Renormalized oscillation theory for Dirac operators

Teschl, Gerald

doi:10.1090/S0002-9939-98-04310-X

Renormalized oscillation theory for Dirac operators
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by Gerald Teschl

Proc. Amer. Math. Soc. 126 (1998), 1685-1695

DOI: https://doi.org/10.1090/S0002-9939-98-04310-X

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Abstract:

Oscillation theory for one-dimensional Dirac operators with separated boundary conditions is investigated. Our main theorem reads: If $\lambda _{0,1}\in \mathbb R$ and if $u,v$ solve the Dirac equation $H u= \lambda _0 u$, $H v= \lambda _1 v$ (in the weak sense) and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection $P_{(\lambda _0, \lambda _1)}(H)$ equals the number of zeros of the Wronskian of $u$ and $v$. As an application we establish finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.

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