Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

In the space L ²0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L ²0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in 0,1] $$ . We prove the trace formula Σ _n=1 ^∞ µ _n ? λ _n ? Σ _k=1 ^m α _k ⁽ⁿ⁾ ] = 0.