general depends on T such that for $\|U\|_{(L^{p}(\Gamma))^{2}}\leq R$,
$\|V\|_{(L^{p}(\Gamma))^{2}}\leq R$, one has

$\displaystyle\hskip 42.67912pt\|e^{(\Delta-\sigma
I)t}\partial_{x}{\mathcal{F}}(U)\|_{L^{p}(\Gamma)}\leq Ce^{-\delta
t}t^{-\frac{p+\gamma-1}{2p}}\left(1+\|U\|_{(L^{p}(\Gamma))^{2}}\right),t\in(0,T],$
(B.20) $\displaystyle\|e^{(\Delta-\sigma
I)t}\partial_{x}{\mathcal{F}}(U)-e^{(\Delta-\sigma
I)t}\partial_{x}{\mathcal{F}}(V)\|_{L^{p}(\Gamma)}\leq Ce^{-\delta
t}t^{-\frac{p+\gamma-1}{2p}}\|U-V\|_{(L^{p}(\Gamma))^{2}},t\in(0,T].$ (B.21)

Furthermore, if $\sigma>\sigma_{0}$, where $\sigma_{0}>0$ is as in Corollary
2.1, then there exists a $T$-independent constant $C>0$ and
$0<\delta\ll\sigma$ such that (B.16)–(B.21) hold with $t\in(0,T]$ replaced by
$t\in(0,\infty)$.

###### Proof.

(1) The inequalities (B.16), (B.17) follow from Theorem 1.3 combined with
(B.3), (B.4) correspondingly.

(2) The inequality (B.18) follows (B.3) and (A.10). To prove (B.19) we write

$\displaystyle\|(e^{(\Delta-\sigma I)h}-I)e^{(\Delta-\sigma
I)t}{\mathcal{F}}(U)\|_{{\mathcal{X}}^{\beta}_{p}}$ $\displaystyle\leq
Ch^{\beta}e^{-\delta t}\|e^{(\Delta-\sigma
I)t}{\mathcal{F}}(U)\|_{{\mathcal{X}}^{2\beta}_{p}}$ $\displaystyle\leq
Ch^{\beta}e^{-\delta
t}t^{-2\beta-\frac{\gamma-1}{2p}}\left(1+\|U\|_{(L^{p}(\Gamma))^{2}}\right).$
(B.22)

(3) The inequalities (B.20), (B.21) follow from (1.23) combined with (B.3),
(B.4) correspondingly.

The last statement follows from Corollary 2.1. ∎

######  Proposition B.4.

Let $\sigma>0$, $p\geq 1$ then there exists $C>0$ such that

$\displaystyle\|(\Delta-\sigma)^{-1}u\|_{L^{\infty}(\Gamma)}\leq
C\|u\|_{L^{p}(\Gamma)},$ (B.23)
$\displaystyle\|\partial_{x}(\Delta-\sigma)^{-1}u\|_{L^{\infty}(\Gamma)}\leq
C\|u\|_{L^{p}(\Gamma)},$ (B.24)
$\displaystyle\|(\Delta-\sigma)^{-1}u\|_{\widehat{W}^{2,p}(\Gamma)}\leq
C\|u\|_{L^{p}(\Gamma)},$ (B.25)

for all $t>0$.

###### Proof.

Let us recall, e.g. from [10, Corollary 7 in Section 4.2], the following
inequality

$\|f\|_{L^{\infty}(\Gamma)}\leq C\left(\|f\|_{L^{p}(\Gamma)}+\|\Delta
f\|_{L^{p}(\Gamma)}\right),f\in\widehat{W}^{2,p}(\Gamma),C>0.$ (B.26)

To prove (B.23) we denote $f:=(\Delta-\sigma)^{-1}u$, that is, $\Delta
f-\sigma f=u$, $u\in L^{p}(\Gamma)$, and employ (B.26) to obtain

$\displaystyle\|f\|_{L^{\infty}(\Gamma)}$ $\displaystyle\leq
C\left(\|f\|_{L^{p}(\Gamma)}+\|\Delta
f\|_{L^{p}(\Gamma)}\right)=C\left((1+|\sigma|)\|f\|_{L^{p}(\Gamma)}+\|u\|_{L^{p}(\Gamma)}\right)$
$\displaystyle\leq
C\left((1+|\sigma|)\|(\Delta-\sigma)^{-1}\|_{{\mathcal{B}}(L^{p}(\Gamma))}\|u\|_{L^{p}(\Gamma)}+\|u\|_{L^{p}(\Gamma)}\right),$
(B.27)

where we implicitly used $p-$independence of the spectrum and resolvent of
${\mathcal{A}}_{p}$ obtained in [3, Theorem 3, Section 4.2]. The third
inequality follows from (B.24) and

$\|\Delta(\Delta-\sigma)^{-1}u\|_{L^{p}(\Gamma)}\leq C\|u\|_{L^{p}(\Gamma)}.$
(B.28)

∎

######  Corollary B.1.

Assume Hypothesis 1.1 and suppose that $\tau=0$. Define

${\mathcal{F}}_{3}(u):=(\Delta-\sigma)^{-1}f(u)\text{ for }u\in
L^{p}(\Gamma).$ (B.29)

