Hamiltonian systems unbounded

To treat real molecules the theory must be extended to typical unbounded Hamiltonian systems, for which we expect there to be long-lived resonances associated with the quasi periodic islands in the classical description. 

Even if in physics and chemistry the Hamiltonian is bounded from below to ensure the existence of a ground state of the system under consideration, the Hamiltonian H as a whole is an unbounded operator. It should be observed, however, that for values of z outside the small circles z -Ek = p, the resolvent R(z) is always a bounded operator, and it is hence much easier to handle than the Hamiltonian itself in addition, it has also much richer and mathematically interesting properties which are of essential value in studying the physical and chemical features of the system under consideration. 

Since the spectrum of the Hamiltonian H in N 1-particle systems will generally be unbounded from above showing a continuous spectrum at the top end, we may conclude that the spectrum of the last component of the generalised excitation energy operator H comprises the full real axis. This can alrecidy be seen in zeroth order where the primary states have the eigenvalues... 

Unlike the above mentioned methods, another Floquet-theorem-based approach, the many-electron many-photon theory (MEMPT) of Mercouris and Nicolaides (71,72) does not involve complex rotated Hamiltonians. The complex coordinate rotation is used only to regularize that part of the wave functions which describes unbound electrons (see the CESE method). This allows efficient description of bound or quasi-bound states, involved in a problem under consideration, by MCHF solutions and therefore enables ab initio application to many-electron systems (71,72,83-87). 

The DF choice is essentially a choice of a particular Hamiltonian, which describes the physics of the system, but it is worth noting that the photoionization process involves an unbound electron whose wave-function must be described with... 

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