Estimates of Eigenfunctions for One Class of Boundary Conditions

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 − 2p 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 { y λ } yield certain restrictions on the spectrum. 1991 Mathematics Subject Classification 34L05.