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This paper presents a nonperturbative method for solving eigenproblems. This method applies to almost all potentials and provides nonperturbative approximations for any energy level. The method converts an eigenproblem into a perturbation problem, obtains perturbation solutions through standard perturbation theory, and then analytically continues the perturbative solution into a nonperturbative solution. Concretely, we follow three main steps: (1) Introduce an auxiliary potential that can be solved exactly and treat the potential to be solved as a perturbation on this auxiliary system. (2) Use perturbation theory to obtain an approximate polynomial of the eigenproblem. (3) Use a rational approximation to analytically continue this approximate polynomial into the nonperturbative region.

It is revealed that there exist duality families of the KdV type equation. A duality family consists of an infinite number of generalized KdV (GKdV) equations. A duality transformation relates the GKdV equations in a duality family. Once a family member is solved, the duality transformation presents the solutions of all other family members. We show some dualities as examples, such as the soliton solution-soliton solution duality and the periodic solution-soliton solution duality.

Some probability distributions have moments, and some do not. For example, the normal distribution has power moments of arbitrary order, but the Cauchy distribution does not have power moments. In this paper, by analogy with the renormalization method in quantum field theory, we suggest a renormalization scheme to remove the divergence in divergent moments. We establish more than one renormalization procedure to renormalize the same moment to prove that the renormalized moment is scheme-independent. The power moment is usually a positive-integer-power moment; in this paper, we introduce nonpositive-integer-power moments by a similar treatment of renormalization. An approach to calculating logarithmic moment from power moment is proposed, which can serve as a verification of the validity of the renormalization procedure. The renormalization schemes proposed are the zeta function scheme, the subtraction scheme, the weighted moment scheme, the cut-off scheme, the characteristic function scheme, the Mellin transformation scheme, and the power-logarithmic moment scheme. The probability distributions considered are the Cauchy distribution, the Levy distribution, the q-exponential distribution, the q-Gaussian distribution, the normal distribution, the Student's t-distribution, and the Laplace distribution.

In this paper, we calculate the energy spectrum of interacting gases by converting the cluster expansion method in statistical mechanics into a method of solving energy eigenvalues. We obtain an explicit expression of the energy eigenvalue, by which we can calculate the eigenvalue of an interacting gas from the interparticle potential directly. As an example, we calculate the energy spectrum for an interacting gas with soft-sphere potentials.

We propose an approach for calculating one-loop effective actions and vacuum energies in quantum field theory. Spectral functions are functions defined by the eigenvalues of an operator. One-loop effective actions and vacuum energies in quantum field theory, as well as scattering phase shifts and scattering amplitudes in quantum mechanics, are all spectral functions. If a transformation between different spectral functions is identified, we can obtain a spectral function from another through the transformation. In this paper, we convert quantum mechanical methods for calculating scattering phase shifts and scattering amplitudes into quantum field theory methods for calculating one-loop effective actions and vacuum energies. As examples, the Born approximation and the WKB approximation in quantum mechanics are converted into quantum field theory methods. We also calculate the one-loop effective action and vacuum energy of scalar fields in the Schwarzschild spacetime and the Reissner-Nordstr\"{o}m spacetime as examples. Some integral representations of the Bessel function are given in appendices.

The aim of this paper is to construct a quantum effective action for gravitational fields by the effective potential method in quantum field theory. The minimum of the quantum effective action gives an equation of quantum fluctuations. We discuss the quantum fluctuation in the flat spacetime and in the Schwarzschild spacetime. It is shown that a baby spacetime may be created from a classical vacuum through a quantum fluctuation., Comment: 20 pages

If we cannot obtain all terms of a series, or if we cannot sum up a series, we have to turn to the partial sum approximation which approximate a function by the first several terms of the series. However, the partial sum approximation often does not work well for periodic functions. In the partial sum approximation of a periodic function, there exists an incorrect oscillation which cannot be eliminated by keeping more terms, especially at the domain endpoints. A famous example is the Gibbs phenomenon in the Fourier expansion. In the paper, we suggest an approach for eliminating such oscillations in the partial sum approximation of periodic functions.

