Some eigenvalue problems for the vectorial Hill'sequation

Abstract

Let Q(x) be a smooth n-by-n real symmetric matrix-valuedperiodic function, Q(x + 1) = Q(x). For 0⩽a⩽1 letλm(a) denote the mth eigenvalue of the Dirichleteigenvalue problem y''(x) + [λIn-Q(x + a)]y(x) = 0,y(0) = y(1) = 0. We prove that if, for each m, λm(a) isa constant function which is of multiplicity n, then Q(x) isof the form qIn, where q is a real constant. Using asimilar method and the stability theory developed by M G Krein for the periodic linear Hamiltonian systems, we prove thatif Q(x) is even, and λm(a) is a constant function ofmultiplicity n for all m exceeding some index m0, thenthe continuous spectrum of the vectorial Hill's operator -(d2/dx2) + Q(x) contains the half-line[λm0,∞). In addition to the previous results,we prove that, for the case n = 2, if Q(x) is a periodic evensolution of the matrix Korteweg-de Vries equationQ'''(x)-3[Q'(x)Q(x) + Q(x)Q'(x)] + Q'(x) = 0, assuming its period tobe one for convenience, then the continuous spectrum of thevectorial Hill's operator also contains a half-line [µnQ,∞), where µnQ is a Dirichlet eigenvalue of theoperator -(d2/dx2) + Q(x) on the interval[0,1].