arXiv subjects

Batu Güneysu

Publications and source records attributed to Batu Güneysu.

At least 37 records · Page 2Linked to original sources

arXiv2020

Molecules as metric measure spaces with Kato-bounded Ricci curvature

Set $Ψ:=-\log(\tildeΨ)$, with $\tildeΨ>0$ the ground state of an arbitrary molecule with $n$ electrons in the infinite mass limit (neglecting spin/statistics). Let $Σ\subset \IR^{3n}$ be the set of singularities of the underlying Coulomb potential. We show that the metric measure space $\IMM$ given by $\IR^{3n}$ with its Euclidean distance and the measure $$ μ(dx)=e^{-2Ψ(x)}dx $$ has a Bakry-Emery-Ricci tensor which is absolutely bounded by the the function $x\mapsto |x-Σ|^{-1}$, which we show to be an element of the Kato class induced by $\IMM$. In addition, it is shown $\IMM$ is stochastically complete, that is, the Brownian motion which is induced by a molecule is nonexplosive, and that the heat semigroup of $\IMM$ has the $L^{\infty}$-to-Lipschitz smoothing property. Our proofs reveal a fundamental connection between the above geometric/probabilistic properties and recently obtained derivative estimates for $e^Ψ$ by Fournais/Sørensen, as well as Aizenman/Simon's Harnack inequality for Schrödinger operators. Moreover, our results suggest to study general metric measure spaces having a Ricci curvature which is synthetically bounded from below/above by a function in the underlying Kato class.

math.MG↗

arXiv2019

Kac regular sets and Sobolev spaces in geometry, probability and quantum physics

Let $Ω\subset M$ be an open subset of a Riemannian manifold $M$ and let $V:M\to \IR$ be a Kato decomposable potential. With $W^{1,2}_{0}(M;V)$ the natural form domain of the Schrödinger operator $-Δ+V$ in $L^2(M)$, in this paper we study systematically the following question: Under which assumption on $Ω$ is the statement $$ \text{ for all $f\in W^{1,2}_{0}(M;V)$ with $f=0$ a.e. in $M\setminus Ω$ one has $f|_Ω\in W^{1,2}_{0}(Ω;V)$} $$ true for every such $V$? We prove that without any further assumptions on $V$, the above property is satisfied, if $Ω$ is Kac regular, a probabilistic property which means that the first exit time of Brownian motion on $M$ from $Ω$ is equal to its first penetration time to $M\setminus Ω$. In fact, we treat more general covariant Schrödinger operators acting on sections in metric vector bundles, allowing new results concerning the harmonicity of Dirac spinors on singular subsets. Finally, we prove that locally Lipschitz regular $Ω$'s are Kac regular.

math.FA↗

arXiv2019

Odd characteristic classes in entire cyclic homology and equivariant loop space homology

Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut-Chern character under the equivariant Chen integral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$ induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ to the odd homology group of $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{T}))$

math.KT↗

arXiv2018

$L^{p}$-interpolation inequalities and global Sobolev regularity results (with an appendix by Ognjen Milatovic)

On any complete Riemannian manifold $M$ and for all $p\in [2,\infty)$, we prove a family of second order $L^{p}$-interpolation inequalities that arise from the following simple $L^{p}$-estimate valid for every $u \in C^{\infty}(M)$: $$ \|\nabla u\|_{p}^p \leq \|u Δ_{p} u\|_1\in [0,\infty], $$ where $Δ_p$ denotes the $p$-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for $L^p$-solutions of the Poisson equation for all $p\in (1,\infty)$, and new global Sobolev regularity results for the singular magnetic Schrödinger semigroups.

math.AP↗

arXiv2018

Covariant Schrödinger semigroups on Riemannian manifolds

This monograph develops the theory of covariant Schrödinger semigroups acting on sections of vector bundles over noncompact Riemannian manifolds from scratch. Contents: I. Sobolev spaces on vector bundles II. Smooth heat kernels on vector bundles III. Basis differential operators in Riemannian manifolds IV. Some specific results for the minimal heat kernel V. Wiener measure and Brownian motion on Riemannian manifolds VI. Contractive Dynkin and Kato potentials VII. Foundations of covariant Schrödinger semigroups VIII. Compactness of $V(H^{\nabla}+1)^{-1}$ IX. $L^q$-properties of covariant Schrödinger semigroups X. Continuity properties of covariant Schrödinger semigroups XI. Integral kernels for covariant Schrödinger semigroups XII. Essential self-adjointness of covariant Schrödinger semigroups XIII. Smooth compactly supported sections as form core XIV. Applications (in quantum mechanics and geometric analysis)