We suggest a scheme for considering the quantum correction of the gravitational constant. In the model, the gravitational constant originates from a coupling of the gravitational field with a scalar field. In this paper, we show that if the scalar field, as it should be in the real physical world, is a quantum field, then the gravitational constant will have a spacetime-dependent quantum correction, so that the quantum corrected physical constant is no longer a constant. The quantum correction of the gravitational constant is different in different spacetime. We calculate the quantum correction in the Schwarzschild spacetime, the $H_{3}$ (Euclidean $AdS_{3}$) spacetime, the $H_{3}/Z$ spacetime, the universe model, the de Sitter spacetime, and the Rindler spacetime.

TWe suggest a method to construct exactly solvable Gross-Pitaevskii type equations, especially the variable-coefficient high-order Gross-Pitaevskii type equations. We show that there exists a relation between the Gross-Pitaevskii type equations. The Gross-Pitaevskii equations connected by the relation form a family. In the family one only needs to solve one equation and other equations in the family can be solved by a transform. That is, one can construct a series of exactly solvable Gross-Pitaevskii type equations from one exactly solvable Gross-Pitaevskii type equation. As examples, we consider the family of some special Gross-Pitaevskii type equations: the nonlinear Schr\"odinger equation, the quintic Gross-Pitaevskii equation, and cubic-quintic Gross-Pitaevskii equation. We also construct the family of a kind of generalized Gross-Pitaevskii type equation.

Dealing with a few-fermion system in the canonical ensemble, rather than in the grand canonical ensemble, shows that a few-fermion system with odd number fermions behaves differently from a few-fermion system with even number fermions. An even-number-fermion system behaves like a Bose system rather than a Fermi system.

We show that there exists a duality family of self-interacting massive scalar fields. The scalar field in a duality family are related by a duality transformation. Such a duality of scalar fields is a field version of the Newton-Hooke duality in classical mechanics. The duality transformation preserves the type of the field equation: transforming a Klein-Gordon type equation to another Klein-Gordon type equation with a different self-interacting potential. Once a field in a duality family is solved, all other family members are solved by the transformation. That is, a series of exactly solvable models can be constructed from one exactly solvable model. The dual field of the power-interaction field, the sine-Gordon field, etc., are considered. Moreover, as a comparison, we show an analogue of the duality in classical and quantum mechanics.

We reveal a duality in classical and quantum mechanics. Dual systems are related by duality transforms. All mechanical systems that are dual to each other form a duality family. In a duality family, once a system is solved, all other potentials are solved by the dual transform. That is, in a duality family, we only need to solve one system.

It is shown that there exists a duality among fields. If a field is dual to another field, the solution of the field can be obtained from the dual field by the duality transformation. We give a general result on the dual fields. Different fields may have different numbers of dual fields, e.g., the free field and the $\phi^{4}$-field are self-dual, the $\phi^{n}$-field has one dual field, a field with an $n$-term polynomial potential has $n+1$ dual fields, and a field with a nonpolynomial potential may have infinite number of dual fields. All fields which are dual to each other form a duality family. This implies that the field can be classified in the sense of duality, or, the duality family defines a duality class. Based on the duality relation, we can construct a high-efficiency approach for seeking the solution of field equations: solving one field in the duality family, all solutions of other fields in the family are obtained immediately by the duality transformation. As examples, we consider some $\phi^{n}$-fields, general polynomial-potential fields, and the sine-Gordon field.

We introduce probability thermodynamics and probability quantum fields. By probability we mean that there is an unknown operator, physical or nonphysical, whose eigenvalues obey a certain statistical distribution. Eigenvalue spectra define spectral functions. Various thermodynamic quantities in thermodynamics and effective actions in quantum field theory are all spectral functions. In the scheme, eigenvalues obey a probability distribution, so a probability distribution determines a family of spectral functions in thermodynamics and quantum field theory. This leads to probability thermodynamics and probability quantum fields determined by a probability distribution. In constructing spectral functions, we encounter a problem. The conventional definition of spectral functions applies only to lower bounded spectra. In our scheme, however, there are two types of spectra: lower bounded spectra, corresponding to the probability distribution with nonnegative random variables, and the lower unbounded spectra, corresponding to probability distributions with negative random variables. To take the lower unbounded spectra into account, we generalize the definition of spectral functions by analytical continuation. In some cases, we encounter divergences. We remove the divergence by a renormalization procedure. Moreover, in virtue of spectral theory in physics, we generalize some concepts in probability theory. For example, the moment-generating function in probability theory does not always exist. We redefine the moment-generating function as the generalized heat kernel introduced in this paper, which makes the concept definable when the definition in probability theory fails. Thermodynamic quantities, vacuum amplitudes, one-loop effective actions, and vacuum energies for various probability distributions are presented.

