Klein–Gordon equations with homogeneous time-dependent electric fields

Abstract

We consider a system associated to Klein–Gordon equations with homogeneous time-dependent electric fields. The upper and lower boundaries of a time-evolution propagator for this system were proven by Veselić (J Oper Theory 25:319–330, 1991) for electric fields that are independent of time. We extend this result to time-dependent electric fields.

Appendices

Appendix A: Klein–Gordon systems with electric fields

In this section, we construct the (Hamilton) system equation in (2). This construction is the same one in [15]. Denote

$$\begin{aligned} \varPsi _0 (t,x) = \left( \begin{array}{cccccccccc}\psi _0(t,x) \\ \psi _{0,1}(t,x)\end{array}\right) , \quad \psi _{0,1}(t,x) := ( i \partial _t +q_E) \psi _0 (t,x), \quad \varPsi _0 = \left( \begin{array}{cccccccccc}\psi _{0,0} \\ \psi _{0,1}\end{array}\right) , \end{aligned}$$

where $\psi _0 (t,x)$, $\psi _{0,0}$, and $\psi _{0,1}$ are the same as those defined in (2). Then $\varPsi _0 (t,x)$ satisfies the following equations:

$$\begin{aligned}&i\frac{\partial }{\partial t} \varPsi _0(t,x) = A_0(t) \varPsi _0(t,x), \quad A_0(t) = \left( \begin{array}{cccccccccc}-q_E &{} 1 \\ L (0,p) &{} -q_E\end{array}\right) , \quad \varPsi _0 (0,x) = \varPsi _0. \end{aligned}$$

(38)

Here, we set $\zeta _j (t, \xi )$ to be that defined in (14) (or (17) and (18)). Focusing on $\zeta ''_j (t,\xi ) = - L(t,\xi ) \zeta _j (t,\xi )$, $j \in \{0,1\}$, a propagator for $A_0 (t)$, $U_{A_0} (t)$ can be described by

$$\begin{aligned} U_{A_0} (t) = \left( e^{ib(t) \cdot x}\right) _M \left( \begin{array}{cccccccccc} \zeta _0 (t,p) &{} \zeta _1 (t,p) \\ i \zeta _0 '(t,p) &{} i \zeta _1 '(t,p)\end{array}\right) . \end{aligned}$$

(39)

Indeed,

$$\begin{aligned} i \frac{\partial }{\partial t} U_{A_0} (t)&= (e^{ib(t) \cdot x})_M (i \partial _t -qE(t) \cdot x)_M ({\mathscr {F}_1^{-1}})_M\left( \begin{array}{cccccccccc}\zeta _0(t,\xi ) &{} \zeta _1 (t,\xi ) \\ i \zeta _0'(t,\xi ) &{} i \zeta _1 '(t,\xi )\end{array}\right) ({\mathscr {F}}_1^{+1})_M \\&=(e^{ib(t) \cdot x})_M ({\mathscr {F}}_1^{-1})_M \Bigg \{\left( \begin{array}{cccccccccc}-qE (t)\cdot x &{} 0 \\ 0 &{} -qE(t) \cdot x\end{array}\right) \left( \begin{array}{cccccccccc}\zeta _0(t,\xi ) &{} \zeta _1 (t,\xi ) \\ i \zeta _0'(t,\xi ) &{} i \zeta _1 '(t,\xi )\end{array}\right) \\&\quad + \left( \begin{array}{cccccccccc}i\zeta _0'(t,\xi ) &{} i\zeta _1'(t,\xi ) \\ L(t,\xi ) \zeta _0(t,\xi ) &{} L(t,\xi ) \zeta _1 (t,\xi )\end{array}\right) \Bigg \}({\mathscr {F}}_1^{+1})_M \\&=(e^{ib(t) \cdot x})_M \left( \begin{array}{cccccccccc}-q_E &{} 1 \\ L(t,p) &{} -q_E\end{array}\right) \left( \begin{array}{cccccccccc}\zeta _0(t,p) &{} \zeta _1 (t,p) \\ i \zeta _0'(t,p) &{} i \zeta _1'(t,p)\end{array}\right) = A_0(t) U_{A_0}(t), \end{aligned}$$

where $e^{ib(t) \cdot x} L(t,p) e^{-ib(t) \cdot x} = L(0,p)$ and $(i \partial _t) e^{ib(t) \cdot x} = e^{ib(t) \cdot x} (i \partial _t -b'(t) \cdot x)$.

