The first order perturbations of the energy levels of a stationary hydrogen atom in static external gravitational field, with Schwarzschild metric, are investigated. The energy shifts are calculated for the relativistic 1S, 2S, 2P, 3S, 3P, 3D, 4S, 4P, 4D and 4F levels. The results show that the energy-level shifts of the states with total angular momentum quantum number 1/2 are all zero, and the ratio of absolute energy shifts with total angular momentum quantum number 5/2 is 1:4:5. This feature can be used to help us to distinguish the gravitational effect from other effect.

The grading Becchi-Rouet-Stora-Tyutin (BRST) method gives a way to construct the integer W2,s strings, where the BRST charge is written as QB= Q0 +Q1. Using this method, we reconstruct the nilpotent BRST charges Q0 for the integer W2,s strings and the half-integer W2,s strings. Then we construct the exact grading BRST charge with spinor fields and give the new realizations of the half-integer W2,s strings for the cases of s= 3/2, 5/2, and 7/2.

In this paper, we construct the nilpotent Becchi-Rouet-Stora-Tyutin(BRST) charges of spinor non-critical W2,s strings. The cases of s = 3, 4 are discussed in detail, and spinor realization for s = 4 is given explicitly. The BRST charges are graded.

In Phys. Rev. D 78 (2008) 104018 [arXiv:0807.1481], the conclusion that“entropy eigenvalues of GB black hole are discrete and equally spaced, but the area spacing is not equidistant” was firstly presented by Kothawala, Padmanabhan and Sarkar. In this paper, using the new physical interpretation of quasinormal modes proposed by Maggiore,we calculate the quantum spectra of entropy for various types of non-rotating black holes with no charge. The spectrum is obtained by imposing Bohr-Sommerfeld quantization condition to the adiabatic invariant quantity. We conjecture that the spacing of entropy spectrum is equidistant and is independent of the dimension of spacetime. However, the spacing of area spectrum depends on gravity theory. In Einstein’s gravity, it is equally spaced, otherwise it is non-equidistant. This conjecture agrees with the result of Kothawala,Padmanabhan and Sarkar.

In this paper, we mainly investigate the WM2,s WL2,s system, in which the matter and the Liouville subsystems generate WM2,s and WL2,s algebras respectively. We first give a brief discussion of the physical states for corresponding W stings. The lower states are given by freezing the spin-2 and spin-s currents.Then, introducing two pairs of ghost-like fields, we give the realizations of W1,2,s algebras. Based on these linear realizations, BRST operators for W2,s algebras are obtained. Finally, we construct new BRST charges of Liouville system for WL2,s strings at the specific values of central charges c: c = −225/5 for WL2,3 algebra,c = −24 for WL2,4 algebra and c = −2,−286 /3 for WL2,6 algebra, at which the corresponding WL 2,s algebras are singular.