In this paper, we solve the massive scalar field in the Reissner-Nordstr\"om spacetime. The scalar field in the Reissner-Nordstr\"om spacetime has both bound states and scattering states. For bound states, we solve the bound-state wave function and the eigenvalue spectrum. For scattering states, we solve the scattering wave function and give an explicit expression for scattering phase shift by the integral equation method. Especially, we introduce the tortoise coordinate for the Reissner-Nordstr\"om spacetime.

A duality between spacetime manifolds, the geodesic duality, is introduced. Two manifolds are geodesic dual, if the transformation between their metrics is also the transformation between their geodesics. That is, the transformation that transforms the metric to the metric of the dual manifold is also the transformation that transforms the geodesic to the geodesic of the dual manifold. On the contrary, for nondual spacetime manifolds, a geodesic is no longer a geodesic after the transformation between the metrics. We give a general result of the duality between spacetime manifolds with diagonal metrics. The geodesic duality of spherically symmetric spacetime are discussed for illustrating the concept. The geodesic dual spacetime of the Schwarzschild spacetime and the geodesic dual spacetime of the Reissner-Nordstr\"om spacetime are presented.

An integral equation method for scalar scattering in Schwarzschild spacetime is constructed. The zeroth-order and first-order scattering phase shift is obtained.

Inspired by general relativity, we suggest an approach for long-range potential scattering. In scattering theory, there is a general theory for short-range potential scattering, but there is no general theory for long-range potential scattering. This is because the scattering boundary conditions for all short-range potentials are the same, but for different long-range potentials are different. In this paper, by introducing tortoise coordinates, we convert long-range potential scattering to short-range potential scattering. This allows us to deal with long-range potential scattering as short-range potential scattering. An explicit expression of the scattering wave function for long-range potential scattering is presented, in which the scattering wave function is represented by the tortoise coordinate and the scattering phase shift. We show that the long-range potential scattering wave function is just the short-range potential scattering wave function with a replacement of a common coordinate by a tortoise coordinate. The approach applies not only to scattering but also applies to bound states. Furthermore, in terms of tortoise coordinates, we suggest a classification scheme for potentials. We also discuss the duality between tortoise coordinates.

Newton revealed an underlying duality relation between power potentials in classical mechanics. In this paper, we establish the quantum version of the Newton duality. The main aim of this paper is threefold: (1) first generalizing the original Newton duality to more general potentials, including general polynomial potentials and transcendental-function potentials, 2) constructing a quantum version of the Newton duality, including power potentials, general polynomial potentials, transcendental-function potentials, and power potentials in different spatial dimensions, and 3) suggesting a method for solving eigenproblems in quantum mechanics based on the quantum Newton duality provided in the paper. The classical Newton duality is a duality among orbits of classical dynamical systems. Our result shows that the Newton duality is not only limited to power potentials, but a more universal duality relation among dynamical systems with various potentials. The key task of this paper is to construct a quantum Newton duality, the quantum version of the classical Newton duality. The quantum Newton duality provides a duality relations among wave functions and eigenvalues. As applications, we suggest a method for solving potentials from their Newtonianly dual potential: once the solution of a potential is known, the solution of all its dual potentials can be obtained by the duality transformation directly. Using this method, we obtain a series of exact solutions of various potentials. In appendices, as preparations, we solve the potentials which is solved by the Newton duality method in this paper by directly solving the eigenequation.

In conventional gravitational wave scattering theory, a large-distance asymptotic approximation is employed. In this approximation, the gravitational wave is approximated by its large-distance asymptotics. In this paper, we establish a gravitational wave scattering theory without the large-distance asymptotic approximation.

