On confining potentials and essential self-adjointness for schrödinger operators on bounded domains in ℝn

Abstract

Let Ω be a bounded domain in ℝn with C 2-smooth boundary, ∥Ω, of co-dimension 1, and let H = -Δ + V(x) be a Schrödinger operator on Ω with potential V ∈ Lloc∞(Ω). We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary ∥Ω which guarantee essential self-adjointness of H on C 0∞(Ω). As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition V(x) ≥ 3/4d(x)2 where d(x) = dist(x, ∥Ω). More precisely, we show that if, as x approaches ∥Ω, V(x) ≥ 1/d(x) 2 (3/4 - 1/ln(d(x)-1) - 1(d(x)-1) ̇ In ln(d(x)-1) - ⋯) where the brackets contain an arbitrary finite number of logarithmic terms, then H is essentially self-adjoint on C 0∞(Ω). The constant 1 in front of each logarithmic term is optimal. The proof is based on a refined Agmon exponential estimate combined with a well-known multidimensional Hardy inequality. © 2009 Birkhäuser Verlag Basel/Switzerland.