Hölder estimates for magnetic Schrödinger semigroups in $${\mathbb {R}}^{d}$$ from mirror coupling

We use the mirror coupling of Brownian motion to show that under a $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) -dependent Kato-type assumption on the possibly nonsmooth electromagnetic potential, the corresponding magnetic Schrödinger semigroup in $${\mathbb {R}}^d$$ R d has a global $$L^{p}$$ L p -to-$$C^{0,\beta }$$ C 0 , β Hölder smoothing property for all $$p\in [1,\infty ]$$ p ∈ [ 1 , ∞ ] ; in particular, his all eigenfunctions are uniformly $$\beta $$ β -Hölder continuous. This result shows that the eigenfunctions of the Hamilton operator of a molecule in a magnetic field are uniformly $$\beta $$ β -Hölder continuous under weak $$L^q$$ L q -assumptions on the magnetic potential.

On the images of Sobolev spaces under the Schrödinger semigroup

In this article, we consider the Schrödinger semigroup for the Laplacian Δ on {\mathbb{R}^{n}} , and characterize the image of a Sobolev space in {L^{2}(\mathbb{R}^{n},e^{u^{2}}du)} under this semigroup as weighted Bergman space (up to equivalence of norms). Also we have a similar characterization for Hermite Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator H on {\mathbb{R}^{n}} .

THE REPRESENTATION OF STRONGLY CONTINUOUS OPERATOR SEMIGROUPS ON VECTOR BUNDLES AS HIDA DISTRIBUTIONS

Let E → M be a vector bundle over a compact manifold. In this short note we prove that the Schrödinger semigroup (complex heat semigroup) associated to a self-adjoint (non-negative self-adjoint) operator in ΓL2(M, E) corresponds to a ΓL2(M, E)*-valued Hida distribution, which depends continuously (holomorphically) on the time parameter.