math.DG↗

arXiv2018

Nonlinear Calderón-Zygmund inequalities for maps

Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calderón-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

math.DG↗

arXiv2018

Scattering the geometry of weighted graphs

Given two weighted graphs $(X,b_k,m_k)$, $k=1,2$ with $b_1\sim b_2$ and $m_1\sim m_2$, we prove a weighted $L^1$-criterion for the existence and completeness of the wave operators $ W_{\pm}(H_{2},H_1, I_{1,2})$, where $H_k$ denotes the natural Laplacian in $\ell^2(X,m_k)$ w.r.t. $(X,b_k,m_k)$ and $I_{1,2}$ the trivial identification of $\ell^2(X,m_1)$ with $\ell^2(X,m_2)$. In particular, this entails a very general criterion for the absolutely continuous spectra of $H_1$ and $H_2$ to be equal.

math-ph↗

arXiv2017

The BMS conjecture

I explain an open conjecture by Braverman/Milatovic/Shubin (BMS) on the positivity of square integrable solutions $f$ of $(-Δ+1)f\geq 0$ on a geodescially complete Riemannian manifold, and its connection to essential self-adjointness problems of covariant Schrödinger operators. The latter conjecture has remained open for more than 14 years now.

math.DG↗

arXiv2017

Feynman path integrals for magnetic Schrödinger operators on infinite weighted graphs

We prove a Feynman path integral formula for the unitary group $ \exp(-itL_{v,θ})$, $t\geq 0$, associated with a discrete magnetic Schrödinger operator $L_{v,θ}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$ |\exp(-itL_{v,θ})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y), $$ which controls the unitary group uniformly in the potentials in terms of a Schrödinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.

math.PR↗

arXiv2017

Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow

We prove a completely new integral criterion for the existence and completeness of the wave operators $W_{\pm}(-Δ_h,-Δ_g, I_{g,h})$ corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators $-Δ_j$, $j=1,2$, that are induced by two quasi-isometric complete Riemannian metrics $g$ and $h$ on an open manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra of $-Δ_g$ and $-Δ_h$ to coincide. Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup. Unlike all previous results, our integral criterion only requires some lower control on the Ricci curvatures and some upper control on the heat kernels, but no control at all on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow.

math.DG↗

arXiv2017

Quantitative C^1 - estimates on manifolds

We prove a $\mathsf{C}^1$-elliptic estimate of the form $ \sup_{B(x,r/2)} |\mathrm{grad} (ψ) | \leq C \left\{ \sup_{B(x,r)} |Δψ| + \sup_{B(x,r)} |ψ| \right\}, $ valid on any complete Riemannian manifold $M$ and for any smooth function $ψ$ which is defined in a nighbourhood of $B(x,r)$, with an explicit quantitative control on the constant $C=C(B(x,r))$ in terms of the curvature of the geodesic ball $B(x,r)\subset M$.

math.DG↗

arXiv2017

On the geometry of semiclassical limits on Dirichlet spaces

This paper is a contribution to semiclassical analysis for abstract Schrödinger type operators on locally compact spaces: Let $X$ be a metrizable seperable locally compact space, let $μ$ be a Radon measure on $X$ with a full support. Let $(t,x,y)\mapsto p(t,x,y)$ be a strictly positive pointwise consistent $μ$-heat kernel, and assume that the generator $H_p\geq 0$ of the corresponding self-adjoint contraction semigroup in $L^2(X,μ)$ induces a regular Dirichlet form. Then, given a function $Ψ: (0,1)\to (0,\infty)$ such that the limit $\lim_{t\to 0+}p(t,x,x)Ψ(t)$ exists for all $x\in X$, we prove that for every potential $w:X\to \mathbb{R}$ one has $$ \lim_{t \to 0+} Ψ(t)\mathrm{tr}\big(\mathrm{e}^{ -t H_p + w}\big)= \int \mathrm{e}^{-w(x) }\lim_{t \to 0+}p(t,x,x) Ψ(t) dμ(x)<\infty $$ for the Schrödinger type operator $H_p + w$, provided $w$ satisfies very mild conditions at $\infty$, that are essentially only made to guarantee that the sum of quadratic forms $ H_p + w/t$ is self-adjoint and bounded from below for small $t$, and to guarantee that $$ \int \mathrm{e}^{-w(x) }\lim_{t\to 0+}p(t,x,x) Ψ(t) dμ(x)<\infty. $$ The proof is probabilistic and relies on a principle of not feeling the boundary for $p(t,x,x)$. In particular, this result implies a new semiclassical limit result for partition functions valid on arbitrary connected geodesically complete Riemannian manifolds, and one also recovers a previously established semiclassical limit result for possibly locally infinite connected weighted graphs.

