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Let      L = ∑  m = 0 n A m ( x ) D m L = \sum \nolimits _{m = 0}^n {{A_m}(x){D^m}}       be a differential operator on      ⊕  k L 2 [ 0 , 1 ] { \oplus ^k}{L^2}[0,1]       with infinitely differentiable      k ×  k k \times k       matrix valued coefficients. Assume that      det A n ( x ) ≠  0 \det \,{A_n}(x) \ne 0       for      x ∈  [ 0 , 1 ] x \in [0,1]      . The domain of  L  is the set of  k -vector valued functions  f  such that      f ∈  C n −  1 ( [ 0 , 1 ] ) , f ( n −  1 ) f \in {C^{n - 1}}([0,1]),{f^{(n - 1)}}       is absolutely continuous on [0, 1] and      L f ∈  ⊕  k L 2 [ 0 , 1 ] Lf \in { \oplus ^k}{L^2}[0,1]      . Let      x 0 ∈  ( 0 , 1 ) {x_0} \in (0,1)      . Then there is a neighborhood      U ( x 0 ) U({x_0})       containing      x 0 {x_0}       such that the restrictions of the eigenfunctions of  L  to      U ( x 0 ) U({x_0})       have dense span in      ⊕  k L 2 [ U ( x 0 ) ] { \oplus ^k}{L^2}[U({x_0})]      . The example      L = e −  i x d / d x L = {e^{ - ix}}d/dx       shows that this is the best possible abstract result.

Local completeness for eigenfunctions of regular maximal ordinary differential operators