Degenerate elliptic operators in one dimension
Abstract
Let H be the symmetric second-order differential operator on L2() with domain Cc∞() and action H=-(c ')' where c∈ W1,2 loc() is a real function which is strictly positive on \0\ but with c(0)=0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if =+- where (x)=∫ 1 x c-1 then H has a unique self-adjoint extension if and only if ∈ L2(0,1) and a unique submarkovian extension if and only if ∈ L∞(0,1). In both cases the corresponding semigroup leaves L2(0,∞) and L2(-∞,0) invariant. In addition we prove that for a general non-negative c∈ W1,∞ loc() the corresponding operator H has a unique submarkovian extension.
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