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1.
2.
The complete proofs of Krein’s theorem on the spectral shift function and the trace formula are given for a pair of self-adjoint
operators such that either (i) their difference is trace-class or (ii) the difference of their resolvents is trace-class.
The proofs, essentially due to Krein, is based on Herglotz’s theorem on the boundary value of the analytic functions whose
imaginary part is non-negative on the upper half plane, and an almost optimal class of functions are obtained for which the
trace formula is valid. Also an alternative method based on Weyl-von Neumann’s theorem for self-adjoint operators, avoiding
the complex function theory and inspired by Voiculescu’s work, is given for the first case. Furthermore, some applications
of the spectral shift function have been discussed. 相似文献
3.
Kalyan Kali Sen Gupta Mrityunjoy Adbikary Subrata Kumar Mandal Shipra Sen Gupta 《Transition Metal Chemistry》1987,12(2):152-155
Summary The oxidation of MeCHO by chromium(VI) has been studied in HClO4 medium over a wide range of experimental conditions and has been found to obey the rate law;v=k[MeCHO][HCrO
4
–
][H+]. The calculated H
and-S
values for the reaction are 30±2kJ mol–1 and 171±7J mol–1deg–1, respectively. The mechanism is discussed in terms of carbon-hydrogen bond cleavage. 相似文献
4.
The influence of a chiral surfactant and a polymer-supported chiral additive on reduction of ketones using sodium borohydride will be described. Initial preparations involved methylation of (S)-leucinol to give (2S)-N , N-dimethyl-2-amino-4-methyl-1-pentanol (1) (67%). The chiral surfactant (2) was synthesized by reacting (1) with bromohexadecane (71%). The functionalized styrene for the polymer-supported chiral additive (5) was synthesized by reacting (1) with 4-vinylbenzyl chloride. Polymerization was carried out with 10% of the functionalized monomer (4), 5% cross-linking agent divinylbenzene, and 85% styrene with AIBN as the initiator. The activity of the chiral surfactant and polymeric additive were examined by using them as additives in a standard reduction of 2-pentanone with sodium borohydride to yield (R)- and (S)-2-pentanol (3) (20%). The resulting alcohol was analyzed by polarimetry (ee 9.5%) and also esterified with (2S)-methylbutyric acid prior to characterization by NMR. 13C NMR indicated an enantiomeric excess of 5.2% when the chiral surfactant was used, and 7% when the polymeric additive was used. 相似文献
5.
Kalyan Kumar Das D. Mukherjee S. P. Bhattacharyya 《International journal of quantum chemistry》1986,29(6):1745-1754
Structural features of a series of tetraatomic carbonyls in the 1,3nπ* states are studied by using a recently proposed MCSCF algorithm at the INDO/2 level. The computed geometrical parameters and the inversion barrier heights agree quite well with the available experimental or ab initio theoretical data. The calculated transition energies turn out to be somewhat lower than the experimental counterparts. The specific MCSCF algorithm used is found to have excellent convergence properties. 相似文献
6.
Summary A stop time S in the boson Fock space over L
2()+ is a spectral measure in [0,] such that {S([0,t])} is an adapted process. Following the ideas of Hudson [6], to each stop time S a canonical shift operator U
Sis constructed in . When S({}) has the vacuum as a null vector U
Sbecomes an isometry. When S({})=0 it is shown that admits a factorisation
S]{S
where {S
is the range of U
Sand
S] is a suitable subspace of called the Fock space upto time S. This, in particular, implies the strong Markov property of quantum Brownian motion in the boson as well as fermion sense and the Dynkin-Hunt property that the classical Brownian motion begins afresh at each stop time. The stopped Weyl and fermion processes are defined and their properties studied. A composition operation is introduced in the space of stop time to make it a semigroup. Stop time integrals are introduced and their properties constitute the basic tools for the subject. 相似文献
7.
Kalyan K. Das D. Mukherjee S. P. Bhattacharyya 《International journal of quantum chemistry》1989,35(4):483-494
Structural parameters of a set of five thiocarbonyl molecules in the lowest nπ* states are calculated by using a generalized orbital optimization algorithm (a variant of the orthogonal gradient method) in an INDO MCSCF framework. Transition energies, singlet-triplet splittings, planar inversion barriers, and dipole moments in nπ* states of different spin multiplicities are reported. Predicted structural features agree reasonably well with available experimental or theoretical data. Some interesting trends are noted in the computed inversion barrier heights, singlet-triplet splittings, and dipole moments in nπ* states. 相似文献
8.
A variety of benzyl halides were converted to the corresponding aldehydes/ketones in respectable yields by IBX in DMSO at 65 °C. The bromohydrin reaction of olefins using NBS-H2O in DMSO can be nicely adapted to IBX-mediated oxidation of benzyl halides in such a way that olefins are converted to the corresponding 1,2-diketones in good isolated yields in one-pot. 相似文献
9.
Applied Biochemistry and Biotechnology - The capacity of polystyrene microspheres with immobilized antibodies against type 1 pili ofE. coli was measured. Using pure IgG-type monoclonal antibodies... 相似文献
10.
Kalyan Kali Sen Gupta Saroj Chandra Kumar Pratik Kumar Sen Amalendu Banerjee 《Reaction Kinetics and Catalysis Letters》1988,36(2):423-428
The kinetics of oxidation of methanol by bromate ion in hydrochloric acid medium has been investigated. A mechanism consistent with the experimental observations is suggested.
. , .相似文献