math.DG↗

arXiv2017

The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs

In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.

math.AP↗

arXiv2016

q-parabolicity of stratified pseudomanifolds and other singular spaces

The main result of this paper is a sufficient condition in order to have a compact Thom-Mather stratified pseudomanifold endowed with a $\hat{c}$-iterated edge metric on its regular part $q$-parabolic. Moreover, besides stratified pseudomanifolds, the $q$-parabolicity of other classes of singular spaces, such as compact complex Hermitian spaces, is investigated.

math.PR↗

arXiv2016

Heat kernel estimates and the relative compactness of perturbations by potentials

We consider a self-adjoint non-negative operator $H$ in a Hilbert space $\mathsf{L}^2(X,{\rm d}μ)$. We assume that the semigroup $(\mathrm{e}^{-t H})_{t>0}$ is defined by an integral kernel, $p$, which allows an estimate of the form $p(t,x,x)\le F_1(x)F_2(t)$ for all $(x,t)\in X\times\mathbb{R_+}$; we refer to $F_1$ as the \emph{control function}. We show that such an estimate leads to rather satisfying abstract results on relative compactness of perturbations of $H$ by potentials. It came as a surprise to us, however, that such an estimate holds for the Laplace-Beltrami operator on \emph{any} Riemannian manifold. In particular, using a domination principle, one can deduce from the latter fact a very general result on the relative compactness of perturbations by potentials of the Bochner Laplacian associated with a Hermitian bundle $(E, h^E,\nabla^E)$ over an arbitrary Riemannian manifold $(M,g)$; in fact, only quantities of order zero in $g$ enter in the estimates. We extend this result to weighted Riemannian manifolds, where under lower curvature bounds on the $α$-Bakry-Émery tensor one can construct quite explicit control functions, and to any weighted graph, where the control function is expressed in terms of the vertex weight function.

math.SP↗

arXiv2016

Heat kernels in the context of Kato potentials on arbitrary manifolds

By introducing the concept of \emph{Kato control pairs} for a given Riemannian minimal heat kernel, we prove that on every Riemannian manifold $(M,g)$ the Kato class $\mathcal{K}(M,g)$ has a subspace of the form $\mathsf{L}^q(M,d\varrho)$, where $\varrho$ has a continuous density with respect to the volume measure $μ_g$ (where $q$ depends on $\dim(M)$). Using a local parabolic $\mathsf{L}^1$-mean value inequality, we prove the existence of such densities for every Riemannian manifold, which in particular implies $\mathsf{L}^q_{loc}(M)\subset\mathcal{K}_{loc}(M,g)$. Based on previously established results, the latter local fact can be applied to the question of essential self-adjointness of Schrödinger operators with singular magnetic and electric potentials. Finally, we also provide a Kato criterion in terms of minimal Riemannian submersions.

math-ph↗

arXiv2016

On the semimartingale property of Brownian bridges on complete manifolds

I prove that every adapted Brownian bridge on a geodesically complete connected Riemannian manifold is a semimartingale including its terminal time, without any further assumptions on the geometry. In particular, it follows that every such process can be horizontally lifted to a smooth principal fiber bundle with connection, including its terminal time. The proof is based on a localized Hamilton-type gradient estimate by Arnaudon/Thalmaier.

math.PR↗

arXiv2016

Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields

We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it is given by a series of operator-valued time-ordered integrals, whose integrands are factorized into annihilation, preservation, creation, and scalar parts. The Feynman-Kac formula implied by these results is new in the matrix-valued case. Furthermore, we discuss stochastic differential equations and Feynman-Kac representations for an operator-valued integral kernel of the semi-group. As a byproduct we obtain analogous results for Nelson's model.

math-ph↗