Essential self-adjointness of symmetric linear relations associated to first order systems

Abstract

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 equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss criteria for essential self-adjointness of such systems. A byproduct of the regularity result is a short proof of a result due to Kogan and Rofe-Beketov (V. Kogan and F. Rofe-Beketov: “On square-integrable solutions of symmetric systems of differential equations of arbitrary order”, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 5-40): the so-called formal deficiency indices of a symmetric first order system are locally constant on ℂ∖ℝ. The regularity and its corollary are based on joint work with Mark Malamud. Details will be published elsewhere. In the second part we consider a complete riemannian manifold, M, and a first order differential operator, D:C 0 ∞ (E)→C 0 ∞ (F), acting between sections of the hermitian vector bundles E,F. Moreover, let V:C ∞ (E)→L loc ∞ (E) be a self-adjoint zero order differential operator. We give a sufficient condition for the Schrödinger operator H=D t D+V to be essentially self-adjoint. This generalizes recent work of I. Oleinik (I. Oleinik: “On the essential self-adjointness of the Schrödinger operator on a complete Riemannian manifold”, Math. Notes 54 (1993), 934-939), (I. Oleinik: “On the connection of the classical and quantum mechanical completeness of a potential at infinity on complete riemannian manifolds”, Math. Notes 55 (1994), 380-386), (I. Oleinik: “On the essential self-adjointness of the general second order elliptic operators”, Proc. Amer. Math. Soc. 127 (1999), 889-900), M. Shubin (M. Shubin: “Classical and quantum completeness for the Schrödinger operators on non-compact manifolds”, In: B. Booß-Bavnbek and K. P. Wojciechowski (eds.), “Geometric aspects of partial differential equations” (Roskilde, 2000), vol. 242 of Contemp. Math. Amer. Math. Soc., Providence, RI (1999), pp. 257-269), (M. Shubin: “Essential self-adjointness for magnetic Schrödinger operators on non-compact manifolds”, In: Seminaire: Équations aux Dérivées Partielles, 2000-1999). École Polytech., Palaiseau (1999), pp. Exp. No. XV, 24), and M. Braverman (M. Braverman: “On self-adjointness of a Schrödinger operator on differential forms”, Proc. Amer. Math. Soc. 126 (2000), 617-623). We essentially use the method of Shubin. Our presentation shows that there is a close link between Shubin’s self-adjointness condition for the Schrödinger operator and Chernoff’s self-adjointness condition for powers of first order operators. We also discuss non-elliptic operators. However, in this case we need an additional assumption. We conjecture that the additional assumption turns out to be obsolete in general. The criteria we are going to present in the first and second part of this note are very closely related. In fact, after we had done the second part, we saw that the theory can be extended to symmetric linear relations associated to symmetric first order systems.