In conventional acoustic scattering theory, a large-distance asymptotic approximation is employed. In this approximation, a far-field pattern, an asymptotic approximation of the exact result, is used to describe a scattering process. The information of the distance between the target and the observer, however, is lost in the large-distance asymptotic approximation. In this paper, we provide a rigorous theory of acoustic scattering without the large-distance asymptotic approximation. The acoustic scattering treatment developed in this paper provides an improved description for the acoustic wave outside the target. Moreover, as examples, we consider acoustic scattering on a rigid sphere and on a nonrigid sphere. We also illustrate the influence of the near target effect on the angular distribution of outgoing waves. It is shown that for long wavelength acoustic scattering, the near target effect must be reckoned in.

We solve a $1+5$-dimensional cylindrical gravitational-wave solution of the Einstein equation, in which there are two curvature singularities. Then we show that one of the curvature singularities can be removed by an extension of the spacetime. The result exemplifies that the curvature singularity is not always a spacetime singularity; in other words, the curvature singularity cannot serve as a criterion for spacetime singularities.

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., Comment: Minor corrections

In the calculation of quantum-mechanical singular-potential scattering, one encounters divergence. We suggest three renormalization schemes, dimensional renormalization, analytic continuation approach, and minimal-subtraction scheme to remove the divergence.

The Hawking radiation emits all species of particles, but the Bekenstein-Hawking entropy is independent of the number of the species of particles. This is the so-called species problem --- a puzzling problem for a long time. In this paper, we suggest a solution to this problem. A result of the scheme is that the black hole atmosphere has a mass equaling 3/8 mass of a classical Schwarzschild black hole, which agrees with 't Hooft's brick wall model.

An exact solution of the three-dimensional spherical symmetry inverse-square-root potential $V\left( r\right) =-\frac{\alpha}{\sqrt{r}}$, including scattering and bound-state solutions, is presented.

Covariant perturbation expansion is an important method in quantum field theory. In this paper an expansion up to arbitrary order for off-diagonal heat kernels in flat space based on the covariant perturbation expansion is given. In literature, only diagonal heat kernels are calculated based on the covariant perturbation expansion., Comment: To appear in International Journal of Theoretical Physics

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 large-distance 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.

A thermo field dynamics approach for calculating the entropy of a spacetime is suggested. It is exemplified through the Rindler spacetime, the Milne spacetime, the Boulware spacetime, and the Minkowski spacetime with a moving mirror that the entropy of a spacetime is equal to the entropy of a thermo quantum field with the same temperature of the spacetime we study. This in fact suggests an thermo field dynamics approach of calculating the entropy of a spacetime. In this approach, the entropy of a spacetime is an expectation value of the entropy operator on a thermo vacuum state. The thermo vacuum state is the vacuum state on the maximal manifold which is the maximal analytic extension of the spacetime we study.

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 the covariant perturbation theory. Moreover, as an alternative application of the relation between heat kernels and partial-wave phase shifts presented in this paper, we give an example of how to calculate a global heat kernel from a known scattering phase shift.

By large-distance asymptotics, in conventional scattering theory, at the cost of losing the information of the distance between target and observer, one arrives at an explicit expression for scattering wave functions represented by a scattering phase shift. In the present paper, together with a preceding paper (T. Liu,W.-D. Li, and W.-S. Dai, JHEP06(2014)087), we establish a rigorous scattering theory without imposing large-distance asymptotics. We show that even without large-distance asymptotics, one can also obtain an explicit scattering wave function represented also by a scattering phase shift, in which, of course, the information of the distance is preserved. Nevertheless, the scattering amplitude obtained in the preceding paper depends not only on the scattering angle but also on the distance between target and observer. In this paper, by constructing a scattering boundary condition without large-distance asymptotics, we introduce a scattering amplitude, like that in conventional scattering theory, depending only on the scattering angle and being independent of the distance. Such a scattering amplitude, when taking large-distance asymptotics, will recover the scattering amplitude in conventional scattering theory. The present paper, with the preceding paper, provides a complete scattering theory without large-distance asymptotics.

In this paper, based on the conformal mapping method and the perturbation theory, we develop a method to solve the statistical problem within general 2-dimensional regions. We consider some examples and the numerical results and fitting results are given. We also give the thermodynamic quantities of the general 2-dimensional regions, and compare the thermodynamic quantities of the different regions., Comment: 16 pages, 10 figures, 3 tables

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.