Levinson's theorem for the Schr\"{o}dinger equation in two dimensions

Levinson's theorem for the Schr\"{o}dinger equation with a cylindrically symmetric potential in two dimensions is re-established by the Sturm-Liouville theorem. The critical case, where the Schr\"{o}dinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison with Levinson's theorem in non-critical case, the half bound state for $P$ wave, in which the wave function for the zero-energy solution does not decay fast enough at infinity to be square integrable, will cause the phase shift of $P$ wave at zero energy to increase an additional $\pi$.Comment: Latex 11 pages, no figure and accepted by P.R.A (in August); Email: [email protected], [email protected]

The Relativistic Levinson Theorem in Two Dimensions

In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number $n_{j}$ of the bound states and the sum of the phase shifts $\eta_{j}(\pm M)$ of the scattering states with the angular momentum $j$: $$\eta_{j}(M)+\eta_{j}(-M)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$~~~=\left\{\begin{array}{ll} (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=M ~~{\rm and}~~ j=3/2~{\rm or}~-1/2\\ (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=-M~~{\rm and}~~ j=1/2~{\rm or}~-3/2\\ n_{j}\pi~&{\rm the~rest~cases} . \end{array} \right.$$ \noindent The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.Comment: Latex 14 pages, no figure, submitted to Phys.Rev.A; Email: [email protected], [email protected]

Effect of carrot puree edible films on quality preservation of fresh-cut carrots

peer-reviewedFinancial support from the high level talent fund of Henan University of Technology Science and Technology (No. 2012BS024) is gratefully acknowledged.The effect of edible films based on carrot puree, chitosan, corn starch, gelatin, glycerol and cinnamaldehyde on fresh-cut carrots was studied during storage. Several parameters, such as firmness, colour, weight loss, total carotenoids, total phenols, polyphenol oxidase (PPO) activity and peroxidase (POD) activity in coated carrots were determined at regular intervals and then compared with the uncoated carrots throughout the storage period. Significant and expected changes were observed in all carrot samples that were compared. The coating treatment significantly (P < 0.05) delayed the senescence, reduced the deterioration of exterior quality and retained total carotenoids well compared with control (P < 0.05). In addition, significant inhibition of PPO activity (P < 0.05) and POD activity (P < 0.05) as well as reduced accumulation of polyphenols (P < 0.05) were observed for all coated samples. All of these favourable responses induced by coating treatment on minimally processed fresh-cut carrots showed beneficial physiological effects, which would give some useful references to the fresh-cut fruit and vegetable processing industry and satisfy people’s requirements allowing for extending product shelf life without negatively affecting the sensory quality or acceptability.Henan University of Technology Science and Technolog

A refined invariant subspace method and applications to evolution equations

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations was analyzed to shed light on the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables.Comment: 16 page

Assessing Ageing Condition of Mineral Oil-Paper Insulation by Polarization/Depolarization Current

Accurately assessing the ageing status of oil-paper insulation in transformer is essential and important. Polarization and Depolarization Current (PDC) technique is effective in assessing the condition of oil-paper insulation system. Though the PDC behaviour of mineral oil-paper insulation has been widely investigated, there is no report about how to make the quantitative analysis of mineral oil-paper insulation ageing condition by PDC. The PDC characteristics of mineral oil-paper insulation samples were investigated over the ageing period at 110°C. A new method for assessing the ageing condition of mineral oil-paper insulation by calculating the depolarization charge quantity was proposed. Results show that the depolarization charge quantity of mineral oil-paper insulation sample is very sensitive to its ageing condition. The stable depolarization charge quantity could be used to predict the ageing condition of mineral oil-paper insulation

Brueckner-Hartree-Fock and its renormalized calculations for finite nuclei

We have performed self-consistent Brueckner-Hartree-Fock (BHF) and its renormalized theory to the structure calculations of finite nuclei. The $G$-matrix is calculated within the BHF basis, and the exact Pauli exclusion operator is determined by the BHF spectrum. Self-consistent occupation probabilities are included in the renormalized Brueckner-Hartree-Fock (RBHF). Various systematics and convergences are studies. Good results are obtained for the ground-state energy and radius. RBHF can give a more reasonable single-particle spectrum and radius. We present a first benchmark calculation with other {\it ab initio} methods using the same effective Hamiltonian. We find that the BHF and RBHF results are in good agreement with other $\it{ab}$ $\it{initio}$ methods