It is well-known that the Vlasov-Poisson system (1.5), (1.3) has many invariances, see [49, p. 427], for instance: if \(f=f(t, x, v)\) is a solution, so is
$$\begin{aligned} \tilde{f}(\tilde{t}, \tilde{x}, \tilde{v}) =\frac{\mu }{\lambda ^2}\,f\Big (\frac{\tilde{t}+t_0}{\mu \lambda }, \frac{\tilde{x}+x_0}{\lambda }, \mu \tilde{v}\Big ), \end{aligned}$$
(6.1)
where \(\mu , \lambda >0\), \(t_0\in \mathbb R\) and \(x_0\in \mathbb R^3\). The associated potential and density are
$$\begin{aligned} U_{\tilde{f}}(\tilde{t}, \tilde{x})=\frac{1}{\mu ^2} \,U_f\Big (\frac{\tilde{t}+t_0}{\mu \lambda }, \frac{\tilde{x}+x_0}{\lambda }\Big ),\quad \rho _{\tilde{f}}(\tilde{t}, \tilde{x})=\frac{1}{\mu ^2\lambda ^2} \,\rho _f\Big (\frac{\tilde{t}+t_0}{\mu \lambda }, \frac{\tilde{x}+x_0}{\lambda }\Big ). \end{aligned}$$
(6.2)
It can be expected that quantities that are invariant will play a particularly important role. It is the purpose of this section to determine several such quantities.
Next let \(Q=Q(e_Q)\) depend only upon \(e_Q(x, v)=\frac{1}{2}\,|v|^2+U_Q(x)\). Then,
$$\begin{aligned} e_Q(x, v)= & {} \frac{1}{2}\,|v|^2+U_Q(x)=\frac{1}{2}\,\mu ^2 |\tilde{v}|^2+U_Q(\lambda ^{-1}\tilde{x}) \nonumber \\= & {} \frac{1}{2}\,\mu ^2 |\tilde{v}|^2+\mu ^2 U_{\tilde{Q}}(\tilde{x}) =\mu ^2 e_{\tilde{Q}}(\tilde{x}, \tilde{v}), \end{aligned}$$
(6.4)
and (6.3) leads to
$$\begin{aligned} \tilde{Q}(e_{\tilde{Q}})=\frac{\mu }{\lambda ^2}\,Q(e_Q)=\frac{\mu }{\lambda ^2}\,Q(\mu ^2 e_{\tilde{Q}}). \end{aligned}$$
Thus, if \(Q=Q(e_Q)\) and \(\tilde{Q}=\tilde{Q}(e_{\tilde{Q}})\) are understood as functions of one variable, then
$$\begin{aligned} \tilde{Q}'(e_{\tilde{Q}})=\frac{\mu ^3}{\lambda ^2}\,Q'(e_Q). \end{aligned}$$
(6.5)
For radial potentials and densities, we have
$$\begin{aligned} U_{\tilde{Q}}(\tilde{r})=\frac{1}{\mu ^2}\,U_Q(r),\quad \rho _{\tilde{Q}}(\tilde{r})=\frac{1}{\lambda ^2\mu ^2}\,\rho _Q(r), \end{aligned}$$
(6.6)
which leads to
$$\begin{aligned} U'_{\tilde{Q}}(\tilde{r})=\frac{1}{\lambda \mu ^2}\,U'_Q(r). \end{aligned}$$
(6.7)
The central densities are related by
$$\begin{aligned} \rho _{\tilde{Q}}(0)=\frac{1}{\lambda ^2\mu ^2}\,\rho _Q(0). \end{aligned}$$
(6.8)
The effective potential from (7.4) is \(U_{\mathrm{eff}}(r, \ell )=U_Q(r)+\frac{\ell ^2}{2r^2}\), which we also write as \(U_{\mathrm{eff}}(r, \beta )=U_Q(r)+\frac{\beta }{2r^2}\) for \(\beta =\ell ^2\). Let
$$\begin{aligned} \tilde{\beta }=\frac{\lambda ^2}{\mu ^2}\,\beta . \end{aligned}$$
Then,
$$ \tilde{U}_{\mathrm{eff}}(\tilde{r}, \tilde{\beta }):=U_{\tilde{Q}}(\tilde{r})+\frac{\tilde{\beta }}{2\tilde{r}^2} =\frac{1}{\mu ^2}\,U_Q(r)+\frac{\lambda ^2}{\mu ^2}\,\beta \,\frac{1}{2\lambda ^2 r^2} =\frac{1}{\mu ^2}\,U_{\mathrm{eff}}(r, \beta ) $$