Next, we define

$$\begin{aligned} {\mathscr {F}} = ({\mathscr {F}}_{1}^{+1})_M , \quad {\mathscr {F}}^{-1} = ({\mathscr {F}}_1^{-1})_M, \end{aligned}$$

(40)

and set

$$\begin{aligned} K_{\alpha } (0,p)&= \left( \begin{array}{cccccccccc}(L (0,p) )^{1/2- \alpha }&{} 0 \\ 0 &{} (L(0,p))^{-1/2 - \alpha }\end{array}\right) , \nonumber \\ K_{\alpha } (0.p)\varPhi&= {\mathscr {F}}^{-1} K_{\alpha }(0 ,\xi ){\mathscr {F}}\varPhi , \end{aligned}$$

(41)

$$\begin{aligned} {\mathscr {K}}_{\alpha }&= L^{1/4-\alpha /2} L^2(\mathbf{R}^n) \times L^{-1/4-\alpha /2}L^2(\mathbf{R}^n) , \end{aligned}$$

(42)

for $\alpha \in \mathbf{R}$ and $\varPhi \in {\mathscr {D}}( K_{\alpha } )$, where $L^{j} L^2(\mathbf{R}^n)$, $j\in \mathbf{R}$ is defined as the norm space with respect to the norm

$$\begin{aligned} \left\| u \right\| _{L^{j}L^2(\mathbf{R}^n)} := \left\| (L(0, \xi ))^j \hat{u} (\xi ) \right\| _{L^2(\mathbf{R}^n_{\xi })}, \quad u\in {\mathscr {F}}_1^{-1} {\mathscr {D}}( (L(0, \xi ))^j ). \end{aligned}$$

Furthermore, we define

$$\begin{aligned} \left( u , v \right) _{{{\mathscr {K}}_{\alpha }}}&:= \left( {K}_{\alpha }(0,\xi ) {\mathscr {F}} u , {\mathscr {F}} v \right) _{{\mathscr {H}}}, \quad {\mathscr {F}}u, \ {\mathscr {F}}v \in {\mathscr {D}}( K_{\alpha } (0, \xi ) ), \nonumber \\ K_{\alpha }^{1/2}&:= (K_{\alpha } (0,p))^{1/2} = \left( \begin{array}{cccccccccc}(L(0,p))^{1/4 - \alpha /2} &{} 0 \\ 0 &{} (L(0,p))^{-1/4 - \alpha /2}\end{array}\right) , \nonumber \\ \varPhi _{0, \alpha } (t,x)&= K_{\alpha }^{1/2} \varPsi _{0, \alpha } (t,x), \quad \varPhi _{0, \alpha } (0,x) = \varPhi _{0,\alpha } = K^{1/2}_{\alpha } \varPsi _{0}. \end{aligned}$$

(43)

It can be shown that for $u = (u_1,u_2)^{\mathrm {T}}$,

$$\begin{aligned} \left( u ,u \right) _{{\mathscr {K}}_{\alpha }}&= \left( (L(0, \xi ))^{1/2 - \alpha } \hat{u}_1, \hat{u}_1 \right) _{L^2(\mathbf{R}^n)} + \left( (L(0, \xi ))^{-1/2 - \alpha } \hat{u}_2 , \hat{u} _2 \right) _{L^2(\mathbf{R}^n)} \\&=\left\| (L(0,\xi ))^{1/4 - \alpha /2} \hat{u}_1 \right\| _{L^2(\mathbf{R}^n)}^2 + \left\| (L(0, \xi ))^{-1/4-\alpha /2} \hat{u}_2 \right\| _{L^2(\mathbf{R}^n)}^2 = \left\| u \right\| _{{\mathscr {K}}_{\alpha }}^2. \end{aligned}$$

Thus, $(\cdot , \cdot )_{{\mathscr {K}}_{\alpha }}$ is the inner product of ${\mathscr {K}}_{\alpha }$. Moreover, notice that for $\varPsi _{0} \in {\mathscr {K}}_{\alpha }$, $\left\| \varPhi _{0,\alpha }\right\| _{{\mathscr {H}}} ^2 = (K _{\alpha } \varPsi _{0}, \varPsi _{0})_{{\mathscr {H}}} = \left\| \varPsi _{0}\right\| _{{\mathscr {K}}_{\alpha }}$, i.e., $\varPhi _{0,\alpha } \in {\mathscr {H}}$. We then define the system

$$\begin{aligned} i \frac{\partial }{ \partial t} \varPhi _{0, \alpha } (t,x)&= H_{0, \alpha } (t) \varPhi _{0, \alpha } (t,x), \nonumber \\ \varPhi _{0, \alpha }(0,x)&= \varPhi _{0,\alpha } , \nonumber \\ H_{0, \alpha } (t)&= K^{1/2}_{\alpha } A_0 (t) (K^{1/2}_{\alpha })^{-1}. \end{aligned}$$

