Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

Abstract

In the space L ²[0, 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 ²[0, 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 ^m_k=1 α ⁽ⁿ⁾ _k ] = 0.