  We interpret the neutrino anomalies in neutrino oscillation experiments and
the high energy neutrino events at IceCube in terms of neutrino oscillations in
an extension of the standard model where three sterile neutrinos are introduced
so as to make two light neutrinos to be Pseudo-Dirac particles and a light
neutrino to be a Majorana particle. Our model is different from the so-called
$3+n$ model with $n$ sterile neutrinos suggested to interpret short baseline
anomalies in terms of neutrino oscillations. While the
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix in $3+n$ model is simply extended
to $n\times n$ unitary matrix, the neutrino mixing matrix in our model is
parameterized so as to keep the $3\times3$ PMNS mixing matrix for three active
neutrinos unitary. There are also no flavor changing neutral current
interactions leading to the conversion of active neutrinos to sterile ones or
vice versa. We derive new forms of neutrino oscillation probabilities
containing the new interference between the active and sterile neutrinos which
are characterized by additional new parameters $\Delta m^2$ and $\theta$. Based
on the new formulae derived, we show how the short baseline neutrino anomalies
can be explained in terms of oscillations, and study the implication of the
high energy neutrino events detected at IceCube on the probe of pseudo-Dirac
neutrinos. New phenomenological effects attributed to the existence of the
sterile neutrinos are discussed.
