Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics

Abstract

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 possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the operator and its higher powers up to the regularity of its coefficients. We also present applications to Dirac operators on Dirac bundles when the metric is non-smooth.

Essential Self-adjointness of Symmetric First-Order Differential Systems and Confinement of Dirac Particles on Bounded Domains in $${\mathbb {R}}^d$$

“…Theorem 2. 5 Let V be a scalar potential, V ∈ C 1 ( ; C k×k ), and choose t 0 as in (2.13) and h as in (2.14). Assume that there exist M < ∞, m < 2, δ 0 ∈ (0, t 0 ), and c > 0 such that M(x) M δ(x) m for all x ∈ , (2.24)…”

“…3. In addition to the classical results of Chernoff [13,14] and the results in [35] for the smooth case, criteria for essential self-adjointness of general first order matrixvalued differential operators with rough coefficients on R d were recently obtained by completely different methods in [11], and in [5] in the elliptic case. In [11], the authors consider abstract operators which, in our case, have the form…”

On the Global Limiting Absorption Principle for Massless Dirac Operators

“…In a broader picture our methods relate to the noncommutative geometry program [17], since our proof is in some sense coordinate-free and can therefore be readily generalized to a much wider array of unbounded operators of the form j A j · D j + B (with D j and D k commuting for all 1 j, k n), provided that an appropriate reference operator, for example j D 2 j , is already well-understood. For a very recent approach to essential self-adjointness of first-order differential operators with applications to Dirac-type operators we refer to [6] (the approach in [6] is quite different, relying on ellipticity conditions which are not used in our setup).…”

“…Let η : R n → R denote a radial function of the formη(p) = h(|p|), p ∈ R n , (3.5) where h : [0, ∞) → [0, 1] is defined by h(r) := ∈ [0, 1/2), k(r), r ∈ [1/2, 1), 1, r ∈ [1, ∞),(3 6). and the function k : [1/2, 1) → [0, ∞) is nondecreasing and chosen so that η ∈ C ∞ (R n \{0}).…”

Self‐adjointness of perturbed biharmonic operators on Riemannian manifolds

“…In the last two decades, there has been a lot of activity on the problem of self‐adjointness of Schrödinger operators on Riemannian manifolds (including operators acting on sections of Hermitian vector bundles), as seen, for instance, in . Recently, the authors of proved, among other things, the essential self‐adjointness of powers of first‐order elliptic operators with low‐regularity coefficients on vector bundles with low‐regularity coefficient metrics over manifolds with low‐regularity metrics.…”