An exact solution of the three-dimensional spherical symmetry inverse-squareroot potential V (r) = − α √ r , including scattering and bound-state solutions, is presented. 1 daiwusheng@tju.edu.cn.

In conventional scattering theory, to obtain an explicit result, one imposes a precondition that the distance between target and observer is infinite. With the help of this precondition, one can asymptotically replace the Hankel function and the Bessel function with the sine functions so that one can achieve an explicit result. Nevertheless, after such a treatment, the information of the distance between target and observer is inevitably lost. In this paper, we show that such a precondition is not necessary: without losing any information of distance, one can still obtain an explicit result of a scattering rigorously. In other words, we give an rigorous explicit scattering result which contains the information of distance between target and observer. We show that at a finite distance, a modification factor -the Bessel polynomial -appears in the scattering amplitude, and, consequently, the cross section depends on the distance, the outgoing wave-front surface is no longer a sphere, and, besides the phase shift, there is an additional phase (the argument of the Bessel polynomial) appears in the scattering wave function.

An approach for solving scattering problems, based on two quantum field theory methods, the heat-kernel method and the scattering spectral method, is constructed. This approach converts a method of calculating heat kernels into a method of solving scattering problems. This allows us to establish a method of scattering problems from a method of heat kernels. As an application, we construct an approach for solving scattering problems based on the covariant perturbation theory of heat-kernel expansions. In order to apply the heat-kernel method to scattering problems, we first calculate the off-diagonal heat-kernel expansion in the frame of covariant perturbation theory. Moreover, as an alternative application of the relation between heat kernels and partialwave phase shifts presented in this paper, we give an example of how to calculate a global heat kernel from a known scattering phase shift.

In conventional scattering theory, by large-distance asymptotics, at the cost of losing the information of the distance between target and observer, one imposes a largedistance asymptotics to achieve a scattering wave function which can be represented explicitly by a scattering phase shift. In this paper, without large-distance asymptotics, we establish an arbitrary-dimensional scattering theory. Arbitrary-dimensional scattering wave functions, scattering boundary conditions, cross sections, and phase shifts are given without large-distance asymptotics. The importance of an arbitrary-dimensional scattering theory is that the dimensional renormalization procedure in quantum field theory needs an arbitrary-dimensional result. Moreover, we give a discussion of one-and two-dimensional scatterings.

The main aim of this paper is twofold. (1) Exact solutions of a scalar field in the Schwarzschild spacetime are presented. The exact wave functions of scattering states and bound-states are presented. Besides the exact solution, we also provide explicit approximate expressions for bound-state eigenvalues and scattering phase shifts. (2) By virtue of the exact solutions, we give a direct calculation for the discontinuous jump on the horizon for massive scalar fields, while in literature such a jump is obtained from an asymptotic solution by an analytic extension treatment.Corresponding to the region outside the horizon, i.e., r ∈ [2M, ∞), the range of the variable z in the confluent Heun equation is z ∈ [1, ∞). The confluent Heun equation (3.1) has two singular points, z = 1 and z → ∞, in the region z ∈ [1, ∞) [46]. These two singular points just correspond to the two singular points of the Schwarzschild spacetime, r = 2M and r → ∞.