In this paper we show how to totally break the fully homomorphic encryption scheme of eprint 2017/458.

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We show how the von Neumann theory of self-adjoint extensions can be used to investigatequantum systems the configuration space of which can be decomposed into parts of different dimensionalities. The method can be applied in many situations; we illustrate it on examples including point contact spectroscopy, nanotube systems, microwave resonators, or spin conductance oscillations.

We consider a two-dimensional electron with an anomalous magnetic moment, <i>g</i>&gt;2, interacting with a nonzero magnetic field <i>B</i> perpendicular to the plane which gives rise to a flux <i>F</i>. Recent results about the discrete spectrum of the Pauli operator are extended to fields with the \mathcal{O}\left( {r^{ - 2 - \delta } } ight) decay at infinity: we show that if \mathcal{O}\left( {r^{ - 2 - \delta } } ight) exceeds an integer <i>N</i>, there is at least \mathcal{O}\left( {r^{ - 2 - \delta } } ight) bound states. Furthermore, we prove that weakly coupled bound states exist under mild regularity assumptions also in the zero flux case.

We consider Schrdinger operators in dimension 2 with a singular interaction supported by an infinite family of concentric spheres, analogous to a system studied by Hempel and coauthors for regular potentials. The essential spectrum covers a half line determined by the appropriate one-dimensional comparison operator; it is dense pure point in the gaps of the latter. If the interaction is nontrivial and radially periodic, there are infinitely many absolutely continuous bands; in contrast to the regular case the lengths of the p.p. segments interlacing with the bands tend asymptotically to a positive constant in the high-energy limit.