Spectral theory

Yakunina G.V. (1981) Smoothness of solutions of variational inequalities. Partial differential equations. Spectral theory. Leningrad Univ. (8), 213-220 (in Russian). 

Bonamy L., Nguyen Minh Hoang P. Far infrared absorption of diatomic polar molecules in simple liquids and statistical properties of the interactions. I. Spectral theory, J. Chem. Phys. 67, 4423-30 (1977) ... 

Dunford, N. and Schwartz, J. (1971b) Linear Operators (second edition). Part II Spectral Theory. Self-Adjoint Operators in Hilbert Space. Wiley New York. 

Stability theory is the central part of the theory of difference schemes. Recent years have seen a great number of papers dedicated to investigating stability of such schemes. Many works are based on applications of spectral methods and include ineffective results given certain restrictions on the structure of difference operators. For schemes with non-self-adjoint operators the spectral theory guides only the choice of necessary stability conditions, but sufficient conditions and a priori estimates are of no less importance. An energy approach connected with the above definitions of the scheme permits one to carry out an exhaustive stability analysis for operators in a prescribed Hilbert space Hh-... 

References Courant, R., and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York (1953) Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM Publications, Philadelphia (1985) Porter, D., and D. S. G. Stirling, Integral Equations A Practical Treatment from Spectral Theory to Applications, Cambridge University Press (1990) Statgold, I., Greens Functions and Boundary Value Problems, 2d ed., Interscience, New York (1997). 

Vol. 1498 R. Lang, Spectral Theory of Random Schrodinger Operators. X, 125 pages. 1991. 

Vol. 1258 J. Weidmann, Spectral Theory of Ordinary Differential Operators. VI, 303 pages. 1987. 

The 8 part in (2.53) is responsible for elastic scattering, whereas the second term, which is proportional to the Fourier transform of C(f), leads to the gain and loss spectral lines. When the system undergoes undamped oscillations with frequency A0, this leads to two delta peaks in the structure factor, placed at spectral line. The spectral theory clearly requires knowing an object different from (o-2(/)), the correlation function [Dattaggupta et al., 1989]. 

W. P. Reinhardt, Complex Scaling in Atomic Physics A Staging Ground for Experimental Mathematics and for Extracting Physics from Otherwise Impossible Computations, In Spectral Theory and Mathematical Physics A Fest Schrift in Honor of Barry Simon s 60th... 

E. Brandas, P. Froelich, M. Hehenberger, Theory of Resonances in Many-Body Systems Spectral Theory of Unbounded Schrodinger Operators, Complex Scaling, and Extended Virial Theorem, Int. J. Quant. Chem. XIV (1978) 419. 

The spectral theory of graphs is well elaborated, both in the case of general graphs [25] and graphs of interest in chemistry [24]. In this section we are concerned with the graph spectra which are specific for benzenoid systems. 

As already discussed in detail, formulas (1) or (4) represent a very important (spectral) property of benzenoid systems. Although this result has been known since 1952, not much progress in the spectral theory of benzenoid systems has been made in the meantime. 

After the completion of the text of this article a remarkable discovery has been made in the spectral theory of benzenoid molecules. In the 1980s serveral authors tried to find isospectral benzenoid systems (i.e. benzenoids having equal spectra, cf Sect. 3). These efforts were, however, not successful. Finally, Cioslowski... 

In the case of the HMO model the success of the search for structure-property relations is much enhanced by the possibility of applying the powerful mathematical apparatus of graph-spectral theory [18, 19]. There is hardly any application of graph (spectral) theory in more sophisticated molecular orbital models. 

The integral formulas for total rr-electron energy were invented by Coulson [2, 3, 6], Later, they were extensively elaborated (see [13], pp. 139-147). Their real usefulness in structure-property analysis became evident only after they were combined with the results of graph spectral theory [13, 18, 65, 66]. 

Modern spectral theory and technology have jointly brought chemical structure research into a new era. Mulidimensional NMR (nD-NMR) theory and instrumentation in particular have provided many new ways for... 

Stockman, M.I., Kurlayev, K.B., George, T.F. Linear and nonlinear optical susceptibilities of Maxwell Garnett composites Dipolar spectral theory. Phys. Rev. B 60, 17071-17083 (1999)... 

Here we use the standard Weyl constructions, described for the spectral theory of molecules, for example in [59]. For more modern constructions see [5,55]. For example, one may use, for the free eigenfunction i/rj of the state with the energy Ej, the modified function ffj with the cutoff function /(r), that is equal to 1 for some large region 2 (A) and vanishes outside a neighborhood of this region, 2(A + 8). In this case... 

Dautray, R., and Lions, J., Spectral Theory and Applications, vol. 3 of Mathematical Analysis and Numerical Methods for Science and Technology, Berlin Springer-Verlag, 1985. 

Davis, E. B., Spectral Theory and Differential Operators, Cambridge, UK Cambridge University Press, 1995. Debnath, L., Nonlinear Partial Differential Equations for Scientists and Engineers, Boston Birkhauser, 1997. Deen, W. M., Analysis of Transport Phenomena, Oxford Oxford University Press, 1998. 

Porter, D., and Stirling, D. S. G., Integral Equations A Practical Treatment from Spectral Theory to Applications, Cambridge, UK Cambridge University Press, 1990. 

Pearson (1988) Quantum Scattering and Spectral Theory Academic Press, New York, page 277... 

In the formulation of Theorem 4.4.1.3 we have used the word unfortunately. This Is because the existence of isospectrai graphs indicates that a substantial part of the information about the structure of a gr h G has been lost in Ch(6). The theory of the characteristic polynomial is thus less rich than the theory of graphs and many problems in graph theory are simply intractable by means of graph spectral theory. As a consequence, the power of polynomial techniques in graph theory is much reduced. Similarly, several difficulties occur in the applications of Ch in chemistry. 

According to Theorem 4.4.2.1, the spectrum of a graph is fully determined by the characteristic polynomial and vice verso [21]. There is thus no essential difference between graph spectral theory and the... 

The above result was known to Cooison and Longuet-Hig ns [56] though it was later overlooked in both graph spectral theory and the... 

The main merit of Theorem 4.S.2.7 is that it reduces the entire theory of the matching polynomial to the much more developed graph spectral theory. It is thus now no longer surprising that there are numerous analogies between Ma(G) and Ch(G). We list two more results of this kind [97-98], and refer the interested reader to reference [99] ... 

Isomorphism of Huckel Theory and Graph Spectral Theory... 

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