Sturm–Liouville operators and their spectral functions

Abstract

Assume that the differential operator − DpD+ q in L 2(0,∞) has 0 as a regular point and that the limit-point case prevails at ∞. If p≡1 and q satisfies some smoothness conditions, it was proved by Gelfand and Levitan that the spectral functions σ( t) for the Sturm–Liouville operator corresponding to the boundary conditions ( pu′)(0)= τu(0), τ∈ R , satisfy the integrability condition ∫ R dσ(t)/(|t|+1)<∞ . The boundary condition u(0)=0 is exceptional, since the corresponding spectral function does not satisfy such an integrability condition. In fact, this situation gives an example of a differential operator for which one can construct an analog of the Friedrichs extension, even though the underlying minimal operator is not semibounded. In the present paper it is shown with simple arguments and under mild conditions on the coefficients p and q, including the case p≡1, that there exists an analog of the Friedrichs extension for nonsemibounded second order differential operators of the form − DpD+ q by establishing the above mentioned integrability conditions for the underlying spectral functions.