1 Introduction

The propagator K considered is defined by the unitary operator \(\Theta (t)\exp [-\textbf{i}H\,t/\hbar ]\) where the Heaviside function \(\Theta (t)=1\) if \(t>0\) and zero if \(t<0\). In position representation an initial wave function \(\psi _0(\textbf{r}^{\prime })\) is propagated to \(\psi (\textbf{r},t)\) by K as

$$\begin{aligned} \psi (\textbf{r},t) =\int \textrm{d} \textbf{r}^{\prime }\, K(\textbf{r}, \textbf{r}^{\prime }, t ) \psi _0( \textbf{r}^{\prime }), \quad t> 0. \end{aligned}$$
(1)

In terms of a complete set of eigenfunctions with \(H \psi _{\nu }=E_{\nu }\psi _{\nu }\), one writes

$$\begin{aligned} K(\textbf{r}, \textbf{r}^{\prime }, t) = \Theta (t) \sum _{\nu }\exp [-\textbf{i}E_{\nu } t/\hbar ] \,\psi _{\nu }(\textbf{r}) \psi _{\nu }^*(\textbf{r}^{\prime }). \end{aligned}$$
(2)

By differentiation with respect to t, one immediately finds that \(K(\textbf{r}, \textbf{r}^{\prime }, t )\) obeys the following partial differential equation (PDE)

$$\begin{aligned} \left[ \frac{\partial }{\partial t}+\frac{\textbf{i}}{\hbar } H\right] K(\textbf{r}, \textbf{r}^{\prime }, t) =\delta (t) \delta ^3(\mathbf{r- r^{\prime }}), \quad H\equiv H(\textbf{r}). \end{aligned}$$
(3)

By time Fourier transformation, one defines the energy amplitude \(K_E\) and subsequently makes use of (2) as follows:

$$\begin{aligned} \begin{aligned} K_E(\mathbf{r, r^{\prime }})&= \int _0^{\infty } \textrm{d}t\,\exp [\textbf{i}E \,t/\hbar ] \,K(\textbf{r}, \textbf{r}^{\prime }, t) = \frac{\hbar }{\textbf{i}} G_E(\mathbf{r,r^{\prime }}), \\ G_E&= \sum _{\nu }\frac{\psi _{\nu }(\textbf{r}) \psi _{\nu }^*(\textbf{r}^{\prime })}{E_{\nu }-E}, \,\,\textrm{Im}(E)>0. \end{aligned} \end{aligned}$$
(4)

For the non-relativistic Hamiltonian H of the Coulomb–Kepler problem, in a milestone achievement, Hostler [1, 2] succeeded to exactly determine the Green function \(G_E\) in a compact form. The Hamiltonian reads

$$\begin{aligned} H=-\frac{\hbar ^2}{2 m}\Delta -\frac{\alpha }{r}, \quad \alpha = \pm \frac{ \left| e_1 e_2 \right| }{4 \pi \epsilon _0} \quad {\mathrm or} \quad \alpha =m_1m_2 G_{grav}. \end{aligned}$$
(5)

In (5), the coupling constant \(\alpha\) is defined for the electric and gravitational case, respectively; it is positive, except for repellent charges; m denotes the reduced mass of the two-body problem. The Hamiltonian H encompasses both the hydrogen atom and the two-body Kepler problem of celestial mechanics which differ in scale up to 80 decimals. Hostler’s solution will be noted further below. We try to derive the propagator K from Hostler’s Green’s function by means of the inverse Fourier transformation as follows:

$$\begin{aligned} K(\mathbf{r},\mathbf{r}^{\prime }, t)=&\lim _{\epsilon \rightarrow 0}\lim _{R\rightarrow \infty }\left\{ \frac{(-\mathbf{i})}{2\pi }\int _{-R}^{R}\mathbf{d}E\right.\\&\left. \exp [-\mathbf{i}E \,t/\hbar ] G_{E+\mathbf{i}\epsilon }( \mathbf{r},\mathbf{r}^{\prime })\right\} , \quad t>0, \quad \epsilon >0. \end{aligned}$$
(6)

In Sect. 3, the potential-free case as noted in [3] will be recovered. By taking the limiting procedure above, we avoid using series representations in Schwartz space [4], and simultaneously define how to approach or sidestep the essential singularity at \(E =\infty\). We remark on different notations: In [5], the propagator is termed kernel, also the notation resolvent is sometimes used instead of Green’s function in operator form.

By its explicit time dependence, the propagator delivers more immediate information than the Green’s function. By knowing \(K(\textbf{r},\mathbf{r^{\prime }}, t)\), one can follow, for instance, the scattering process of a wave packet through the whole time interval, finite or infinite, including the behavior near the scattering center, at variance with the asymptotic scattering description.

