Operators complex symmetric

It is also straightforward to generalize the off-diagonal interaction to incorporate the previously mentioned resonance picture of unstable states by using a complex symmetric operator. For general discussions on this issue, we refer to the proceedings of the Uppsala-, Lertorpet- and the Nobel-Satellite workshops and references therein [13-15]. Thus one may arrive at a complex symmetric secular problem (note that the same matrix construction may be derived from a suitable hermitean matrix in combination with a nonpositive definite metric [9] see also below), which surprisingly leads to a comparable secular equation as the one obtained from Eq. (1). To be more specific we write... 

The simplest example of a complex symmetric operator is the non-relativistic many-particle Hamiltonian H for an atomic, molecular, or solid-state system, which consists essentially of the kinetic energy of the particles and their mutual Coulomb interaction. Since such a Hamiltonian is both self-adjoint and real, one obtains... 

The Hartree-Fock Scheme for a Complex Symmetric Operator. 

Special Case when T = T = T. Let us now consider the special case when a complex symmetric operator is real, so that T = T. In this case, the operator T is also self-adjoint, T = T, and one can use the results of the conventional Hartree-Fock method 7. The eigenvalues are real, X = X. and - if an eigenvalue X is non-degenerate, the associated eigenfunction C is necessarily real or a real function multiplied by a constant phase factor exp(i a). In both cases, one has D = C 1 = C. In the conventional Hartree-Fock theory, the one-particle projector p takes the form... 

We note finally that, in the numerical studies of the complex symmetric operators Tu and Tueff, it is usually convenient to use orthonormal basis sets which are real, since the associated matrices will then automatically be symmetric with complex elements. In the case of a truncated basis of order m, most of the eigenvalues will usually turn out to be complex, but we observe that one has to study their behaviour when m goes to infinity and the set becomes complete, before one can make any definite conclusions as to the existence of true complex eigenvalues to Tu and Tuefj, respectively. The connection between the results of approximate numerical treatments and the exact theory is still a very interesting but mostly unsolved problem. 

Ha et al. [18] have measured the equilibrium quotients for the formation of a complex between the lac repressor protein and a symmetric operator sequence of DNA as a function of temperature. Their results are given below. (The standard state is 1 mol dm. )... 

Fig. 1.15. Bending of the DNA in the CAP protein-DNA complex. The CAP protein ( . coli) binds as a dimer to the two-fold symmetric operator sequence. The DNA is bent nearly 90deg in the complex. The turns are centered around two GT sequences (shown in black) of the recognition element.

The complex scalar product lets us dehne an analog of Euclidean orthogonal projections. First we need to dehne Hermitian operators. These are analogous to symmetric operators on R". 

Dehnition 3.10 Suppose V is a complex scalar product space. A Hermihan linear operator (also known as a Hermihan symmetric operator or self-adjoint operator) on V is a linear operator T V V such that for all Vi, V2 E V we have... 

As already discussed at the end of Section 2.2.3, we derived a universal superposition principle from a complex symmetric ansatz arriving at a Klein-Gordon-like equation relevant for the theory of special relativity. This approach, which posits a secular-like operator equation in terms of energy and momenta, was adjoined with a conjugate formal operator representation in terms of time and position. As it will be seen, this provides a viable extension to the general theory [7, 82]. We will hence recover Einstein s laws of relativity as construed from the overall global superposition, demonstrating in addition the independent choice of a classical and/or a quantum representation. In this way, decoherence to classical reality seems always possible provided that appropriate operator realizations are made. 

Equation (97) serves as a boundary condition for the operator matrix model, which in the complex symmetric representation, making the replacement v/c = k(t), reads (note that x(r) < b see below)... 

For our example we assume, for more relevant details see further below, that the operator W(r) and the functions y(r), e/>(r) are aptly defined for the scaling to be meaningful. Such a family of potentials will be denoted by jH0 [53]. If the family has an extension, see more details below, to arg r] < 0 analytic in the interior and up to the boundary, the class is denoted by jy0. At this point, it is trivial to deduce from Eq. (D.l) that the matrix element is analytic in the parameter r] provided that the complex conjugate of i] is inserted on the "bra" side of Eq. (D.2). Hence, many complex scaling treatments in quantum chemistry are operationally derived from complex symmetric representations. 

This is a nonsymmetric construction, which could be derived from a traditional Hermitean version via a nonpositive definite metric A An = — A22 = 1, A12 = A21 = 0. As we will see analogous constructions also follow from our complex symmetric realizations, cf. previous developments above. We emphasize that these realizations are required with the intention to mimic our dissipative system, the environment" here being made up of the other wffs of the formal system. The probability operator p is represented in an abstract space spanned by the vectors true and false). Diagonalising T p > i), i.e.,... 

The adjoint operator is defined only on the domain D(Tf) inside the L2 Hilbert space. In the important case, when the operator T is complex symmetric within the domain Z)(Tt), so that T+ = T, one has first of all... 

The puzzle depended on the simple fact that most physicists using the method of complex scaling had not realized that the associated operator u - the so-called dilatation operator - was an unbounded operator, and that the change of spectra -e.g. the occurrence of complex eigenvalues - was due to a change of the boundary conditions. Some of these features have been clarified in reference A, and in this paper we will discuss how these properties will influence the Hartree-Fock scheme. The existence of the numerical examples finally convinced us that the Hartree-Fock scheme in the complex symmetric case would not automatically reduce to the ordinary Hartree-Fock scheme in the case when the many-electron Hamiltonian became real and self-adjoint. Some aspects of this problem have been briefly discussed at the 1987 Sanibel Symposium, and a preliminary report has been given in a paper4 which will be referred to as reference D. 

In concluding this section, a few words should be said about the case when the operator T is complex symmetric ... 

Operator. - Let us now consider the special case when the many-particle operator T is complex symmetric, so that T = T. If the eigenvalue X is non-degenerate, one has according to (2.26) the simple relation D = C 1 between the eigen-... 

The many-particle operator T defined by (2.35) is complex symmetric because the various terms in (2.35) are assumed to be complex symmetric, so that... 

Since T12 = T21, the operator T12 = T12 (1-P12) is also complex symmetric. This follows from the fact that... 

One sees immediately that, if the operator T is originally complex symmetric, so that T = T. it will keep this property invariant under the similarity transformation provided that the operator U satisfies the condition... 

For restricted similarity transformations, the transformed operator Tu= UTU 1 is still complex symmetric, and the same applies - according to (3.33) and (3.42) - to the one-particle operators pu and Tueff. If one starts from the assumption that... 

It is evident that, if one wants to avoid this trivial case, one has to abandon relation (3.71) and assume that the set has more general properties. For this purpose, we will start from the complex symmetric many-particle operator Tu and consider the associated one-particle operators... 

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