Scattering Theory Without Asymptotic Approximation,Heat-kernel Approach For Scattering,and Scattering Of(?)and (?)Potentials,Gravitational Waves,and Black Holes

Scattering plays an important role in physics.In the thesis,we present a series of exact treatments on scattering and some exact scattering solutions,including the fundamental theory,calculation methods,and some exact solutions of scattering.(1)The standard scattering theory in quantum mechanics textbooks is an ap-proximate theory which assumes that the distance between target and observer is infinite.This approximation loses all the information of the distance.The reason why one employs such an approximation is that one want to express the result by the scattering phase shift explicitly.In the thesis,we show that even without such an approximation,one can still achieve a rigorous scattering theory which is expressed explicitly by the scattering phase shift and,of course,contains the information of dis-tance that is lost in standard scattering theory.(2)Furthermore,we generalize the three-dimensional scattering theory without large-distance asymptotics given above to arbitrary dimensions.(3)We establish a gravitational wave scattering without large-distance asymptotics.(4)We establish the Lippmann-Schwinger equation for scattering without large-distance asymptotics.(5)We provide an approach for cal-culating scattering phase shifts based on the heat-kernel method and the scattering spectral method in quantum field theory.By this approach,each method of cal-culating heat kernels is converted into a method of solving scattering phase shifts.We also construct two approaches for solving scattering phase shifts based on two heat-kernel expansions:the Seeley-DeWitt expansion and the covariant perturbation theory as applications.(6)Moreover,based on the relation between heat kernels and scattering phase shifts,we also construct a method of calculating a heat kernel from a given scattering phase shift.(7)All large-distance asymptotic behaviors of short-range potential are the same,but the scattering boundary for short-range potentials differs from one potential to another.In this thesis,we give a general discussion on the large-distance asymptotic behavior of various three-dimensional long-rangepower potentials.(8)We obtain an exact solution of the three-dimensional 1/(?)-potential.(9)We obtain an exact solution of the three-dimensional 1/r3/2-potential.(10)The scalar scattering on the Schwarzschild black hole is solved.The scattering theory without large-distance asymptotics obtained in the thesis improves the conventional scattering theory in text books.Furthermore,based the result given in the thesis,we can directly establish the acoustic wave scattering and electromagnetic wave scattering theory without large-distance asymptotics.The heat-kernel method for scattering presented in the thesis is not only one single approach;it is indeed a set of methods for scattering,which converts a method of calculating heat kernels to a method of calculating a scattering phase shift.There are mature theories on the large-distance asymptotic behavior of short-range potentials,but for long-range potentials,there are no systemic discussions.The result of the large-distance asymptotic behavior of long-range power potentials given in the thesis will contribute to study of long-range potential scattering.The exact solution is rare.The exact solutions of 1/(?)and 1/r3/2-potentials are valuable results in thearea.Scattering on black holes is an important problem and a fundamental method in gravitation problems,the result on scattering on the Schwarzschild black hole given in the thesis can be directly applied to various gravitation problems.