Stimulated by Richard Feynman, there have been attempts to calculate the propagator by the method of path integrals; see [6, 7] and for a critical study [8]. Also encouraged by Feynman, Hostler’s Green’s function was used to calculate the propagator in [9]. Recently, a numerical investigation was published [10], where the propagator is represented in a basis of square-integrable Sturmian functions. All these attempts that can be said were only partially successful and so far did not lead to important applications.

The present study is part of a further attempt to calculate the propagator from Hostler’s Green’ s function. It was motivated by our previous study of the quantum mechanics of rectilinear orbits [11, 12]. There, the Kustaanheimo-Stiefel transformation [13] was applied which leads to a model Hamiltonian of four decoupled harmonic oscillators. One could then apply coherent states of the harmonic [14] and of the reversed harmonic oscillator [15, 16]. As a shortcoming of the method, the evolution of the initial state is controlled by the eccentric anomaly and not by the Schrödinger time t. The necessary condition that eccentric anomaly and time are in one-one correspondence failed to be true before the simulated rectilinear orbit entered the physically interesting region.

In the next Section, we deform the integration path of the E integral in (6) to a quarter arc circle close to the singularity at \(E=\infty\). One observes a highly oscillating exponential phase in the integrand. In Sect. 3, as a kind of litmus test, the propagator of the potential-free case is determined from the Green’s function: (i) by exact integration of the modified Fourier transformation in terms of the complex error function and its asymptotic form, (ii) by the method of stationary phase. In Sect. 4, the method of stationary phase is generalized to include integrands which will be relevant to the problem of non-zero potential, work which is still in progress. Appendices 1 and 2 contain proofs on the equivalence of the method of stationary phase and the exact calculation of certain improper integrals.

2 Transformation of the Fourier Integral in Complex Energy Plane

Hostler’s Green’s function \(G_\textrm{Ho}\) differs from \(G_E\), defined in (4), by a constant factor as

$$\begin{aligned} \gamma ^2 G_\textrm{Ho}= - G_E, \quad \gamma =\sqrt{2 m/\hbar ^2}. \end{aligned}$$
(7)

To write down the exact solution of the Fourier transformed PDE (3), given by Eq. (8) of [2], we state the symbols there used

$$\begin{aligned} x&=r+r^{\prime }+r_{12},\,\, y=r+r-r_{12}, \\ r_{12}&=\left| \mathbf{r-r^{\prime }}\right| , \,\, k=\gamma \sqrt{E}, \,\ 2\nu =\gamma ^2 \alpha /k, \end{aligned}$$
(8)

where x, y are the Lambert coordinates, k is a complex wave number, and \(\nu\) a complex number which is proportional to the coupling constant \(\alpha\) and, via k, depends on the energy E (by the notation of [2], \(\alpha =Z e^2/(4 \pi ))\). The functions M and W denote the Whittaker functions as defined in [17], \(\Gamma\) is the Gamma function. The solution, which factorizes in the Lambert coordinates, reads

$$\begin{aligned} G_\mathrm{Ho}(\mathbf{r, r^{\prime }},E)=&-\frac{\Gamma (1-\mathbf{i}\nu )}{4 \pi r_{12}} \frac{1}{\mathbf{i}k} \left( -\frac{\partial }{\partial y}+\frac{\partial }{\partial x}\right)\\& W_{\mathbf{i}\nu ; 1/2}(-\mathbf{i}k x) M_{\mathbf{i}\nu ; 1/2}(-\mathbf{i}k y). \end{aligned}$$
(9)

In the potential-free case, the coupling constant \(\alpha =0\) and thus \(\nu =0\). The Whittaker functions are then elementary functions as

$$\begin{aligned} M_{0; 1/2}(z) =2 \sinh (z/2),\quad W_{0; 1/2}(z) =\exp (-z/2), \end{aligned}$$
(10)

which give rise to the well known free-particle Green’s function; see e.g. [18, 19],

$$\begin{aligned} G_\textrm{Ho}^{(0)}(\mathbf{r, r^{\prime }},E)= -\frac{\exp \left( \textbf{i}kr_{12}\right) }{4 \pi r_{12}} \quad \textrm{or} \quad G_{E}^{(0)}(\mathbf{r, r^{\prime }},E)= \gamma ^2\,\frac{\exp \left( \textbf{i}kr_{12}\right) }{4 \pi r_{12}}. \end{aligned}$$
(11)

