Deficiency indices and spectrum of self-adjoint extensions of some classes of differential operators

I. N. Dolgikh ; K. A. Mirzoev

Sbornik. Mathematics, Tome 197 (2006) no. 4, pp. 525-546

Résumé

Problems relating to the asymptotic behaviour in the neighbourhood of the point $+\infty$ and in the neighbourhood of the origin of a solution of an equation $l_ny=\lambda y$ of arbitrary (even or odd) order with complex-valued coefficients are studied. It is assumed here that the coefficients of the quasidifferential expression $l_n$ have the following property: if one reduces the equation $l_ny=\lambda y$ to a system of first-order differential equations, then one can transform that system to a system of differential equations with regular singular point at $x=\infty$ or $x=0$. The results obtained allow one to determine the deficiency indices of the corresponding minimal symmetric differential operators and the structure of the spectrum of self-adjoint extensions of these operators. In addition, on the basis of refined asymptotic formulae for solutions to the equation $l_ny=\lambda y$ the deficiency numbers of a certain differential operator generated by a differential expression with leading coefficient vanishing in the interior of the interval in question are found. Bibliography: 14 titles.