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On Hamiltonians for Kerov functions

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.