SPECTRAL PROPERTIES OF HIGHER ORDER ANHARMONIC OSCILLATORS

Abstract

Abstract. We discuss spectral properties of the self-adjoint operator − d2 tk+1 ” 2 + − α dt2 k + 1 in L 2 (R) for odd integers k. We prove that the minimum over α of the ground state energy of this operator is attained at a unique point which tends to zero as k tends to infinity. Moreover, we show that the minimum is non-degenerate. These questions arise naturally in the spectral analysis of Schrödinger operators with magnetic field. This extends or clarifies previous results by Pan-Kwek [11], Helffer-Morame [8], Aramaki [1], Helffer-Kordyukov [4, 6, 7] and Helffer [3]. 1

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