Regularized Traces of the Airy Operator Perturbed by the Dirac Delta Function

Resumo

We consider the Sturm-Liouville operator generated in the space L²[0, +∞) by the expression l_a,b:= −d²/dx² + x + aδ(x ™ b) and the boundary condition y(0) = 0, where δ is the Dirac delta function and a and b are positive numbers. Regularized trace formulas for this operator are obtained, and some identities for the eigenvalues are found. In particular, we prove that the sum of reciprocal squares of zeros of the Airy function Ai is 4π²/(3¹/³Γ⁴(1/3)), where Γ is the Euler gamma function.