(44)

on the Hilbert space ${\mathscr {H}}$. In the same way, $U_{0, \alpha }(t)$, the propagator for $H_{0,\alpha } (t)$, can be written as

$$\begin{aligned} U_{0,\alpha }(t)&= K^{1/2}_{\alpha } U_{A_0} (t) (K^{1/2}_{\alpha })^{-1}, \nonumber \\ U_{0, \alpha }(t) ^{-1}&= K_{\alpha }^{1/2} U_{A_0} (t) ^{-1} (K_{\alpha }^{1/2})^{-1}, \end{aligned}$$

(45)

and we obtain the system

$$\begin{aligned} i \frac{d}{dt} U_{0, \alpha } (t) \varPhi _{0, \alpha } = H_{0, \alpha }(t) U_{0, \alpha } (t) \varPhi _{0, \alpha } , \quad \varPhi _{0, \alpha } \in {\mathscr {H}} \end{aligned}$$

(46)

with Hilbert space ${\mathscr {H}}$ and complex valued energy $H_{0, \alpha } (t)$. Straightforward calculations show that $H_{0, \alpha }(t)$ can be written as

$$\begin{aligned} \left( \begin{array}{cccccccccc}(L(0,p))^{1/4- \alpha /2}(-q_E) (L(0,p))^{-1/4 + \alpha /2} &{} (L(0,p))^{1/2} \\ (L(0,p))^{1/2} &{} (L(0,p))^{-1/4 - \alpha /2}(-q_E)(L(0,p))^{1/4 + \alpha /2} \end{array}\right) . \end{aligned}$$

Noting that for an invertible smooth function F and its inverse $F^{-1 }$,

$$\begin{aligned} F(p)^{-1} x F(p)&= {\mathscr {F}}_1^{-1} F(\xi )^{-1} {\mathscr {F}}_1^{+1} x {\mathscr {F}}_1^{-1} F(\xi ) {\mathscr {F}}_1^{+1} , \quad ({\mathscr {F}}_{1}^{+1} x {\mathscr {F}}_1^{-1} = i \nabla _{\xi }), \\&= {\mathscr {F}}_1^{-1} F(\xi )^{-1} (i \nabla F)(\xi ) {\mathscr {F}}_1^{+1} + {\mathscr {F}}_1^{-1} F(\xi )^{-1} F(\xi ) {\mathscr {F}}_1^{+1} x {\mathscr {F}}_1^{-1} {\mathscr {F}}_1^{+1} \\&= i F(p)^{-1}(\nabla F)(p) + x \end{aligned}$$

holds. Hence, $(L(0,p))^{- \theta } qE \cdot x (L(0,p))^{\theta } = qE \cdot x + 2 i c^2 \theta qE \cdot p (L(0,p))^{-1}$, and $H_{0, \alpha } (t)$ can be decomposed into $\hat{H}_{0, \alpha } (t) = \hat{H}_0 (t) + P_{0, \alpha } (t)$ ; $\hat{H}_0 (t)$ and $P_{0, \alpha } (t)$ are the same as those defined in (5) and (6), respectively. Here, $\hat{H}_0 (t)$ is a symmetric operator (self-adjoint operator for every fixed t, see Lemma 2.1. of [15]), but $P_0(t)$ is a non-symmetric operator (clearly, it is a complex valued operator).

Appendix B: models of time-dependent electric fields

Here, we give examples of electric fields satisfying Assumption (E1). First, we assume that b(t) satisfies $b(t) = (0, 0, \dots , 0,b_j(t),0,\ldots ,0)$, $j \in \{1,2,\ldots ,n\}$, and $b_j(t)$ can be written as

$$\begin{aligned} b_j(t) = {\left\{ \begin{array}{ll} C_{\gamma } t^{\gamma } + \rho _{\gamma } (t) &{} 0< \gamma < 1, \\ C_1 t + \rho _1 (t) + \theta _1 (t) &{} \gamma = 1, \end{array}\right. } \end{aligned}$$

(47)

where $C_{\gamma } \ne 0$ is a constant, $\rho _{\gamma } \in C^2(\mathbf{R}^n)$ satisfies $| \rho ^{(l)}_{\gamma } (t) | = o(t^{\gamma - l})$ for $l \in \{0,1,2\}$, and $| \theta ^{(l)}_{1} (t) | \le C$ for $l\in \{0,1,2\}$. It can easily be shown that

$$\begin{aligned} \int _{|a + b(s)| \le 2E_{0,0}/(mc^2)} |b'(s)| ds&\le \int _{|a_j +b_j(s) | \le 2E_{0,0}/(mc^2)} |b_j'(s)| ds \nonumber \\&\le \left| \int _{|\tau | \le 2E_{0,0}/(mc^2)} \frac{|b_j'(s)|}{b_j'(s)} d \tau \right| \le C \end{aligned}$$