The Green function depends on \(k=k(E)\). We set

$$\begin{aligned} k=+\gamma \sqrt{E+\textbf{i}\epsilon }\quad \textrm{if}\quad E>0, \,\,\textrm{and} \,\,k=+\textbf{i}\gamma \sqrt{-E-\textbf{i}\epsilon }\quad \textrm{if}\quad E<0. \end{aligned}$$
(12)

In (12), the signs of the square roots are chosen such that in the case \(E>0\) one has outgoing radial waves, whereas, if \(E<0\), the free-particle Green’s function (11) describes evanescent radial waves. In (6), we apply the transformation \(E \rightarrow \kappa =\textbf{i}/k(E)\), which is well defined, since

$$\begin{aligned} \frac{\textrm{d} E }{\textrm{d} \kappa } =\frac{2}{\gamma ^2 \kappa ^3} = \frac{ 2\,\textbf{i}k^3}{\gamma ^2}, \quad \kappa =\frac{\textbf{i}}{k}, \end{aligned}$$
(13)

differs from zero for \(\mid E \mid \, \le R\), and \(\epsilon >0\).

We assume that the integrand in (6) is analytic within the closed contour of Fig. 1. Then, by Cauchy’s theorem, the original integral is equivalent to the integral along the quarter arc of the \(\rho\) circle:

$$\begin{aligned} \int _{E_1\rightarrow (E=0) \rightarrow E_2} \textrm{d}E ... = \int _{E_1\rightarrow E_2} \textrm{d}\kappa \, \frac{\textrm{d} E }{\textrm{d}\kappa } .... \end{aligned}$$
(14)

We have to comment on the analyticity assumption. The closed path lies in the first quadrant with Re\((\kappa )\equiv \kappa _R\ge 0\) and Im\((\kappa )\equiv \kappa _I\ge 0\), whereby \(\kappa _I\) can get zero only close to the origin where also \(\kappa _R\) gets small. The \(\Gamma\) factor which appears in the integrand (6) through \(G_\textrm{Ho}\), depends on \(\kappa\) through \(\Gamma (1-\textbf{i}\nu ) = \Gamma (1-\alpha \gamma ^2 \kappa )\); since \(\kappa _I >0\), poles cannot appear; the poles of \(\Gamma (z)\) lie on the negative real axis at integer values \(z=-n\), \(n=0,1,...\), and are connected with the discrete part of the energy spectrum. As to the Whittaker functions which constitute \(G_\textrm{Ho}\), we use their connections with the confluent hypergeometric functions \(\Phi\) (Kummer function) and U (Tricomi function) as defined in [20]:

$$\begin{aligned} \begin{aligned} M_{\textbf{i}\nu ,1/2}(z) =&z \exp (-z/2)\, \Phi (1-\textbf{i}\nu ,2,z),\\ W_{\textbf{i}\nu ,1/2}(z) =&z \exp (-z/2)\, U(1-\textbf{i}\nu ,2,z), \end{aligned} \end{aligned}$$
(15)

where \(z=-\textbf{i}k \,x\) or \(z=-\textbf{i}k \,y\) with the Lambert coordinates x and y larger zero. The function \(\Phi (a,b,z)\) is analytic in all three complex entries, whereas U(a, b, z) is multivalued in z with the branch cut defined along the negative real axis of the complex variable z. But on the \(\kappa\) contour, Im\((k)\equiv k_I >0\) which implies that Real\((z) >0\); \(k_I>0\) follows from the relations \(k_I=\kappa _I/\left| \kappa \right| ^2 > 0\). By (13), \(\textrm{d} E/ \textrm{d}\kappa\) is analytic within the contour. In concluding, the integrand in (6) is analytic within the closed path and, thus, justifies the application of Cauchy’s integral theorem.

In view of (12), we have the following relations on the \(\rho\) circle:

$$\begin{aligned} \kappa =\rho \exp (\textbf{i}\varphi ), \quad E(\varphi )=-\textbf{i}\epsilon - \frac{\exp (-2 \textbf{i}\varphi )}{\eta ^2}, \quad \eta ^2=\gamma ^2 \rho ^2. \end{aligned}$$
(16)

The endpoints \(E_{1,2}=\mp R\) are real numbers. For small \(\epsilon\), the corresponding angles \(\varphi _{1,2}\) are close to 0 and \(\pi /2\), respectively. The conditions Im\([E(\varphi _{1,2})]=0\) imply

$$\begin{aligned} \varphi _1 = \epsilon \, \eta ^2/2 +\mathcal{O}(\epsilon ^3 \eta ^6),\quad \varphi _2 = \pi /2 -\epsilon \,\eta ^2/2 +\mathcal{O}(\epsilon ^3 \eta ^6), \end{aligned}$$
(17)

and

$$\begin{aligned} E_1\equiv E(\varphi _1) &=-1/\eta ^2+\epsilon \,\mathcal{O}(\epsilon \,\eta ^2), \\ E_2\equiv E(\varphi _2) &=+1/\eta ^2+\epsilon \, \mathcal{O}(\epsilon \,\eta ^2). \end{aligned}$$
(18)

We remark that \(\epsilon \,\eta ^2\) is a dimensionless number with both \(\epsilon >0\) and \(\eta >0\) arbitrary small.

