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Ognjen Milatovic

Publications and source records attributed to Ognjen Milatovic.

At least 19 recordsLinked to original sources
arXiv2026

Extended Sobolev Scale on Non-Compact Manifolds

Adapting the definition of ``extended Sobolev scale" on compact manifolds by Mikhailets and Murach to the setting of a (generally non-compact) manifold of bounded geometry $X$, we define the ``extended Sobolev scale" $H^φ(X)$, where $φ$ is a function which is $RO$-varying at infinity. With the help of the scale $H^φ(X)$, we obtain a description of all Hilbert function-spaces that serve as interpolation spaces with respect to a pair of Sobolev spaces $[H^{(s_0)}(X), H^{(s_1)}(X)]$, with $s_0<s_1$. We use this interpolation property to establish a mapping property of proper uniform pseudo-differential operators (PUPDOs) in the context of the scale $H^φ(X)$. Additionally, using a first-order positive-definite PUPDO $A$ of elliptic type we define the ``extended $A$-scale" $H^φ_{A}(X)$ and show that it coincides, up to norm equivalence, with the scale $H^φ(X)$. Besides the mentioned results, we show that further properties of the $H^φ$-scale, originally established by Mikhailets and Murach on $\mathbb{R}^n$ and on compact manifolds, carry over to manifolds of bounded geometry.

math.AP↗
arXiv2025

Essential self-adjointness of non-semibounded Schrödinger operators on infinite graphs

We work in the setting of infinite, not necessarily locally finite, weighted graphs. We give a sufficient condition for the essential self-adjointness of (discrete) Schrödinger operators $\mathcal{L}_{V}$ that are not necessarily lower semi-bounded. As a corollary of the main result, we show that $\mathcal{L}_{V}$ is essentially self-adjoint if the potential $V$ satisfies $V(x)\geq -b_1-b_2[ρ(0,x)]^2$, for all vertices $x$, where $o$ is a fixed vertex, $b_1$ and $b_2$ are non-negative constants, and $ρ$ is an intrinsic metric of finite jump size, such that the restriction of the weighted vertex degree to every ball corresponding to $ρ$ is bounded (not necessarily uniformly bounded).

math.SP↗
arXiv2025

On two properties of positively perturbed discrete Schrödinger operators

We show that if we start from a symmetric lower semi-bounded Schrödinger operator $\mathcal{H}$ on finitely supported functions on a discrete weighted graph (satisfying certain conditions), apply the Friedrichs construction to get a self-adjoint extension $H$, and then perturb $H$ by a non-negative function $W$, then the resulting form-sum $H\widetilde{+}W$ coincides with the Friedrichs extension of $\mathcal{H}+W$. Additionally, we consider a non-negative perturbation of an essentially self-adjoint lower semi-bounded Schrödinger operator $H$ on a discrete weighted graph. We show that, under certain conditions on the graph and the perturbation, the essential self-adjointness of $H$ remains stable under the given perturbation.

math.SP↗
arXiv2024

The essential adjointness of pseudo-differential operators on $\mathbb{Z}^n$

In the setting of the lattice $\mathbb{Z}^n$ we consider a pseudo-differential operator $A$ whose symbol belongs to a class defined on $\mathbb{Z}^n\times \mathbb{T}^n$, where $\mathbb{T}^n$ is the $n$-torus. We realize $A$ as an operator acting between the discrete Sobolev spaces $H^{s_j}(\mathbb{Z}^n)$, $s_j\in\mathbb{R}$, $j=1,2$, with the discrete Schwartz space serving as the domain of $A$. We provide a sufficient condition for the essential adjointness of the pair $(A,\,A^{\dagger})$, where $A^{\dagger}$ is the formal adjoint of $A$.

math.FA↗
arXiv2024

Covariant Schrödinger Operator and $L^2$-Vanishing Property on Riemannian Manifolds