is the corresponding transformation rule. The points \(r_\pm =r_\pm (e, \beta )\) are determined by the relation \(U_{\mathrm{eff}}(r_\pm (e, \beta ), \beta )=e\). Owing to
$$\begin{aligned} \tilde{U}_{\mathrm{eff}}(\tilde{r}_\pm (\tilde{e}, \tilde{\beta }), \tilde{\beta })=\tilde{e} \quad \Longleftrightarrow \quad \frac{1}{\mu ^2}\,U_{\mathrm{eff}}(\lambda ^{-1}\tilde{r}_\pm (\tilde{e}, \tilde{\beta }), \beta )=\frac{1}{\mu ^2}\,e \end{aligned}$$
we obtain
$$\begin{aligned} \tilde{r}_\pm (\tilde{e}, \tilde{\beta })=\lambda \,r_\pm (e, \beta ). \end{aligned}$$
Next, \(r_0=r_0(\beta )\) is the point where \(U_{\mathrm{eff}}(\cdot , \beta )\) attains its minimum. Since \(\tilde{U}_{\mathrm{eff}}(\tilde{r}, \tilde{\beta })=\mu ^{-2} U_{\mathrm{eff}}(r, \beta ) =\mu ^{-2} U_{\mathrm{eff}}(\lambda ^{-1}\tilde{r}, \beta )\), we get
$$ \tilde{U}'_{\mathrm{eff}}(\tilde{r}, \tilde{\beta }) =\lambda ^{-1}\mu ^{-2}\,U'_{\mathrm{eff}}(\lambda ^{-1}\tilde{r}, \beta ), $$
and this implies that
$$\begin{aligned} \tilde{r}_0(\tilde{\beta })=\lambda \,r_0(\beta ). \end{aligned}$$
In terms of the variables e and \(\beta \), the period function from (A.20) is
$$ T_1(e, \beta )=2\int _{r_-(e, \beta )}^{r_+(e, \beta )} \frac{dr}{\sqrt{2(e-U_{\mathrm{eff}}(r, \beta ))}}. $$
Using the transformation \(\tilde{r}=\lambda r\), \(d\tilde{r}=\lambda dr\), it follows that
$$\begin{aligned} \tilde{T}_1(\tilde{e}, \tilde{\beta })= & {} 2\int _{\tilde{r}_-(\tilde{e}, \tilde{\beta })}^{\tilde{r}_+(\tilde{e}, \tilde{\beta })} \frac{d\tilde{r}}{\sqrt{2(\tilde{e}-\tilde{U}_{\mathrm{eff}}(\tilde{r}, \tilde{\beta }))}} \\= & {} 2\lambda \int _{\lambda ^{-1}\tilde{r}_-(\tilde{e}, \tilde{\beta })}^{\lambda ^{-1} \tilde{r}_+(\tilde{e}, \tilde{\beta })} \frac{dr}{\sqrt{2(\tilde{e}-\tilde{U}_{\mathrm{eff}}(\lambda r, \tilde{\beta }))}} \\= & {} 2\lambda \int _{r_-(e, \beta )}^{r_+(e, \beta )} \frac{dr}{\sqrt{2(\mu ^{-2}e-\mu ^{-2} U_{\mathrm{eff}}(r, \beta ))}} \\= & {} \lambda \mu \,T_1(e, \beta ). \end{aligned}$$
In particular, \(\tilde{\omega }_1(\tilde{e}, \tilde{\beta })=\frac{1}{\lambda \mu }\,\omega _1(e, \beta )\) for \(\tilde{\omega }_1=\frac{2\pi }{\tilde{T}_1}\), and if we denote \(\delta _1=\inf \omega _1\), then also
$$\begin{aligned} \tilde{\delta }_1=\frac{1}{\lambda \mu }\,\delta _1. \end{aligned}$$
(6.9)
Next we consider the space \(L^2_{\mathrm{sph},\,\frac{1}{|Q'|}}(K)=X^0\) of spherically symmetric functions with the Q-dependent inner product
$$\begin{aligned} {(u_1, u_2)}_Q=\iint \limits _K\frac{1}{|Q'(e_Q)|}\,\overline{u_1(x, v)}\,u_2(x, v)\,dx\,dv, \end{aligned}$$
