Regularized Trace of a Sturm–Liouville Operator on a Curve with a Regular Singularity on the Chord

Abstract

For a Sturm–Liouville operator on a piecewise smooth curve, we study the effect that the spectrum of a nonintegrable singularity of the potential on the segment joining the endpoints of this curve has on the operator asymptotics. It is shown that in the case where the singular point does not give rise to the branching of solutions in its neighborhood (the case of trivial monodromy), the spectral asymptotics and the formula for the regularized trace look the same as for the classical Sturm–Liouville operator on a segment with smooth potential. Further, it is shown that in the case of nontrivial monodromy the spectral asymptotics substantially depends on the commensurability of the parts into which the segment is partitioned by the singular point $$\theta $$ : if $$\theta $$ is rational, then the spectrum is divided into finitely many series, each going to infinity along “its own” parabola. In this case, the regularized trace formula is significantly more complicated and does not show any similarity with the classical formula.