(3+1)-spectrum of neutrino masses: A chance for LSND?

If active to active neutrino transitions are dominant modes of the atmospheric ($\nu_{\mu} \to \nu_{\tau}$) and the solar neutrino oscillations ($\nu_{e}\to \nu_{\mu}/\nu_{\tau}$), as is indicated by recent data, the favoured scheme which accommodates the LSND result -- the so called $(2+2)$-scheme -- should be discarded. We introduce the parameters $\eta_s^{atm}$ and $\eta_s^{sun}$ which quantify an involvement of the sterile component in the solar and atmospheric neutrino oscillations. The $(2+2)$-scheme predicts $\eta_s^{atm} + \eta_s^{sun} = 1$ and the experimental proof of deviation from this equality will discriminate the scheme. In this connection the $(3+1)$-scheme is revisited in which the fourth (predominantly sterile) neutrino is isolated from a block of three flavour neutrinos by the mass gap $\Delta m^2_{LSND} \sim (0.4-10)$ eV$^2$. We find that in the $(3+1)$-scheme the LSND result can be reconciled with existing bounds on $\nu_e$- and $\nu_{\mu}$ - disappearance at $95-99 %$ C.L.. The generic prediction of the scheme is the $\nu_e$- and $\nu_{\mu}$ - disappearance probabilities at the level of present experimental bounds. The possibility to strengthen the bound on $\nu_{\mu}$- disappearance in the KEK -- front detector experiment is studied. We consider phenomenology of the $(3 + 1)$-scheme, in particular, its implications for the atmospheric neutrinos, neutrinoless double beta decay searches, supernova neutrinos and primordial nucleosynthesis.