Estimates of Eigenfunctions for One Class of Boundary Conditions

Abstract

Requirements for boundary conditions are found which imply sharp by order estimates in L∞-metrics of eigenfunctions of the ordinary differential operator l ( y ) = D n y + Σ k = 0 n − 2 p n − k ( x ) D k y , p n − k ( x ) ∈ L 1 ( − 1 , 1 ) ; D = 1 i d d x , x ∈ [ − 1 , 1 ] . It is shown that the inequalities ‖ y λ ( k ) ‖ ∞ ⩽ c ∣ λ ∣ k / n ‖ y λ ‖ 2 , c = const . for the eigenfunctions {yλ} yield certain restrictions on the spectrum. 1991 Mathematics Subject Classification 34L05.