  Kerov Hamiltonians are defined as a set of commuting operators which have
Kerov functions as common eigenfunctions. In the particular case of Macdonald
polynomials, well known are the exponential Ruijsenaars Hamiltonians, but the
exponential shape is not preserved in lifting to the Kerov level.
Straightforwardly lifted is a bilinear expansion in Schur polynomials, the
expansion coefficients being factorized and restricted to single-hook diagrams.
However, beyond the Macdonald locus, the coefficients do not celebrate these
properties, even for the simplest Hamiltonian in the set. The coefficients are
easily expressed in terms of the eigenvalues, and, if so defined, the set of
commuting Hamiltonians is enormously large: one can build one for each
arbitrary set of eigenvalues $\{E_R\}$, specified independently for each Young
diagrams $R$. A problem with these Hamiltonians is that they are constructed
with the help of Kostka matrix instead of defining it, and thus are less
powerful than the Ruijsenaars ones.