as in Remark B.2. Defining
$$\begin{aligned} \tilde{u}(\tilde{x}, \tilde{v}) =\frac{\mu }{\lambda ^2}\,u\Big (\frac{\tilde{x}}{\lambda }, \mu \tilde{v}\Big ) \end{aligned}$$
(6.10)
in accordance with (6.3), we calculate, using \(dx=\lambda ^{-3}d\tilde{x}\) and \(dv=\mu ^3 d\tilde{v}\) as well as (6.5):
$$\begin{aligned} {\Vert \tilde{u}\Vert }_{\tilde{Q}}^2= & {} \int \int \frac{d\tilde{x}\,d\tilde{v}}{|\tilde{Q}'(e_{\tilde{Q}})|}\,|\tilde{u}(\tilde{x}, \tilde{v})|^2 \nonumber \\= & {} \frac{\lambda ^2}{\mu ^3}\frac{\mu ^2}{\lambda ^4} \int \int \frac{\lambda ^3 dx\,\mu ^{-3} dv}{|Q'(e_Q)|}\,|u(x, v)|^2 =\frac{\lambda }{\mu ^4}\,{\Vert u\Vert }_Q^2. \end{aligned}$$
(6.11)
Let the operator \((\mathcal{T}g)(x, v)=v\cdot \nabla _x g(x, v)-\nabla _v g(x, v)\cdot \nabla _x U_Q(x)\) be as in (1.11). From the above relations, it follows that
$$\begin{aligned} (\mathcal{T}\tilde{u})(\tilde{x}, \tilde{v})= & {} \tilde{v}\cdot \nabla _{\tilde{x}}\tilde{u}-\nabla _{\tilde{v}}\tilde{u}\cdot \nabla _{\tilde{x}} U_{\tilde{Q}} \nonumber \\= & {} \mu ^{-1}\mu \lambda ^{-2}\lambda ^{-1}\,v\cdot \nabla _x u -\mu \lambda ^{-2}\mu \mu ^{-2}\lambda ^{-1}\,\nabla _v u\cdot \nabla _x U_Q \nonumber \\= & {} \lambda ^{-3}(\mathcal{T}u)(x, v). \end{aligned}$$
(6.12)
Alternatively, \(\mathcal{T}u=\{u, e_Q\}\) can be used. From (6.4) and (6.10), we get
$$\begin{aligned} (\mathcal{T}\tilde{u})(\tilde{x}, \tilde{v})= & {} \{\tilde{u}, e_{\tilde{Q}}\} =\nabla _{\tilde{x}}\,\tilde{u}\cdot \nabla _{\tilde{v}}\,e_{\tilde{Q}} -\nabla _{\tilde{x}}\,e_{\tilde{Q}}\cdot \nabla _{\tilde{v}}\,\tilde{u} \\= & {} \mu \lambda ^{-3}\nabla _x u\cdot \mu ^{-2}\mu \nabla _v e_Q -\mu ^{-2}\lambda ^{-1}\nabla _x e_Q\cdot \mu ^2\lambda ^{-2}\nabla _v u \\= & {} \lambda ^{-3}\,\{u, e_Q\}=\lambda ^{-3}(\mathcal{T}u)(x, v). \end{aligned}$$
This in turn leads to
$$\begin{aligned} (\mathcal{T}^2\tilde{u})(\tilde{x}, \tilde{v})= & {} \{\mathcal{T}\tilde{u}, e_{\tilde{Q}}\} =\nabla _{\tilde{x}}\,(\mathcal{T}\tilde{u})\cdot \nabla _{\tilde{v}}\,e_{\tilde{Q}} -\nabla _{\tilde{x}}\,e_{\tilde{Q}}\cdot \nabla _{\tilde{v}}\,(\mathcal{T}\tilde{u}) \nonumber \\= & {} \lambda ^{-4}\nabla _x (\mathcal{T}u)\cdot \mu ^{-2}\mu \nabla _v e_Q -\mu ^{-2}\lambda ^{-1}\nabla _x e_Q\cdot \lambda ^{-3}\mu \nabla _v (\mathcal{T}u) \nonumber \\= & {} \lambda ^{-4}\mu ^{-1} \{\mathcal{T}u, e_Q\}=\lambda ^{-4}\mu ^{-1} (\mathcal{T}^2u)(x, v). \end{aligned}$$
(6.13)
Alternatively, if we put \(\hat{u}(\tilde{x}, \tilde{v})=u(\frac{\tilde{x}}{\lambda }, \mu \tilde{v})\), then \(\tilde{u}=\mu \lambda ^{-2}\hat{u}\), so (6.13) may be re-expressed as
