The Classical Moment Problem

The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

It is possible to make evident some properties of the resolvent of a Hermitian operator, by using the identity 

Gqo( ) is analytic and ImC7oo( ) 0. We also see that any approximant of (5.3), where a are real and are positive, satisfy the Herglotz property since it represents a half linear chain of variables of energies a and nearest-neighbor interaction b . 

A physical quantity of remarkable interest is the projected density of states, defined for Hermitian operators as 

The Herglotz property of Gqo(E) and of the approximants G E) guarantees that n E) given by (5.20) or any approximate expression of the type(-l/ r)ImG H + ie), e 0 (often used in practice) are definite positive. 

---

N. I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965. 

We have not yet specified if the operator to be handled is Hermitian (real eigenvalues) or whether it is a relaxation operator (eigenvalues either real or in the lower half of the complex plane). Uie moment problem related to a Hermitian operator is addressed as the classical moment problem, while by relaxation moment problem we mean the treatment of relaxation operators. 

It is not possible to extend right away the results of the classical moment problem to the relaxation moment problem. However, our survey of Section V.B has been done in such a way that it is possible to select which relations maintain their validity in the relaxation moment problem and which are to be disregarded. Thus little remains to be said except for a few comments. 

Pq( ) = 1]. The expression of D can be recognized as the standard expression of Hankel determinants, wtiich are known to be essentially positive quantities (in the classical moment problem). 

In their pioneering work. Handy and Bessis showed that by imposing the positivity conditions originating from the classic Moment Problem in mathematics, one could quantize the low lying discrete states of (multidimensional) rational fraction potential problems. 

Consider collisions between two molecules A and B. For the moment, ignore the structure of the molecules, so that each is represented as a particle. After separating out the centre of mass motion, the classical Hamiltonian that describes tliis problem is... 

We consider first the polarizability of a molecule consisting of two or more polarizable parts which may be atoms, bonds, or other units. When the molecule is placed in an electric field the effective field which induces dipole moments in various parts is not just the external field but rather the local field which is influenced by the induced dipoles of the other parts. The classical theory of this interaction of polarizable units was presented by Silberstein36 and others and is summarized by Stuart in his monograph.40 The writer has examined the problem in quantum theory and finds that the same results are obtained to the order of approximation being considered. 

A through-space electrostatic effect (field effect) due to the charge on X. This model was developed by Kirkwood and Westheimer who applied classical electrostatics to the problem. They showed that this model, the classical field effect (CFE), depended on the distance d between X and Y, the cosine of the angle 6 between d and the X—G bond, the effective dielectric constant and the bond moment of X. 

As was also previously noted in Section 9.3.1, the completely general rigid-rotor Schrodinger equation for a molecule characterized by three unique axes and associated moments of inertia does not lend itself to easy solution. However, by pursuing a generalization of the classical mechanical rigid-rotor problem, one can derive a quantum mechanical approximation that is typically quite good. Within that approximation, the rotational partition function becomes... 

Detailed balance. Classical line shapes are symmetric so that all classical, odd spectral moments M of the spectral function vanish. The odd moments of actual measurements are, however, non-vanishing because measured spectral density profiles satisfy the principle of detailed balance, Eq. 5.73. This problem of classical relationships may be largely alleviated by symmetrizing the measured profile prior to determining the moments, using the inverse Egelstaff procedure (P-4) discussed on p. 254 this generates a close approximation to the classical profile from the measurement and use of classical formulae is then justified. 

These atomic charges are usually capable of reproducing the calculated dipole moment and the quantum chemical MEP map of the molecule outside the van der Waals surface. The MEP map of the classical EP system is not so rich in information, however, and there are problems, e.g., with the lone pairs. The local anisotropies due to the lone pairs largely disappear. In order to obtain a more accurate description further point charges, situated in appropriate sites, are usually taken into account during optimization [68],... 

Thus, in this section we have described the manner in which absorption of light by a molecule leads to polarization of the angular momenta of the absorbing level. We have also shown how to calculate the multipole moments created on the lower level. It is important to stress that the adopted model of description enables us to obtain precise analytical expressions for the multipole moments, including both cases, namely those for arbitrary values of angular momenta and those for the classic limit J — oo. Our subsequent discussion will concern problems connected with the manifestation of ground state angular momenta anisotropy in experimentally observable quantities. 

The Lanczos Method. The use of classical moment theory involves large determinants, thereby implying delicate problems of numerical stability. This instability arises from very severe cancellations in the determinants, which in turn come from the properties of the moments themselves. Whitehead and Watt bypassed this problem by using the Lanczos al-... 

With the known 2n moments, the singularities of Ig e), in accordance with Eqn. (2.34), may be examined with a respective function q Z). Standard relations from the classical problem of moments and formulas from the Kristoffel-Darbu theory of orthogonal polynomials [14] allow us to establish the functions... 

---