  It is known that self-adjoint Hamiltonians with purely discrete eigenvalues
can be written as (infinite) linear combination of mutually orthogonal
projectors with eigenvalues as coefficients of the expansion. The projectors
are defined by the eigenvectors of the Hamiltonians. In some recent papers,
this expansion has been extended to the case in which these eigenvectors form a
Riesz basis or, more recently, a $\D$-quasi basis, \cite{bell,bit}, rather than
an orthonormal basis. Here we discuss what can be done when these sets are
replaced by Parseval frames. This interest is motivated by physical reasons,
and in particular by the fact that the {\em mathematical } Hilbert space where
the physical system is originally defined, contains sometimes also states which
cannot really be occupied by the {\em physical} system itself. In particular,
we show what changes in the spectrum of the observables, when going from
orthonormal bases to Parseval frames. In this perspective we propose the notion
of $E$-connection for observables. Several examples are discussed.
