Hilbert space eigenvalue equation

From the A -electron Hilbert-space eigenvalue equation, Eq. (2), follows a hierarchy of p-electron reduced eigenvalue equations [13, 17, 18, 47] for 1 < p < N — 2. The pth equation of this hierarchy couples Dp,Dp+, and and can be expressed as... 

The remarkable fact, first demonstrated by Nakatsuji [18], is that for each p >2, CSE(p) is equivalent (in a necessary and sufficient sense) to the original Hilbert-space eigenvalue equation, Eq. (2), provided that CSE(p) is solved subject to boundary conditions (A -representability conditions) appropriate for the (p + 2)-RDM. CSE(p), in other words, is a closed equation for the (p+ 2)-RDM (which determines the (p + 1)- and p-RDMs by partial trace) and has a unique A -representable solution Dp+2 for each electronic state, including excited states. Without A -representability constraints, however, this equation has many spurious solutions [48, 49]. CSE(2) is the most tractable reduced equation that is still equivalent to the original Hilbert-space equation, and ultimately it is CSE(2) that we wish to solve. Importantly, we do not wish to solve CSE(2) for... 

Neither CSE(l) nor ICSE(l) is equivalent to the original Hilbert-space eigenvalue equation for that we need CSE(2). The unconnected part of 0.2 is [11]... 

According to (1.22) and (1.26), the eigenvalues v of the Liouvillian L are distributed symmetrically around the point v = 0, and this implies that, even if the Hamiltonian H in physics is bounded from below, H > a 1, the Liouvillian L is as a rule unbounded. Except for this difference, practically all the Hilbert-space methods developed to solve the Hamiltonian eigenvalue problem in exact or approximate form may be applied also to the Liouvillian eigenvalue problem. In the time-dependent case, the L2 methods developed to solve the Schrodinger equation are now also applicable to solve the Liouville equation (1.7). 

Calculation of the Floquet states We now solve the eigenvalue equation (8.5) for Floquet states By virtue of the periodicity, each Floquet state ua(t)) can be expanded in a Fourier series. We take an arbitrary time-independent orthonormal basis set that expands the Hilbert space of our interest, then the Floquet state can be expanded using the basis... 

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