Degenerate elliptic operators in one dimension |
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Authors: | Derek W Robinson Adam Sikora |
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Institution: | 1. Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT, 0200, Australia 2. Department of Mathematics, Macquarie University, Sydney, NSW, 2109, Australia
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Abstract: | Let H be the symmetric second-order differential operator on L 2(R) with domain ${C_c^\infty({\bf R})}Let H be the symmetric second-order differential operator on L
2(R) with domain Cc¥(R){C_c^\infty({\bf R})} and action Hj = -(c j¢)¢{H\varphi=-(c\,\varphi^{\prime})^{\prime}} where c ? W1,2loc(R){ c\in W^{1,2}_{\rm loc}({\bf R})} is a real function that is strictly positive on R\{0}{{\bf R}\backslash\{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 n = n+ún-{\nu=\nu_+\vee\nu_-} where n±(x)=±ò±1±x c-1{\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}} then H has a unique self-adjoint extension if and only if n ? L2(0,1){\nu\not\in L_2(0,1)} and a unique submarkovian extension if and only if n ? L¥(0,1){\nu\not\in L_\infty(0,1)}. In both cases, the corresponding semigroup leaves L
2(0,∞) and L
2(−∞,0) invariant. In addition, we prove that for a general non-negative c ? W1,¥loc(R){ c\in W^{1,\infty}_{\rm loc}({\bf R})} the corresponding operator H has a unique submarkovian extension. |
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