Asymptotics of Zeros of an Entire Function With an Integral Representation, Connected by a Regular Loaded Differentiation Operator on an Interval

Abstract

In this paper, we construct a characteristic determinant of the spectral problem for a loaded first-order differential equation on an interval with a periodic boundary value condition, which is an entire analytical function of spectral parameter. Based on the formula of the characteristic determinant, conclusions about the asymptotic behavior of the spectrum of the loaded first-order differential equation are drawn on an interval. Adjoint operator is constructed. Moreover, we show that the spectral questions of the adjoint operator have a similar structure. A special feature of the considered operator is the non-self-adjointness of the operator in L 2 (−1, 1).