Exact results for non-integrable systems

Quantum mechanical many-body systems described by an arbitrary Hamiltonian (H-circumflex) are analyzed. It is shown how positive semidefinite operator (P-circumflex) properties are able to lead in this case to exact results related to the ground state and the low lying part of the excitation spectrum. This is done independent on dimensionality and integrability. The technique first casts the Hamiltonian in a positive semidefinite form in exact terms (H-circumflex = P-circumflex + C, where C is a scalar). Second, the ground state is deduced by constructing the most general Hilbert space vector, on which applying P-circumflex, one obtains zero as a result. It is underlined, that the procedure, if applied for variable total number of particles N, allows to obtain information also related to the low lying part of the excitation spectrum. The uniqueness of the ground states can be demonstrated via the study of the kernel of H-circumflex' = H-circumflex − C. The physical properties of the obtained phases are deduced based on ground state expectation values calculated in terms of the constructed ground states. Since for a fixed structure of P-circumflex, usually the transformation H-circumflex = P-circumflex + C is exact only in a restricted parameter space domain (D), the deduced ground states are present only in D. A global view on the phase diagram is obtained by different transformations of H-circumflex in positive semidefinite form.