Let $M$ be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let $\mathcal{E}$ be a Hermitian vector bundle over $M$ equipped with a metric covariant derivative $\nabla$. We consider the operator $H_{X,V}=\nabla^{\dagger}\nabla+\nabla_{X}+ V$, where $\nabla^{\dagger}$ is the formal adjoint of $\nabla$ with respect to the inner product in the space of square-integrable sections of $\mathcal{E}$, $X$ is a smooth (real) vector field on $M$, and $V$ is a fiberwise self-adjoint, smooth section of the endomorphism bundle $\textrm{End }\mathcal{E}$. We give a sufficient condition for the triviality of the $L^2$-kernel of $H_{X,V}$. As a corollary, putting $X\equiv 0$ and working in the setting of a Clifford module equipped with a Clifford connection $\nabla$, we obtain the triviality of the $L^2$-kernel of $D^2$, where $D$ is the Dirac operator corresponding to $\nabla$. In particular, when $\mathcal{E}=Λ_{\mathbb{C}}^{k}T^*M$ and $D^2$ is the Hodge--deRham Laplacian on (complex-valued) $k$-forms, we recover some recent vanishing results for $L^2$-harmonic (complex-valued) $k$-forms.

math.DG↗
arXiv2023

Extended Sobolev Scale on $\mathbb{Z}^n$

In analogy with the definition of ``extended Sobolev scale" on $\mathbb{R}^n$ by Mikhailets and Murach, working in the setting of the lattice $\mathbb{Z}^n$, we define the ``extended Sobolev scale" $H^φ(\mathbb{Z}^n)$, where $φ$ is a function which is $RO$-varying at infinity. Using the scale $H^φ(\mathbb{Z}^n)$, we describe all Hilbert function-spaces that serve as interpolation spaces with respect to a pair of discrete Sobolev spaces $[H^{(s_0)}(\mathbb{Z}^n), H^{(s_1)}(\mathbb{Z}^n)]$, with $s_0<s_1$. We use this interpolation result to obtain the mapping property and the Fredholmness property of (discrete) pseudo-differential operators (PDOs) in the context of the scale $H^φ(\mathbb{Z}^n)$. Furthermore, starting from a first-order positive-definite (discrete) PDO $A$ of elliptic type, we define the ``extended discrete $A$-scale" $H^φ_{A}(\mathbb{Z}^n)$ and show that it coincides, up to norm equivalence, with the scale $H^φ(\mathbb{Z}^n)$. Additionally, we establish the $\mathbb{Z}^n$-analogues of several other properties of the scale $H^φ(\mathbb{R}^n)$.

math.FA↗
arXiv2022

Self-adjointness of non-semibounded covariant Schrödinger operators on Riemannian manifolds

In the context of a geodesically complete Riemannian manifold $M$, we study the self-adjointness of $\nabla^{\dagger}\nabla+V$ where $\nabla$ is a metric covariant derivative (with formal adjoint $\nabla^{\dagger}$) on a Hermitian vector bundle $\mathcal{V}$ over $M$, and $V$ is a locally square integrable section of $\textrm{End }\mathcal{V}$ such that the (fiberwise) norm of the "negative" part $V^{-}$ belongs to the local Kato class (or, more generally, local contractive Dynkin class). Instead of the lower semiboundedness hypothesis, we assume that there exists a number $\varepsilon \in [0,1]$ and a positive function $q$ on $M$ satisfying certain growth conditions, such that $\varepsilon \nabla^{\dagger}\nabla+V\geq -q$, the inequality being understood in the quadratic form sense over $C_{c}^{\infty}(\mathcal{V})$. In the first result, which pertains to the case $ε\in [0,1)$, we use the elliptic equation method. In the second result, which pertains to the case $\varepsilon=1$, we use the hyperbolic equation method.

math.AP↗
arXiv2021

Generalized Ornstein--Uhlenbeck Semigroups in weighted $L^p$-spaces on Riemannian Manifolds

Let $\mathcal{E}$ be a Hermitian vector bundle over a Riemannian manifold $M$ with metric $g$, let $\nabla$ be a metric covariant derivative on $\mathcal{E}$. We study the generalized Ornstein-Uhlenbeck differential expression $P^{\nabla}=\nabla^{\dagger}\nabla u+\nabla_{(dϕ)^{\sharp}}u-\nabla_{X}u+Vu$, where $\nabla^{\dagger}$ is the formal adjoint of $\nabla$, $(dϕ)^{\sharp}$ is the vector field corresponding to $dϕ$ via $g$, $X$ is a smooth real vector field on $M$, and $V$ is a self-adjoint locally integrable section of the bundle $\textrm{End }\mathcal{E}$. We show that (the negative of) the maximal realization $-H_{p,\max}$ of $P^{\nabla}$ generates an analytic quasi-contractive semigroup in $L^p_μ(\mathcal{E})$, $1<p<\infty$, where $dμ=e^{-ϕ}dν_{g}$, with $ν_{g}$ being the volume measure. Additionally, we describe a Feynman-Kac representation for the semigroup generated by $-H_{p,\max}$. For the Ornstein-Uhlenbeck differential expression acting on functions, that is, $P^{d}=Δu+(dϕ)^{\sharp}u-Xu+Vu$, where $Δ$ is the (non-negative) scalar Laplacian on $M$ and $V$ is a locally integrable real-valued function, we consider another way of realizing $P^{d}$ as an operator in $L^p_μ(M)$ and, by imposing certain geometric conditions on $M$, we prove another semigroup generation result.

