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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.
T. Wanjun L. Yuwen Z. Hen W. Zhiyong W. Cunxin 《Journal of Thermal Analysis and Calorimetry》2003,74(1):309-315
A new approximate formula for temperature integral is proposed. The linear dependence of the new fomula on x has been established. Combining this linear dependence and integration-by-parts, new equation for the evaluation of kinetic
parameters has been obtained from the above dependence. The validity of this equation has been tested with data from numerical
calculating. And its deviation from the values calculated by Simpson's numerical integrating was discussed. Compared with
several published approximate formulae, this new one is much superior to all other approximations and is the most suitable
solution for the evaluation of kinetic parameters from TG experiments.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
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A. Chafiq 《Optics Communications》2007,278(1):142-146
In order to avoid the severe axial intensity oscillations which appear when an ideal Mathieu beam is truncated, we propose to modulate it by a flattened multi-Gaussian envelope. The obtained beam, which is referred as flattened Mathieu-Gauss (FTMG) beam, can be expanded into a finite series of Mathieu-Gauss beams with various waists. The propagation study of this beam reveals that the axial intensity is unchanged within a certain propagation distance as for super-Gaussian-Bessel beams or Gori’s flattened Bessel beams. By using Collins formula, we derive closed-form expressions of FTMG beam propagating through a paraxial axisymmetric ABCD optical system. Some numerical calculations and discussions are also given. 相似文献
6.
A. Chafiq 《Optics Communications》2006,265(2):594-602
Using Collins formula and the expansion of Mathieu beams in terms of Bessel beams we derive the exact propagation equations of Mathieu-Gauss beams through an apertured paraxial ABCD optical system. A comparison between the exact propagation equations and the approximated ones, which are derived by expanding the circ function into a finite sum of Gaussian functions, is presented. The propagation characteristics of zeroth-order Mathieu-Gauss beams in (y-z) and (x-z) planes are analyzed with detailed numerical calculations. It is found that the profile of the propagated Mathieu-Gauss beam is similar to that of Bessel-Gauss beam. Furthermore, the focalization of the Mathieu-Gauss beams through a thin lens is illustrated and analyzed with numerical results. 相似文献
7.
The band artifact in steady-state free precession can be reduced by synthesizing the multiple images obtained through different phase increments of successive radiofrequency pulses. Even though the complex summation method was reported to be effective in reducing the band artifact, it has the pitfalls of intensity abnormality and sensitivity to the phase abnormality. Two new methods have been developed for more reliable reduction of the band artifact than the complex summation method. One method is to sum the complex images partially and to take the maximum intensity of the partially summed images. The other method is to sum the free induction decay (FID) and primary echo components of the Fourier series that are obtained through Fourier analysis of the complex base images. Both proposed methods were compared with other magnitude (maximum intensity projection, spectrally decomposed synthesis, sum-of-squares, nonlinear averaging) and complex-based (complex summation, magnitude-weighted complex summation) methods experimentally at 3 T for the phantom and volunteer's head imaging. Both proposed methods were confirmed to maintain the advantage of the complex summation in reducing both the dark and bright band artifacts while reducing the intensity abnormality and sensitivity to the phase abnormality from that of the complex summation method over a wide range of flip angles and relaxation times. 相似文献
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By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”. 相似文献