Regularized sums of half-integer powers of a Sturm-Liouville operator

Abstract

We investigate the question of the regularized sums of part of the eigenvalues z_n (lying along a direction) of a Sturm-Liouville operator. The first regularized sum is

$$\sum\nolimits_{n = 1}^\infty {(z_n - n - \frac{{c_1 }}{n} + \frac{2}{\pi } \cdot z_n arctg \frac{1}{{z_n }} - \frac{2}{\pi }) = \frac{{B_2 }}{2} - c_1 \cdot \gamma + \int_1^\infty {\left[ {R(z) - \frac{{l_0 }}{{\sqrt z }} - \frac{{l_1 }}{z} - \frac{{l_2 }}{{z\sqrt z }}} \right]} } \sqrt z dz,$$

where the z_n are eigenvalues lying along the positive semi-axis, z ²_n =λ_n,

$$l_0 = \frac{\pi }{2}, l_1 = - \frac{1}{2}, l_2 = - \frac{1}{4}\int_0^\pi {q(x) dx,} c_1 = - \frac{2}{\pi }l_2 ,$$

, B₂ is a Bernoulli number, γ is Euler's constant, and$R(z)$ is the trace of the resolvent of a Sturm-Liouville operator.