy operators on the half-space, that is, ordinary and fractional Schr\"odinger operators with potentials given by the appropriate power of the distance to the boundary. We show that the scales of homogeneous Sobolev spaces generated by the Hardy operators and by the fractional Laplacian are comparable with each other when the coupling constant is not too large in a quantitative sense. Our results extend those in the whole Euclidean space and rely on recent heat kernel bounds.

math.AP↗
arXiv2022

The Scott conjecture for large Coulomb systems: a review

We review some older and more recent results concerning the energy and particle distribution in ground states of heavy Coulomb systems. The reviewed results are asymptotic in nature: they describe properties of many-particle systems in the limit of a large number of particles. Particular emphasis is put on models that take relativistic kinematics into account. While non-relativistic models are typically rather well understood, this is generally not the case for relativistic ones and leads to a variety of open questions.

math-ph↗
arXiv2022

Random Schr\"odinger operators with complex decaying potentials

We prove that the eigenvalues of a continuum random Schr\"odinger operator $-\Delta+ V_{\omega}$ of Anderson type, with complex decaying potential, can be bounded (with high probability) in terms of an $L^q$ norm of the potential for all $q\leq d+1$. This shows that in the random setting, the exponent $q$ can be essentially doubled compared to the deterministic bounds of Frank (Bull. Lond. Math. Soc., 2011). This improvement is based on ideas of Bourgain (Discrete Contin. Dyn. Syst., 2002) related to almost sure scattering for lattice Schr\"odinger operators.

math.SP↗
arXiv2020

Relativistic Strong Scott Conjecture: A Short Proof

We consider heavy neutral atoms of atomic number $Z$ modeled with kinetic energy $(c^2p^2+c^4)^{1/2}-c^2$ used already by Chandrasekhar. We study the behavior of the one-particle ground state density on the length scale $Z^{-1}$ in the limit $Z,c\to\infty$ keeping $Z/c$ fixed. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale.

math-ph↗
arXiv2020

Weak coupling limit for Schr\"odinger-type operators with degenerate kinetic energy for a large class of potentials

We improve results by Frank, Hainzl, Naboko, and Seiringer [12] and Hainzl and Seiringer [20] on the weak coupling limit of eigenvalues for Schr\"odinger-type operators whose kinetic energy vanishes on a codimension one submanifold. The main technical innovation that allows us to go beyond the potentials considered in [12, 20] is the use of the Tomas-Stein theorem.

math-ph↗
arXiv2019

Proof of the Strong Scott Conjecture for Chandrasekhar Atoms

We consider a large neutral atom of atomic number $Z$, taking relativistic effects into account by assuming the dispersion relation $\sqrt{c^2p^2+c^4}$. We study the behavior of the one-particle ground state density on the length scale $Z^{-1}$ in the limit $Z,c\to\infty$ keeping $Z/c$ fixed and find that the spherically averaged density as well as all individual angular momentum densities separately converge to the relativistic hydrogenic ones. This proves the generalization of the strong Scott conjecture for relativistic atoms and shows, in particular, that relativistic effects occur close to the nucleus. Along the way we prove upper bounds on the relativistic hydrogenic density.

math-ph↗