$$\begin{aligned} (\mathcal{T}^2\hat{u})(\tilde{x}, \tilde{v})=\lambda ^{-2}\mu ^{-2} (\mathcal{T}^2u)(x, v). \end{aligned}$$
(6.14)
For the density induced by \(\mathcal{T}\tilde{u}\), (6.12) yields
$$ \rho _{\mathcal{T}\tilde{u}}(\tilde{x}) =\int (\mathcal{T}\tilde{u})(\tilde{x}, \tilde{v})\,d\tilde{v} =\lambda ^{-3}\mu ^{-3}\int (\mathcal{T}u)(x, v)\,dv=\lambda ^{-3}\mu ^{-3}\rho _{\mathcal{T}u}(x), $$
so that
$$\begin{aligned} U_{\mathcal{T}\tilde{u}}(\tilde{x})=\lambda ^{-1}\mu ^{-3}\,U_{\mathcal{T}u}(x) \end{aligned}$$
for the potential. In particular,
$$ \nabla _{\tilde{x}} U_{\mathcal{T}\tilde{u}}(\tilde{x}) =\lambda ^{-2}\mu ^{-3}\,\nabla _x U_{\mathcal{T}u}(x), $$
and hence
$$\begin{aligned} \int |\nabla _{\tilde{x}} U_{\mathcal{T}\tilde{u}}(\tilde{x})|^2\,d\tilde{x}= & {} \lambda ^{-4}\mu ^{-6}\lambda ^3\int |\nabla _x U_{\mathcal{T}u}(x)|^2\,dx \nonumber \\= & {} \lambda ^{-1}\mu ^{-6}\int |\nabla _x U_{\mathcal{T}u}(x)|^2\,dx. \end{aligned}$$
(6.15)
For
$$ {(Lu, u)}_Q=\int \int \frac{dx\,dv}{|Q'(e_Q)|}\,|\mathcal{T}u|^2 -\frac{1}{4\pi }\int _{\mathbb R^3} |\nabla _x U_{\mathcal{T}u}|^2\,dx $$
as given by (1.18), we then obtain from (6.5), (6.12) and (6.15):
$$\begin{aligned} {(L\tilde{u}, \tilde{u})}_{\tilde{Q}}= & {} \int \int \frac{d\tilde{x}\,d\tilde{v}}{|\tilde{Q}'(e_{\tilde{Q}})|}\,(\mathcal{T}\tilde{u})^2 -\frac{1}{4\pi }\int |\nabla _{\tilde{x}} U_{\mathcal{T}\tilde{u}}|^2\,d\tilde{x} \nonumber \\= & {} \lambda ^3\mu ^{-3}\lambda ^2\mu ^{-3}\lambda ^{-6} \int \int \frac{dx\,dv}{|Q'(e_Q)|}\,(\mathcal{T}u)^2 -\frac{1}{4\pi }\,\lambda ^{-1}\mu ^{-6}\int |\nabla _x U_{\mathcal{T}u}|^2\,dx \nonumber \\= & {} \lambda ^{-1}\mu ^{-6}\,{(Lu, u)}_Q. \end{aligned}$$
(6.16)
In (1.20), the quantity
$$\begin{aligned} \lambda _*=\inf \,\{{(Lu, u)}_Q: u\in X^2_{\mathrm{odd}}, {\Vert u\Vert }_Q=1\} \end{aligned}$$
is introduced. Therefore, owing to (6.16) and (6.11),
$$\begin{aligned} \tilde{\lambda }_*= & {} \inf \,\{{(L\tilde{u}, \tilde{u})}_{\tilde{Q}}: \tilde{u}\in X^2_{\mathrm{odd}}, {\Vert \tilde{u}\Vert }_{\tilde{Q}}=1\} \nonumber \\= & {} \lambda ^{-1}\mu ^{-6}\inf \,\{{(Lu, u)}_Q: u\in X^2_{\mathrm{odd}}, \lambda \mu ^{-4}\,{\Vert u\Vert }_Q=1\} \nonumber \\= & {} \lambda ^{-1}\mu ^{-6}\lambda ^{-1}\mu ^4 \inf \,\{{(L\hat{u}, \hat{u})}_Q: \hat{u}\in X^2_{\mathrm{odd}}, {\Vert \hat{u}\Vert }_Q=1\} \nonumber \\= & {} \lambda ^{-2}\mu ^{-2}\lambda _*, \end{aligned}$$
(6.17)
by setting \(u=\lambda ^{-1/2}\mu ^2\,\hat{u}\); it maybe checked that \(u\in X^2_{\mathrm{odd}}\) if and only if \(\tilde{u}\in X^2_{\mathrm{odd}}\) w.r. to the transformed variables.
