We consider a model of leaky quantum wires in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction.

In this paper we obtain sharp LiebThirring inequalities for a Schrdinger operator on semiaxis with a matrix potential and show how they can be used to other related problems. Among them are spectral inequalities on star graphs and spectral inequalities for Schrdinger operators on half-spaces with Robin boundary conditions.

We consider an electron with an anomalous magnetic moment, <i>g</i> &gt; &gt; 2, confined to a plane and interacting with a nonhomogeneous magnetic field <i>B</i>, and investigate the corresponding Pauli Hamiltonian. We prove a lower bound on the number of bound states for the case when <i>B</i> is of a compact support and the related flux is <i>N</i> + <i></i>, <i></i>(0, 1]. In particular, there are at least <i>N</i> + 1 bound states if <i>B</i> does not change sign. We also consider the situation where the magnetic field is due to a localized rotationally symmetric electric current vortex in the plane. In this case the flux is zero; there is a pair of bound states for a weak coupling, and higher orbital-momentum spin-down states appearing as the current strength increases.

We propose and analyze a multivariate encryption scheme that uses odd characteristic and an embedding in its construction. This system has a very simple core map F(X) = X^2, allowing for efficient decryption. We also discuss ways to make this decryption faster with specific parameter choices. We give heuristic arguments along with experimental data to show that this scheme resists all known attacks.

We discuss formulations of boundary conditions in a quantum graph vertex and demonstrate that the so-called ST-form can be further reduced up to a form more effective in certain applications: In particular, in identifying the number of independent parameters for given ranks of two connection matrices, or in calculating the scattering matrix when both matrices are singular. The new form of boundary conditions, called the P Q R S-form, also gives a natural scheme to design generalized low and high pass quantum filters.