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Batu Güneysu

Publications and source records attributed to Batu Güneysu.

49 records · Page 3Linked to original sources
arXiv2015

Scattering theory of the Hodge-Laplacian under a conformal perturbation

Let $g$ and $\tilde{g}$ be Riemannian metrics on a noncompact manifold $M$, which are conformally equivalent. We show that under a very mild \emph{first order} control on the conformal factor, the wave operators corresponding to the Hodge-Laplacians $Δ_g$ and $Δ_{\tilde{g}}$ acting on differential forms exist and are complete. We apply this result to Riemannian manifolds with a bounded geometry and more specifically, to warped product Riemannian manifolds with a bounded geometry. Finally, we combine our results with some explicit calculations by Antoci to determine the absolutely continuous spectrum of the Hodge-Laplacian on $j$-forms for a large class of warped product metrics.

math.DG↗
arXiv2015

A Feynman-Kac-Itô Formula for magnetic Schrödinger operators on graphs

In this paper we prove a Feynman-Kac-Itô formula for magnetic Schrödinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrödinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman-Kac-Itô formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman-Kac-Itô formula, we prove a Kato inequality, a Golden-Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials.

math-ph↗
arXiv2014

Sequences of Laplacian cut-off functions

We derive several new applications of the concept of sequences of Laplacian cut-off functions on Riemannian manifolds (which we prove to exist on geodesically complete Riemannian manifolds with nonnegative Ricci curvature): In particular, we prove that this existence implies $\mathsf{L}^q$-estimates of the gradient, a new density result of smooth compactly supported functions in Sobolev spaces on the whole $\mathsf{L}^q$-scale, and a slightly weaker and slightly stronger variant of the conjecture of Braverman, Milatovic and Shubin on the nonnegativity of $\mathsf{L}^2$-solutions $f$ of $(-Δ+1)f\geq 0$. The latter fact is proved within a new notion of positivity preservation for Riemannian manifolds which is related to stochastic completeness.

math.DG↗
arXiv2014

The Calderón-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds

We introduce the concept of Calderón-Zygmund inequalities on Riemannian manifolds. For $1 0$. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calderón-Zygmund inequality (like new denseness results for second order $\mathsf{L}^p$-Sobolev spaces and gradient estimates), we establish sufficient geometric criteria for the validity of these inequalities on possibly noncompact Riemannian manifolds. These results in particular apply to many noncompact hypersurfaces of constant mean curvature.

math.DG↗
arXiv2014

$\mathsf{L}^1$-elliptic regularity and $H=W$ on the whole $\mathsf{L}^p$-scale on arbitrary manifolds

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sections in these spaces under a generalized ellipticity condition on the underlying family. In particular, this implies a covariant version of Meyers-Serrin\rq{}s theorem on the whole $\mathsf{L}^p$-scale, for arbitrary Riemannian manifolds. Furthermore, we prove a new local elliptic regularity result in $\mathsf{L}^1$ on the Besov scale, which shows that the above generalized ellipticity condition is satisfied on the whole $\mathsf{L}^p$-scale, if some differential operator from $\mathfrak{P}$ that has a sufficiently high (but not necessarily the highest) order is elliptic.

math.AP↗
arXiv2014

Semiclassical limits of quantum partition functions on infinite graphs

We prove that if $H$ denotes the operator corresponding to the canonical Dirichlet form on a possibly locally infinite weighted graph $(X,b,m)$, and if $v:X\to \mathbb{R}$ is such that $H+v/\hbar$ is well-defined as a form sum for all $\hbar >0$, then the quantum partition function $\mathrm{tr}(\mathrm{e}^{-β\hbar ( H + v/\hbar)})$ satisfies $$ \mathrm{tr}(\mathrm{e}^{-β\hbar ( H + v/\hbar)})\xrightarrow[]{\hbar\to 0+}\sum_{x\in X} \mathrm{e}^{-βv(x)} \text{ for all $β>0$}, $$ regardless of the fact whether $\mathrm{e}^{-βv}$ is apriori summable or not. We also prove natural generalizations of this semiclassical limit to a large class of covariant Schrödinger operators that act on sections in Hermitian vector bundle over $(X,m,b)$, a result that particularly applies to magnetic Schrödinger operators that are defined on $(X,m,b)$.

math-ph↗
arXiv2013

Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below

After establishing some new global facts (like a measure theoretic structure theorem and approximation results) about complex-valued functions with bounded variation on arbitrary noncompact Riemannian manifolds, we extend results of Miranda/the second author/Paronetto/Preunkert and of Carbonaro/Mauceri on the heat semigroup characterization of the variation of L^1-functions to a class of Riemannian manifolds with possibly unbounded from below Ricci curvature.

math.DG↗
arXiv2012

Path integrals and the essential self-adjointness of differential operators on noncompact manifolds

We consider Schrödinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on the space of smooth functions with compact support, and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli-Dirac operators that describe the energy of Hydrogen type atoms on Riemannian 3-manifolds.

math-ph↗
arXiv2012

On generalized Schrödinger semigroups

We extend the Feynman-Kac formula for Schrödinger type operators on vector bundles over noncompact Riemannian manifolds to possibly very singular potentials that appear in hydrogen like quantum mechanical problems and that need not be bounded from below or locally square integrable. This path integral formula is then used to prove several L^p-type results, like bounds on the ground state energy and L^2 -> L^p smoothing properties of the corresponding Schrödinger semigroups. As another main result, we will prove that with a little control on the Riemannian structure, the latter semigroups are also L^2->{bounded continuous} smoothing for Kato decomposable potentials. These results in particular apply to a very general class of magnetic Schrödinger operators on Riemannian manifolds.

math-ph↗
arXiv2011

Kato's inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold with Laplace-Beltrami operator $-Δ$ and let $E\to M$ be a Hermitian vector bundle with a Hermitian covariant derivative $\nabla$. Furthermore, let H(0) denote the Friedrichs realization of $\nabla^*\nabla$ and let $V$ be a potential. We prove that $V^-$ is H(0)-form bounded with bound $<1$, if the function $\maxσ(V^-)$ is in the Kato class of $(M,g)$. In particular, this gives a sufficient condition under which one can define the form sum $H(V):=H(0)\dotplus V$ on arbitrary Riemannian manifolds.

math-ph↗
arXiv2011

Multiplicative matrix-valued functionals and the continuity properties of semigroups correspondings to partial differential operators with matrix-valued coefficients

We define and examine certain matrix-valued multiplicative functionals with local Kato potential terms and use probabilistic techniques to prove that the semigroups of the corresponding partial differential operators with matrix-valued coefficients are spatially continuous and have a jointly continuous integral kernel. These partial differential operators include Yang-Mills type Hamiltonians and Pauli type Hamiltonians, with "electrical" potentials that are elements of the matrix-valued local Kato class.

math-ph↗