Now, we blow up the circle arc by the transformation \(\kappa \rightarrow A\) where

$$\begin{aligned} A= (1/\eta ) \exp [-\textbf{i}\varphi ], \quad 0< \varphi <\pi /2, \quad \eta = \gamma \,\rho , \end{aligned}$$
(19)

which implies

$$\begin{aligned} \kappa = 1/(\gamma \,A), \quad E= - A^2 +\mathcal{O}(\epsilon ), \quad \textrm{d}E/\textrm{d}A=-2 A, \quad k = \textbf{i}\gamma \,A. \end{aligned}$$
(20)

The end points \(E_{1,2}\) on the \(\rho\)-circle correspond to the values

$$\begin{aligned} A_1 = 1/\eta +\mathcal{O}(\epsilon \, \eta ), \quad A_2 = -\textbf{i}/\eta +\mathcal{O}(\epsilon \,\eta ). \end{aligned}$$
(21)

In conclusion, the Fourier integral (6) is transformed as follows;

$$\begin{aligned} K(\textbf{r},\textbf{r}^{\prime }, t) = \lim _{\epsilon \rightarrow 0}\lim _{\eta \rightarrow 0}\left\{ \frac{(-\textbf{i})}{2\pi }\int _{1/\eta }^{-\textbf{i}/\eta }\textrm{d}A\, (-2 A)\exp [\textbf{i}A^2 \,t/\hbar ] G_{E+\textbf{i}\epsilon }( \textbf{r},\textbf{r}^{\prime })\right\} , \end{aligned}$$
(22)

with \(E=-A^2\), \(k=\textbf{i}\gamma A\) and \(t >0\). It is observed that, through the factor \(\exp (\textbf{i}A^2 t/\hbar )\), there is an unbounded variation of the integrand in the limit \(\eta \rightarrow 0\).

3 The Free-Particle Propagator

From Eq. (3.113) of [3], we quote the propagator as

$$\begin{aligned} K^{(0)}(\textbf{r}, \textbf{r}^{\prime }, t) =\Theta (t)\exp (- 3 \pi \,\textbf{i}/4)\left( \frac{m}{2\pi \hbar t}\right) ^{3/2} \exp \left[ \frac{\textbf{i}m (\mathbf{r-r^{\prime }})^2}{2 \hbar t}\right] . \end{aligned}$$
(23)

As compared to the method of Fourier transformation applied below, the above result is quite easily derived from the spectral representation (2).

3.1 Applying the Method of Fourier Transformation

In the following, we reproduce the standard result (23) by the method of Fourier transformation (FT), using the free-particle Green’s function \(G_E^{(0)}\) given in (11). The FT method cannot be avoided in the present context of the given Hostler’s Green’s function. The application of (22) to the potential-free case constitutes a kind of litmus test. In view of (22) and (11), we have to consider

$$\begin{aligned} K^{(0)}(\textbf{r}, \textbf{r}^{\prime }, t) &= +\textbf{i}\frac{\gamma ^2}{4 \pi ^2 r_{12}} F(c,s), \\ F(c,s)&=\lim _{\eta \rightarrow 0}\int _{1/\eta }^{-\textbf{i}/\eta }\textrm{d}A\, A\exp [\textbf{i}A^2 \,c-s A], \end{aligned}$$
(24)

with \(c=t/\hbar\) and \(s= \gamma \,r_{12}\). The integral F(c, s) can be expressed in terms of the complex error function erf. Subsequently, in the limit \(\eta \rightarrow 0\), asymptotic forms of erf are applied with due attention paid to the occurrence of a sign function which is decisive and depends on the phases of the complex boundary values \(A_1\) and \(A_2\), respectively. Actually, the sign function is mentioned in [21], but we did not find it in standard handbooks of mathematical physics like [20, 22, 23] nor in the monograph [24]. Furthermore, for the existence of the integral F in the limit \(\eta \rightarrow 0\), we introduce an infinitesimal dissipation as

$$\begin{aligned} c \rightarrow c-\textbf{i}\eta ^{3/2}, \quad \eta >0,\quad c=t/\hbar . \end{aligned}$$
(25)

We give details in Appendix 1 including a numerical confirmation of the sign function in the asymptotic forms of the error function. As is shown in Appendix 1, the standard result (23) for the free-particle propagator is recovered by the Fourier integral (24).

