A new representation of solutions to the equation −y ′′ + q(x)y = ω 2 y is obtained. For every x the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter ω. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to ω which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.

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Abstract. We obtain a general class of polynomial potentials for which the Schröedinger operator has a discrete spectrum. This class includes all the scalar potentials in membrane, 5-brane, p-branes, multiple M2 branes, BLG and ABJM theories. We provide a proof of the discreteness of the spectrum of the associated Schröedinger operators. This a a first step in order to analyze BLG and ABJM supersymmetric theories from a non-perturbative point of view.

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