Randomized Hamiltonian mechanics is the Hamiltonian mechanics which is determined by a time-dependent random Hamiltonian function. Corresponding Hamiltonian system is called random Hamiltonian system. The Feynman formulas for the random Hamiltonian systems are obtained. This Feynman formulas describe the solutions of Hamilton equation whose Hamiltonian is the mean value of random Hamiltonian function. The analogs of the above results is obtained for a random quantum system (which is a random infinite dimensional Hamiltonian system). This random quantum Hamiltonians are the part of Hamiltonians of open quantum system.
Abstract Hamiltonian mechanics determined by time-dependent random Hamiltonian functions is called randomized Hamiltonian mechanics. The corresponding Hamiltonian systems are called random. Feynman formulas for random Hamiltonian systems are obtained. It is shown that these formulas describe solutions of a Hamiltonian equation whose Hamiltonian is the mean value of the random Hamiltonian function. Analogous results are obtained for random quantum systems (which are known to be infinite-dimensional random Hamiltonian systems). The random quantum Hamiltonians can be used to describe the so-called open quantum systems (which are part of some lager quantum systems).
We introduce and study a class of non-Hermitian Hamiltonians which have velocity dependent potentials. Since stability can not be advocated directly from the classical potential, we show that the energy spectra are real and bounded from below which proves the stability of the spectra of all members in the class. We find that the introduced class of non-Hermitian Hamiltonians do have a corresponding superpartner class of non-Hermitian Hamiltonians. We were able to introduce supercharges which in conjunction with the corresponding super Hamiltonians constitute a closed super algebra. Among the introduced Hamiltonians, we show that non-
PT-symmetric Hamiltonians can be transformed into their corresponding superpartner Hamiltonians via a specific canonical transformation while the
PT-symmetric ones failed to be mapped to their corresponding superpartner Hamiltonians via the same canonical transformation. Since canonical transformations preserve the spectrum, we conclude that non-
PT-symmetric Hamiltonians out of the introduced class of Hamiltonians have the same spectrum as the corresponding superpartner Hamiltonians and thus Susy is broken for such Hamiltonians. This kind of intertwining of
PT-symmetry and Supersymmetry is new as all the so far discussed cases concentrate on Hamiltonians of broken $\mathcal{PT }%
−symmetrythathavebrokenSupersymmetrytoowhileweshowedthatSusycanbealsobrokenfornon−\mathcal{PT }$-symmetric and non-Hermitian Hamiltonians .