On Scales of Sobolev spaces associated to generalized Hardy operators

We consider the fractional Laplacian with Hardy potential and study the scale of homogeneous $L^p$ Sobolev spaces generated by this operator. Besides generalized and reversed Hardy inequalities, the analysis relies on a H\"ormander multiplier theorem which is crucial to construct a basic Littlewood--Paley theory. The results extend those obtained recently in $L^2$ but do not cover negative coupling constants in general due to the slow decay of the associated heat kernel.Comment: Corrected some misprints. This is a pre-print of an article published in Mathematische Zeitschrift, available online at https://doi.org/10.1007/s00209-020-02651-

The Atomic Density on the Thomas--Fermi Length Scale for the Chandrasekhar Hamiltonian

We consider a large neutral atom of atomic number $Z$, modeled by a pseudo-relativistic Hamiltonian of Chandrasekhar. We study its suitably rescaled one-particle ground state density on the Thomas--Fermi length scale $Z^{-1/3}$. Using an observation by Fefferman and Seco (1989), we find that the density on this scale converges to the minimizer of the Thomas--Fermi functional of hydrogen as $Z\to\infty$ when $Z/c$ is fixed to a value not exceeding $2/\pi$. This shows that the electron density on the Thomas--Fermi length scale does not exhibit any relativistic effects

On complex-time heat kernels of fractional Schr\"odinger operators via Phragm\'en-Lindel\"of principle

We consider fractional Schr\"odinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragm\'en-Lindel\"of theorem for polynomially bounded functions on the right complex half-plane. The interpolation leads to possibly nonoptimal off-diagonal bounds.Comment: 26 pages, added section on applications of obtained bounds. This is a pre-print of an article published in Journal of Evolution Equations, available online at https://doi.org/10.1007/s00028-022-00819-

Equivalence of Sobolev norms involving generalized Hardy operators

We consider the fractional Schr\"odinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schr\"odinger operators and have an important application in the treatment of large relativistic atoms.Comment: 16 pages; v2 contains improved results for positive coupling constant

Proof of the Strong Scott Conjecture for Heavy Atoms: the Furry Picture

We prove the convergence of the density on the scale $Z^{-1}$ to the density of the Bohr atom (with infinitely many electrons) (strong Scott conjecture) for a model that is known to describe heavy atoms accurately

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.Comment: 18 page

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

We improve results by Frank et al. [11] and Hainzl-Seiringer [19] on the weak coupling limit of eigenvalues for Schr\"odinger operators whose kinetic energy vanishes on a codimension one submanifold. The main technical innovation that allows us to go beyond the potentials considered in [11, 19] is the use of the Tomas-Stein theorem.Comment: 12 page

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.Comment: 42 page

Lieb-Thirring-type inequalities for random Schr\"odinger operators with complex potentials

We review some results and proofs on eigenvalue bounds for random Schr\"odinger operators with complex-valued potentials. We also include new Schatten norm estimates for the resolvent and use them to obtain bounds for sums of eigenvalues.Comment: 14 page