Spectral problem

Reproduced from R, Davis and C. H. J. Wells. Spectral Problems In Organic Chemistry, International Textbook Co Chapman and Hall, New York, 1984. 

The absence of a space group makes the spectral problem a difficult one. Our work in this area1- 3 is far from complete,... 

Zelditch, S. Survey on the inverse spectral problem, to appear in J. Diff. Geom. Surveys, 2004. 

Bates, R. B. (1981). Carbon-13 NMR Spectral Problems. The Human Press, Crescent Manor. 

References for spectral problems for partial differential equations are e.g. Vladimirov (1996) and Wloka (1987). 

To be sure, linear methods have value where fast computation is necessary. They perform reasonably well when the experimental data are not band limited, and in trials with computer-generated data devoid of noise. Spectroscopic data are often band limited, however, and computation time is becoming less of a problem with advances in computer hardware. The quantity of data required in spectroscopy is far less than that in image processing, for example, another field that has given much attention to deconvolution. Image processing problems are two and sometimes three-dimensional, whereas spectral problems are usually one dimensional. 

In order to prove this statement we assume the second functional derivative of Stot evaluated in the reference configuration to equal the integral over Q of a suitable quadratic form multiplied by a scalar quantity a and determine under which conditions the corresponding spectral problem admits positive eigenvalues. In particular the following quadratic form is assumed for the second functional derivative of Stot... 

First of all, the compounds for which mass spectral data have been used just for identification will be listed, and a more detailed description will be given of papers dealing with specific mass spectral problems. 

On rewriting the first formula in Eq. (4.137) in terms of a single orientational variable x = (e n), the spectral problem takes the form... 

Fuchs, P.L., and Bunnell, C.A. (1979). Carbon-13 NMR-Based Organic Spectral Problems. New York Wiley. 

On the applied side of quantum chaology we find serious efforts to forge the semiclassical method into a handy tool for easy use in connection with arbitrary classically chaotic systems. Quite frankly, the current status of semiclassical methods is such that they are immensely helpful in the interpretation of quantum spectra and wave functions, but are only of limited power when it comes to accurately predicting the quantum spectrum of a classically chaotic system. In this case numerical methods geared toward a direct numerical solution of the Schrodinger equation are easier to handle, more transparent, more accurate and cheaper than any known semiclassical method. It should be the declared aim of applied semiclassics to provide methods as handy and universal as the currently employed numerical schemes to solve the spectral problem of classically chaotic quantum systems. 

The complexity of the ICAP emission spectra present a definite limitation to the design of an ICAP system and to the practical application to analyses (15). While this situation may be common to most or all emission spectroscopic techniques, it must be recognized in the design of ICAP instrumentation, line selection, and data review. Fortunately most spectral problems can be eliminated for most elements in most samples with the appropriate use of interference filters and computer correction techniques (10). 

HI. Equivalence between Elementary Molecular Orbital Theory and the Graph Spectral Problem... 

Therefore, it is clear that the spectrum of the graph is rather important in elementary MO calculations. The elementary MO problem, i. e. Hiickel problem, is in fact fully equivalent to the graph spectral problem. This was first emphasized by Gtinthard and Primas 56> and later by Schmidt-ke 5 ). 

The eigenfunctions are determined only up to a constant factor. To obtain uniquely chosen solutions of the spectral problems (3.5.9), (3.5.10), we state the following normalization condition on the flow axis ... 

Straightforward verification shows that the change of variables u = Xmg2, F = exp(it/2)fm transforms (3.5.10) into a degenerate hypergeometric equation for the function F = F(u) [28], Therefore, the solution of the spectral problem... 

The back spectral problem (the inverse problem of a parameter set of the effective Hamiltonian determination) is inaccurate. The possibility of improving it, taking into account that the D parameter of separated C-F bond in CH3F coincides with dissociation energy, will be discussed. 

Davis, R., and Wells, C.H.J. (1984). Spectral Problems in Organic Chemistry. New York Chapman and Hall. 

Formation of dimeric structures is ruled out as an explanation of the enhanced LF band intensities because some of the blue proteins contain only one Cu(II). In fact, the spectral problem is nicely illustrated by... 

The Born-Oppenheimer approximation is based on this assumption and enables reduction of the mathematically intractable spectral eigenvalue problem to a set of separable spectral problems for each type of motion. According to this approximation, energy levels associated with each type of motion are proportional to the ratio of electronic mass (mg) to the nuclear mass (Mn). This ratio, f quite smaller than unity, is given by Eq. (2) ... 

Books That Contain Combined Spectral Problems... 

The preceding example was presented because it shows the difficulty of recognizing the continuous spectrum even in a case of almost trivial simphcity. One can find in the literature a variety of interesting cases in which operators related to the B of the Boltzmann equation partly have, partly do hot have, continuous spectra [8]. Similarly, the spectrum, which is confined in the preceding case to the real axis, may or may not have points in the complex plane. The spectral problem will be discussed, in the present Symposium, by Drs. Habetler and Martino and Dr. Wing. We shall also hear, I believe, about the approach of the solutions of the various approximation methods. 

Gladkov, L.L. and K.N. Solovyov (1985). The normal coordinate analysis of porphin and its derivatives based on the solution of the inverse spectral problem for porphin and Cu porphin—II. A valence force field for in-plane vibrations of the Cu porphin molecule. Spectrochim. Acta A 41, 1443. 

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