math.AP↗
arXiv2020

Self-adjointness of perturbed bi-Laplacians on infinite graphs

We give a sufficient condition for the essential self-adjointness of a perturbation of the square of the magnetic Laplacian on an infinite weighted graph. The main result is applicable to graphs whose degree function is not necessarily bounded. The result allows perturbations that are not necessarily bounded from below by a constant.

math.FA↗
arXiv2020

Essential self-adjointness of perturbed biharmonic operators via conformally transformed metrics

We give sufficient conditions for the essential self-adjointness of perturbed biharmonic operators acting on sections of a Hermitian vector bundle over a Riemannian manifold with additional assumptions, such as lower semi-bounded Ricci curvature or bounded sectional curvature. In the case of lower semi-bounded Ricci curvature, we formulate our results in terms of the completeness of the metric that is conformal to the original one, via a conformal factor that depends on a minorant of the perturbing potential $V$. In the bounded sectional curvature situation, we are able to relax the growth condition on the minorant of $V$ imposed in an earlier article. In this context, our growth condition on the minorant of $V$ is consistent with the literature on the self-adjointness of perturbed biharmonic operators on $\mathbb{R}^n$.

math.AP↗
arXiv2018

Inequalities and separation for covariant Schrödinger operators

We consider a differential expression $L^{\nabla}_{V}=\nabla^{\dagger}\nabla+V$, where $\nabla$ is a metric covariant derivative on a Hermitian bundle $E$ over a geodesically complete Riemannian manifold $(M,g)$ with metric $g$, and $V$ is a linear self-adjoint bundle map on $E$. In the language of Everitt and Giertz, the differential expression $L^{\nabla}_{V}$ is said to be separated in $L^p(E)$ if for all $u\in L^p(E)$ such that $L^{\nabla}_{V}u\in L^p(E)$, we have $Vu\in L^p(E)$. We give sufficient conditions for $L^{\nabla}_{V}$ to be separated in $L^2(E)$. We then study the problem of separation of $L^{\nabla}_{V}$ in the more general $L^p$-spaces, and give sufficient conditions for $L^{\nabla}_{V}$ to be separated in $L^p(E)$, when $1<p<\infty$.

math.AP↗
arXiv2015

Self-adjointness of the Gaffney Laplacian on vector bundles

We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator.

math.FA↗
arXiv2015

Self-adjoint extensions of differential operators on Riemannian manifolds

We study $H=D^*D+V$, where $D$ is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold $M$, and $V$ is a Hermitian bundle endomorphism. In the case when $M$ is geodesically complete, we establish the essential self-adjointness of positive integer powers of $H$. In the case when $M$ is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of $H$, expressed in terms of the behavior of $V$ relative to the Cauchy boundary of $M$.

math.SP↗
arXiv2014

Maximal accretive extensions of Schrödinger operators on vector bundles over infinite graphs

Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study the essential self-adjointness of a perturbation of this Laplacian by an operator-valued potential. Additionally, we give a sufficient condition for the resulting Schrödinger operator to serve as the generator of a strongly continuous contraction semigroup in the corresponding l^{p}-space.

math-ph↗
arXiv2013

Generalized Schrödinger semigroups on infinite graphs

With appropriate notions of Hermitian vector bundles and connections over weighted graphs which we allow to be locally infinite, we prove Feynman-Kac-type representations for the corresponding semigroups and derive several applications thereof.

math-ph↗
arXiv2012

Self-adjoint extensions of discrete magnetic Schrödinger operators

Using the concept of intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven in the context of graphs of bounded degree.

math-ph↗