Similarly, denoting \(B(r)=4\pi \rho _Q(r)+A(r)\) as in Lemma A.7(d), owing to (6.18) and (6.6) one gets
$$\begin{aligned} \tilde{B}(\tilde{r})=\lambda ^{-2}\mu ^{-2} B(r). \end{aligned}$$
Now we turn to the operators \(Q_\nu \) from Chap. 4 and their first eigenvalues \(\mu _1(\nu )\) for \(\nu \in ]-\infty , \delta _1^2[\); note the change in notation here for the parameter of the operators, since the letter \(\lambda \) is already occupied from \(\tilde{x}=\lambda x\), \(\tilde{r}=\lambda r\). Let
$$\begin{aligned} \tilde{\nu }=\frac{1}{\lambda ^2\mu ^2}\,\nu . \end{aligned}$$
If \(\nu \in ]-\infty , \delta _1^2[\), then \(\tilde{\nu }\in ]-\infty , \tilde{\delta }_1^2[\) due to (6.9). For \(\Psi =\Psi (r)\) let \(\tilde{\Psi }(\tilde{r})=\Psi (\frac{\tilde{r}}{\lambda })\). Since \(\tilde{p}_r=\frac{\tilde{x}\cdot \tilde{v}}{|\tilde{x}|}=\mu ^{-1}\,\frac{x\cdot v}{|x|}=\mu ^{-1} p_r\), we obtain from (6.5):
$$\begin{aligned} \tilde{\psi }(\tilde{r}, \tilde{p}_r, \tilde{\ell })= & {} |\tilde{Q}'(e_{\tilde{Q}})|\,\tilde{p}_r\,\tilde{\Psi }(\tilde{r}) =\mu ^3\lambda ^{-2}|Q'(e_Q)|\,\mu ^{-1}p_r\,\Psi (r) \nonumber \\= & {} \mu ^2\lambda ^{-2}\,|Q'(e_Q)|\,p_r\,\Psi (r) =\mu ^2\lambda ^{-2}\,\psi (r, p_r, \ell ). \end{aligned}$$
(6.19)
First we determine the scaling of \((-\mathcal{T}^2-z)^{-1}\psi \). Defining
$$\begin{aligned} \tilde{z}=\frac{1}{\lambda ^2\mu ^2}\,z, \end{aligned}$$
we assert that
$$\begin{aligned} ((-\mathcal{T}^2-\tilde{z})^{-1}\tilde{\psi })(\tilde{x}, \tilde{v}) =\mu ^4\,((-\mathcal{T}^2-z)^{-1}\psi )(x, v). \end{aligned}$$
(6.20)
To see this, let \(\tilde{g}=(-\mathcal{T}^2-\tilde{z})^{-1}\tilde{\psi }\) and \(g=(-\mathcal{T}^2-z)^{-1}\psi \). Then, (6.20) is equivalent to \(\tilde{g}=\mu ^4 g\), but \(\tilde{g}\) and g are not necessarily related by (6.10); in fact \(\tilde{g}=\mu ^4\hat{g}\) or \((-\mathcal{T}^2-\tilde{z})\tilde{g}=\mu ^4 (-\mathcal{T}^2-\tilde{z})\hat{g}\) is to be shown. For, owing to (6.14) and (6.19) we have
$$\begin{aligned} \mu ^4 (-\mathcal{T}^2-\tilde{z})\hat{g}= & {} \mu ^4 (-\lambda ^{-2}\mu ^{-2} \mathcal{T}^2 g -\lambda ^{-2}\mu ^{-2}zg)=\mu ^2\lambda ^{-2} (-\mathcal{T}^2-z)g =\mu ^2\lambda ^{-2}\,\psi \\= & {} \tilde{\psi }=(-\mathcal{T}^2-\tilde{z})\tilde{g}, \end{aligned}$$
