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In dimension greater than or equal to three, we investigate the spectrum of aSchr{\"o}dinger operator with a $\delta$-interaction supported on a cone whosecross section is the sphere of co-dimension two. After decomposing into fibers,we prove that there is discrete spectrum only in dimension three and that it isgenerated by the axisymmetric fiber. We get that these eigenvalues arenon-decreasing functions of the aperture of the cone and we exhibit the preciselogarithmic accumulation of the discrete spectrum below the threshold of theessential spectrum.

On the Bound States of Schr\"odinger Operators with $\delta$-interactions on Conical Surfaces