Stability for a class of bilocal Hamiltonians (Q1308459)

Problems containing non-local Hamiltonians (e.g. in regularized supersymmetric quantum field theories) do not fit into the conventional proofs of stability. The authors extend them to the Hamiltonians of the form \[ H = H_ 0 + :\int P(x) V(x - y) P(y) dx dy:, \] where \(P(x) = P(\varphi(x))\) is a polynomial of degree \(n\) in the real scalar field defined on a circle. It is shown that if the potential \(V\) has the slow decrease at infinity in the momentum space, then the non-local interaction can be bounded from below by a local interaction with a momentum dependent coupling constant. Combining this with an estimate on the measure of configurations is proved: 1) that the Hamiltonian is bounded from below and that the heat kernel \(\text{exp}(-TH)\) is a Hilbert-Schmidt operator; 2) that the results are stable under restricted change of coefficients in \(P(x)\), by adding (particular) polynomial \(Q(x)\) of the degree less than \(2n\) in the field, and under rescaling the potential as \(V(z) \to V(\rho z)\).