which completes the proof of (6.20). From (4.22) together with (6.20), we obtain
$$\begin{aligned} (\widetilde{\mathcal{Q}}_{\tilde{z}}\tilde{\Psi })(\tilde{r})= & {} 4\pi \int \tilde{p}_r\,((-\mathcal{T}^2-\tilde{z})^{-1}\tilde{\psi })(\tilde{x}, \tilde{v})\,d\tilde{v} \\= & {} 4\pi \mu ^4\int \frac{\tilde{x}\cdot \tilde{v}}{|\tilde{x}|} \,((-\mathcal{T}^2-z)^{-1}\psi )(\lambda ^{-1}\tilde{x}, \mu \tilde{v})\,d\tilde{v} \\= & {} 4\pi \int \frac{\lambda ^{-1}\tilde{x}\cdot v}{|\lambda ^{-1}\tilde{x}|} \,((-\mathcal{T}^2-z)^{-1}\psi )(\lambda ^{-1}\tilde{x}, v)\,dv \\= & {} (\mathcal{Q}_z\Psi )(r). \end{aligned}$$
Thus, if we define
$$ \tilde{\mu }_1(\tilde{\nu })=\mu _1(\lambda ^2\mu ^2\tilde{\nu }), \quad \tilde{\nu }\in ]-\infty , \tilde{\delta }_1^2[, $$
then \(\tilde{\mu }_1(\tilde{\nu })\) is the first eigenvalue of \(\widetilde{\mathcal{Q}}_{\tilde{\nu }}\), and \(\tilde{\Psi }=\tilde{\Psi }(\tilde{r})\) is an associated eigenfunction if and only if \(\Psi =\Psi (r)\) is an eigenfunction of \(Q_\nu \) for the eigenvalue \(\mu _1(\nu )\). Due to (4.33) it follows that
$$ \tilde{\mu }_*=\lim _{\tilde{\nu }\rightarrow \tilde{\delta }_1^2-}\tilde{\mu }_1(\tilde{\nu }) =\lim _{\nu \rightarrow \delta _1^2-}\mu _1(\nu )=\mu _*. $$
As already noted at the beginning of this chapter, it can be expected that quantities that are unaffected by the scaling do have a special relevance. Hence, \(\mu _*\) is one such quantity. In addition, the condition \(\lambda _*<\delta _1^2\) is invariant, as a consequence of (6.17) and (6.9). Further, we would like to mention
$$\begin{aligned} \frac{2\pi }{\sqrt{\lambda _*}}\,\sqrt{\rho _Q(0)}, \end{aligned}$$
cf. [59, Remark, p. 555], for which we deduce from (6.17) and (6.8):
$$ \frac{2\pi }{\sqrt{\tilde{\lambda }_*}}\,\sqrt{\rho _{\tilde{Q}}(0)} =\frac{2\pi }{\lambda ^{-1}\mu ^{-1}\sqrt{\lambda _*}}\,\lambda ^{-1}\mu ^{-1}\sqrt{\rho _Q(0)} =\frac{2\pi }{\sqrt{\lambda _*}}\,\sqrt{\rho _Q(0)}. $$
This is called the Eddington-Ritter relation; also see [17, (27), p. 15] and [70, Section 4]. The relevance of the number \(\frac{2\pi }{\sqrt{\lambda _*}}\) is that it is the ‘linear period’ of the system, in the sense that the linearized system about Q has a periodic solution of this period (if \(\lambda _*\) is an eigenvalue of L); recall Lemma 1.3.
