Spurious eigenvalues

The Cullum-Willoughby test,27,37 on the other hand, was designed to identify the so-called spurious eigenvalues, rather than the converged eigenvalues. In particular, the tridiagonal matrix T and its submatrix, obtained... 

This completes the definition of the stability problem for the mixed convection flow over the horizontal plate. For a given K and Re, one would be required to solve (6.4.19)-(6.4.38), starting with the initial conditions (6.4.39)-(6.4.58) and satisfy (6.4.63) for particular combinations of the eigenvalues obtained as the complex k and u>. We will use the procedure adopted in Sengupta et al. (1994) to obtain the eigen-spectrum for the mixed convection case, when the problem is in spatial analysis framework. In the process, it is possible to scan for all the eigenvalues in a limited part of the complex k- plane, without any problem of spurious eigenvalues. 

The kinetic energy operator corresponding to G z) is hermitian above. Loss of hermiticity may lead to spurious eigenvalues appearing in the calculations[42]. To avoid this, a proper basis should be chosen. For instance, to treat the term involving G z) in the RCL Hamiltonian, a Fourier D R basis was uscd[38]. 

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... 

Now we easily see the origin of the spurious singularities that plague the FW approach. Let be an eigenfunction of Hq with eigenvalue Then... 

The possible severity of the problem has been shown by M. Stanke and J. Karwowski, Variationalprinciples in the Dirac theory Spurious solutions, unexpected extrema and other traps in A eir Trends in Quantum Systems in Chemistry and Physics, vol. I, pp. 175 190, eds. J. Maniani et al., Kluwer Academic Publishers, Dordrecht (2001). Sometimes an eigenfunction corresponds to a quite different eigenvalue. Nothing of that sort appears in non-relativistic calculations. 

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