(48)

and

$$\begin{aligned}&\int _0^t \frac{|b'(s)|^2 + |b''(s)|}{Q(s, a) ^2} ds \\&\le C_{R} + \int _{R}^t \frac{ |b'_j(s)|^2+ |b_j''(s)| }{c^2(a_j + C_{\gamma } s^{\gamma } + \rho _{\gamma } (s) + \theta _{\gamma } (s) )^2 + (mc^2)^2} ds \\&\le C_{R} + C \sup _{s > R} \left| s^{1- \gamma } (|b'_j(s)|^2 + |b''_j(s) |) \right| \int _{-\infty }^{\infty } \frac{d \tau }{c^2 (\tau + \theta _{\gamma } (s))^2) + (mc^2)^2 } d \tau \end{aligned}$$

hold, where $\theta _{\gamma } (s) \equiv 0$ for $\gamma < 1$. By dividing the limits of integration into two regions, $|\tau | \le 2 |\theta _{\gamma } (s)| \le C$ and $|\tau | \ge 2 |\theta _{\gamma } (s)|$, notice that the last term of the above inequality is smaller than

$$\begin{aligned} C \sup _{s > R} \left| s^{1- \gamma } ( |b'_j(s)|^2 + |b''_j(s) | ) \right| \left( \int _{|\tau | \le C} d \tau + \int _{|\tau | \ge 2|\theta _{\gamma } (s)|} \frac{d \tau }{c^2 \tau ^2 /4 + (mc^2)^2} \right) \le C, \end{aligned}$$

where (47) is utilized. Next, assume $b(t) = (0,\ldots ,0,b_j(t),0\ldots ,0)$ and $b_j(t)$ can be written as

$$\begin{aligned} b_j(t) = e_3 (\log (1 + e_4 |t|)), \end{aligned}$$

where $e _3 \ne 0$ and $e_4 > 0$ are constants. By the same approach as (48),we obtain the left-hand side of (7) for this particular b(t). Moreover, by using the fact that $(b'_j (s)) ^2$ and $b''_j(s)$ are integrable on $[R, \infty )$, the right-hand side of (7) can also be obtained for this b(t).

Remark 1

Suppose b(t) satisfies $b(t) = (0,\ldots ,0,b_{j1}(t),0,\ldots ,0,b_{j2} (t),0,\dots ,0 )$ and $b_{j1} (t)$ and $b_{j2} (t)$ are written in the same form as (47) by replacing $\gamma \rightarrow \gamma _1$ and $\gamma \rightarrow \gamma _2$, respectively. Then it is sufficient to consider the same approach as above for the maximum of $\{ \gamma _1 , \gamma _2\}$; indeed, suppose $\gamma _1 \ge \gamma _2$. Noting that

$$\begin{aligned} \int _{| a + b(s)| \le 2E_{0,0}/(mc^2)} |b'(s)| ds \le C_R + C \int _{\mathop { |a_{j1} + b_{j1}(s)| \le 2E_{0,0}/(mc^2)}\limits _{s \ge R} |b_{j1}'(s)| ds} \end{aligned}$$

and

$$\begin{aligned} \int _R^t \frac{|b'(s)|^2 + |b''(s)|}{Q(s, a)^2} ds \le C \int _{R}^{t} \frac{|b'_{j1} (s)|^2 + |b''_{j1} (s)|}{c^2(a_{j1} + b_{j1} (s))^2 + (mc^2)^2} ds, \end{aligned}$$

it is straightforward to prove that (7) mimics the above approach. Similarly, we consider the case when $b(t) = (b_1 (t), \ldots , b_n(t))$. However, if AC electric fields are included in E(t), (7) is difficult to prove. For example, consider the case when $b_{j1}(t) = t^{ \gamma }$ and $b_{j2} (t) = t^{ \gamma /2} + \cos t$ with $0< \gamma <1$, i.e., $| b_{j1}(t) | \ge |b_{j2} (t)|$ holds for $t \gg 1$, but $|b^{(1)}_{j1} (t)| \ge |b^{(l)}_{j2} (t)|$, $l \in \{1,2\}$, is not always true. Clearly, $s^{1- \gamma } (|b''(s)| + |b'(s)|)$ is not bounded; hence, our proof fails. Other approaches must be established to consider more general electric fields including AC electric fields.