3.2 Applying the Approximation of Stationary Phase

In the integrand of F in (24), the exponent \(\Phi = \textbf{i}A^2 c-s A\) is highly oscillating near the integration boundaries \(A_{1,2}\) when \(\eta\) comes close to zero. The phase \(\Phi\) is stationary when d\(\Phi /\)dA=0, which is the case at \(A=A_s\) with

$$\begin{aligned} A_s=-\textbf{i}s/(2 c). \end{aligned}$$
(26)

We assume, at first, that both c and s are larger than zero, so \(A_s\) lies on the negative imaginary axis. The integration path is deformed such that it passes through \(A_s\) along the imaginary axes. With \(A=A_s-\textbf{i}\alpha\), we have the identity

$$\begin{aligned} \Phi =-\textbf{i}c \,A_s^2-\textbf{i}c\, \alpha ^2. \end{aligned}$$
(27)

It is elementary to determine the following stationary phase (sp) integral \(F_0^{sp}\) for \(c>0\)

$$\begin{aligned} F_0^{sp} = (-\textbf{i}) \exp \left( -\textbf{i}c \,A_s^2 \right) \int _{-\infty }^{\infty }\textrm{d}\alpha \,\exp \left( -\textbf{i}c\, \alpha ^2)\right) \end{aligned}$$
(28)
$$\begin{aligned} = -\textbf{i}\exp \left[ -\textbf{i}c\,A_s^2 \right] \exp (-\textbf{i}\pi /4) \,\sqrt{\frac{\pi }{c}} =(-1)^{5/4}\sqrt{\frac{\pi }{c}}\exp \left[ \textbf{i}\frac{s^2}{4 c}\right] , \end{aligned}$$
(29)

where the last expression coincides with \(F_0\) obtained in (39) by means of the complex error function. The procedure above was inspired by Sec.3 of the monograph [24].

The method of stationary phase is exact in the given case. Apparently, the method picks out here the relevant interval of the integrand around the stationary point whereby the contribution of the remaining interval oscillates to zero.

4 Generalization of the Method of Stationary Phase

A theorem will be proved for the following integrand \(F_1\) which is generalized by an analytic amplitude function f(A) as

$$\begin{aligned} F_1 = \lim _{\eta \rightarrow 0}\int _{A_1}^{A_2}\textrm{d}A\,f(A)\,\exp \left( \textbf{i}c A^2-s\,A\right) , \end{aligned}$$
(30)
$$\begin{aligned} f(A) = \sum _{n=0,1,...} (A_s-\textbf{i}\alpha )^n f_n, \,\,\, \alpha \in \textbf{R}, \quad (\textbf{i}A_s)= s/(2 c)>0. \end{aligned}$$
(31)

Theorem

With the definitions (30) and (31), the application of the method of stationary phase is exact as follows:

$$\begin{aligned} F_1=(-\textbf{i}) \exp \left( -\textbf{i}c \,A_s^2 \right) \int _{-\infty }^{\infty }\textrm{d}\alpha \,f(A_s-\textbf{i}\alpha )\, \exp \left( -\textbf{i}c\, \alpha ^2)\right) . \end{aligned}$$

The proof is given in Appendix 2.

Fig. 1
figure 1

The integration path \(\kappa (E)\) is exemplified by a parametric plot in the interval \(E_1 \equiv -R \le E \le R\equiv E_2\). The magnitudes are non-dimensionlized. The singularity at \(\kappa (E=\pm \infty )=0\) is circumvented by the quarter arc of a circle of radius \(\rho\) which eventually shrinks to zero. For demonstration, the imaginary part \(\epsilon >0\) of the energy E has some finite value

Full size image

5 Summary

The route was outlined to determine the propagator in position representation by means of energy Fourier transformation (FT) from the exact Green’s function of the Coulomb-Kepler problem as derived by Hostler [1, 2]. The FT was defined as a principal value and in addition deformed in the complex energy plane in order to deal with the singularity at the energy \(E=\infty\). The method was tested for the case of the potential-free case whereby the standard textbook result was recovered (i) by means of the analytic performance of the FT, and (ii) by means of the method of stationary phase. In the latter case, a theorem was proved which gives exact results for a certain class of integrands which are relevant to the problem with non-zero potential. The latter problem is still in progress.