arXiv research

The study confirms essential self-adjointness for certain differential operators on manifolds.

We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components …

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator…

Essential self-adjointness and spectrum of CR GJMS operator proved.

The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential eq…

We discuss the essential self-adjointness of wave operators, as well as the limiting absorption principle, in generalizations of asymptotically Minkowski settings. This is obtained via using a Fredholm framework for inverting the spectral family first, and then refining its conclusions to show its dense range in L^2 wh…

The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.

For a Riemannian covering $p \colon M_{2} \to M_{1}$, we compare the spectrum of an essentially self-adjoint differential operator $D_{1}$ on a bundle $E_{1} \to M_{1}$ with the spectrum of its lift $D_{2}$ on $p^{*}E_{1} \to M_{2}$. We prove that if the covering is infinite sheeted and amenable, then the spectrum of $…

Study essential spectrum of differential operators on geometrically finite orbifolds.

In this article we prove a generalization of Weyl's criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.

We consider perturbed quadharmonic operators, $Δ^4 + V$, acting on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential $V$ satisfying a bound from below by a non-positive function depending on the distance from a point. Under a bounded geometry assumption on the Hermitian vecto…

The paper proves positivity preservation and self-adjointness for Schrödinger operators on incomplete Riemannian manifolds.

Study Dirac operators on incomplete cusp edge spaces, proving self-adjointness and Fredholm properties.

We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms of completeness of certain metrics on M; these metrics are naturally associated …

Generalizes results for Riemannian manifolds with boundary to those without.

In this paper we show that, for a sub-Laplacian $Δ$ on a $3$-dimensional manifold $M$, no point interaction centered at a point $q_0\in M$ exists. When $M$ is complete w.r.t. the associated sub-Riemannian structure, this means that $Δ$ acting on $C^\infty_0(M\setminus\{q_0\})$ is essentially self-adjoint. A particular …

Researchers create a parametrix for resolvents on manifolds with ends.

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 f…

Study on removing sets and uniqueness of diffusion operators on various spaces.

We study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl's analysis in each fibre, thus fully characterising the regimes of presence and absence of…

In this article we prove a generalization of Weyl's criterion for the spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with Cheeger-Fukaya-Gromov and Cheeger-Colding theory to study the $k$-form essential spectrum over a complete manifold with vanishing…

We show that any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Lapla…

Graph continuous operators become Riesz continuous after multiplication by unitary operators.

In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp-Lieb's type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the…

Proves formal self-adjointness of certain differential operators.

Abstracts a construction of boundary triplets for self-adjoint elliptic problems.

John Lott has computed an integer-valued signature for the orbit space of a compact orientable $(4k+1)$ manifold with a semi-free $S^1$-action, which is a homotopy invariant of that space, but he did not construct a Dirac type operator which has this signature as its index. In this Thesis, we construct such operator on…

Quantizes Stäckel integrable systems into self-adjoint operators.

We consider an open domain with a compact boundary in an Euclidean space and a Schroedinger operator with magnetic field on this domain. We give sufficient conditions on the rate of growth of the magnetic field near the boundary which guarantees essential self-adjointness of this operator. From the physical point of vi…

Paper proves index theorem for self-adjoint elliptic boundary problems.

Let $Δ$ be a linear differential operator acting on the space of densities of a given weight $\lo$ on a manifold $M$. One can consider a pencil of operators $\hPi(Δ)=\{Δ_ł\}$ passing through the operator $Δ$ such that any $Δ_ł$ is a linear differential operator acting on densities of weight $ł$. This pencil can be iden…

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 boun…

John Lott defined an integer-valued signature $σ_{S^1}(M)$ for the orbit space of a compact orientable manifold with a semi-free $S^1$-action but he did not construct a Dirac-type operator which has this signature as its index. We construct such operator on the orbit space and we show that it is essentially unique and …

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_γ$, the Riemann moduli space of surfaces of genus $γ> 1$. This space has a singular compactification with respect to $g_{\mathrm{WP}}$, and this metric has crossing…

In these survey lectures, we investigate the geometric and analytic properties of transverse Dirac operators. In particular, we define a transverse Dirac operator associated to a distribution that is essentially self-adjoint (Prokhorenkov-R result). We describe the Habib-R Theorem showing that the invariance of the spe…

Study on CR Paneitz operator on non-embeddable CR manifolds.

Classifies local boundary conditions for Dirac-type operators on manifolds.

We introduce the notion of formally self-adjoint conformally covariant polydifferential operators and give some constructions of families of such operators. In one direction, we show that any homogeneous conformally variational scalar Riemannian invariant (CVI) induces one of these operators. In another direction, we u…

We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed.

The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.

Let $M$ be a complete Riemannian manifold and let $Ω^*(M)$ denote the space of differential forms on $M$. Let $d:Ω^*(M) \to Ω^{*+1}(M)$ be the exterior differential operator and let $\Del=dd^*+d^*d$ be the Laplacian. We establish a sufficient condition for the Schroedinger operator $H=\Del+V(x)$ (where the potential $V…

We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold $M$ equipped with a smooth measure $ω$, possibly degenerate or singular near the metric boundary of $M$, and in presence of a real-valued potential $V\in L^2_{\mathrm{loc}}(M)$. The main …

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 specif…

Resolution of a compact group action in the sense described by Albin and Melrose is applied to the conjugation action by the unitary group on self-adjoint matrices. It is shown that the eigenvalues are smooth on the resolved space and that the trivial bundle smoothly decomposes into the direct sum of global one-dimensi…

Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove…

Researchers develop weighted GJMS operators for smooth metric measure spaces.

Study Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds.

The aim of this paper is the study of the geodesic distance in operator groups with several Riemannian metrics. More precisely we study the geodesic distance in self-adjoint operator groups with the left invariant Riemannian metric induced by the infinite trace and extend known results about the completeness of some cl…