We consider the Bloch–Torrey operator in
, where
. After normalization, this
operator takes the form
,
where
and
represents a magnetic
vector field. For
we give
natural conditions under which this operator can be defined as a maximally accretive operator, characterize
its domain and obtain its spectral properties in some special cases where we manage to show that the essential
spectrum is
. This result
lies in contrast with the
case considered in previous works.
In the asymptotic limit
and for
,
assuming that
is an affine function, we give accurate estimates for the location of the discrete spectrum in
the cases
or when
is a finite interval. Resolvent estimates are established as well.