MgB2: superconductivity and pressure effects

We present a Ginzburg-Landau theory for a two-band superconductor with emphasis on MgB$_{2}$. Experiments are proposed which lead to identification of the possible scenarios: whether both $\sigma $- and $\pi $-bands superconduct or $\sigma $% -alone. According to the second scenario a microscopic theory of superconducting MgB$_{2}$ is proposed based on the strongly interacting $% \sigma $-electrons and non-correlated $\pi $-electrons of boron ions. The kinematic and Coulomb interactions of $\sigma $% -electrons provide the superconducting state with an anisotropic gap of $s$% *-wave symmetry. The critical temperature $T_{c}$ has a non-monotonic dependence on the distance $r$ between the centers of gravity of $\sigma $- and $\pi $-bands. The position of MgB$_{2}$ on a bell-shaped curve $T_{c}$ ($% r)$ is identified in the overdoped region. The derived superconducting density of electronic states is in agreement with available experimental and theoretical data. It is argued that the effects of pressure are crucial to identify the microscopic origin of superconductivity in MgB$_{2}$. Possibilities for $T_{c}$ increase are discussed.