Then ${\mathcal{F}}_{3}$ is a bounded mapping between $L^{p}(\Gamma)$ and
$\widehat{W}^{1,p}(\Gamma)$, that is, there exists a constant $C>0$ such that
if $\|u\|_{L^{p}(\Gamma)}\leq R$ then one has

$\|{\mathcal{F}}_{3}(u)\|_{\widehat{W}^{1,p}(\Gamma)}\leq
C(1+\|u\|_{L^{p}(\Gamma)}).$ (B.30)

Moreover, ${\mathcal{F}}_{3}$ is locally Lipschitz continuous, that is, for
$\|u\|_{L^{p}(\Gamma)}\leq R$, $\|v\|_{L^{p}(\Gamma)}\leq R$ one has

$\|{\mathcal{F}}_{3}(u)-{\mathcal{F}}_{3}(v)\|_{\widehat{W}^{1,p}(\Gamma)}\leq
C\|u-v\|_{L^{p}(\Gamma)}.$ (B.31)

###### Proof.

By Proposition B.1 with $F(u,v)=f(u)$, the mapping $u\mapsto f(u)$ from
$L^{p}(\Gamma)$ to $L^{\frac{p}{\gamma}}(\Gamma)$ is bounded and locally
Lipschitz. Next, by Proposition B.4, $(\Delta-\sigma)^{-1}$ maps
$L^{\frac{p}{\gamma}}(\Gamma)$ into $\widehat{W}^{1,\infty}(\Gamma)$ which is
boundedly embedded into $\widehat{W}^{1,p}(\Gamma)$. The assertions follow
from the fact that ${\mathcal{F}}_{3}$ is a composition of these mappings. ∎

######  Proposition B.5.

Assume Hypothesis B.1 and let $T\in(0,\infty)$, $\sigma>0$,
$\delta\in(0,\sigma)$. For $U=(u_{1},u_{2})\in
L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)$ let

${\mathcal{H}}(U)(x):=F\left(u_{1}(x),u_{2}(x)\right)\partial_{x}u_{2}(x),x\in\Gamma.$
(B.32)

Then ${\mathcal{H}}$ maps $L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)$ into
$L^{\frac{p}{2\gamma}}(\Gamma)$, $e^{(\Delta-\sigma I)t}{\mathcal{H}}$ maps
$L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)$ into $L^{p}(\Gamma)$ and both
mappings are bounded and locally Lipschitz continuous. That is, for $R>0$
there exists $C>0$, which in general depends on $T$, such that for $U,V\in
L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)$ with
$\|U\|_{L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)}\leq R$,
$\|V\|_{L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)}\leq R$ one has

$\|{\mathcal{H}}(U)-{\mathcal{H}}(V)\|_{{L^{\frac{p}{2\gamma}}(\Gamma)}}\leq
C\|U-V\|_{L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)}.$ (B.33)

In addition, one has

$\|e^{(\Delta-\sigma I)t}{\mathcal{H}}(U)-e^{(\Delta-\sigma
I)t}{\mathcal{H}}(V)\|_{L^{p}(\Gamma)}\leq Ce^{-\delta
t}t^{-\frac{2\gamma-1}{2p}}\|U-V\|_{{}_{L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)}},t\in(0,T].$
(B.34)

Furthermore, the mapping $e^{(\Delta-\sigma I)t}\partial_{x}$, originally
defined on $C_{0}^{\infty}(\Gamma)$, can be extended to a bounded linear
operator in ${\mathcal{B}}(L^{\frac{p}{2\gamma}}(\Gamma),L^{p}(\Gamma))$ so
that $e^{(\Delta-\sigma I)t}\partial_{x}{\mathcal{H}}$ maps
$L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)$ into $L^{p}(\Gamma)$ and, for
$U,V$ as above, one has

$\|e^{(\Delta-\sigma I)t}\partial_{x}{\mathcal{H}}(U)-e^{(\Delta-\sigma
I)t}\partial_{x}{\mathcal{H}}(V)\|_{L^{p}(\Gamma)}\leq Ce^{-\delta
t}t^{-\frac{p+2\gamma-1}{2p}}\|U-V\|_{{L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)}},t\in(0,T].$
(B.35)

In addition, for $\beta\geq 0$, $h>0$ one has $e^{(\Delta-\sigma
I)t}\partial_{x}{\mathcal{H}}(U)\in{\mathcal{X}}^{\beta}_{p}$ and the
following inequalities hold

$\displaystyle\hskip 28.45274pt\|e^{(\Delta-\sigma
I)t}\partial_{x}{\mathcal{H}}(U)\|_{{\mathcal{X}}^{\beta}_{p}}\leq Ce^{-\delta
t}t^{-\beta-\frac{p+2\gamma-1}{2p}}\|U\|_{{L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)}},t\in(0,T],$
(B.36) $\displaystyle\|(e^{(\Delta-\sigma I)h}-I)e^{(\Delta-\sigma
I)t}\partial_{x}{\mathcal{H}}(U)\|_{{\mathcal{X}}^{\beta}_{p}}\leq
Ch^{\beta}e^{-\delta
t}t^{-2\beta-\frac{p+2\gamma-1}{2p}}\|U\|_{{L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)}},t\in(0,T].$
(B.37)

Furthermore, if $\sigma>\sigma_{0}$, where $\sigma_{0}>0$ is as in Corollary
2.1, then there exists a $T$-independent constant $C>0$ and
$0<\delta\ll\sigma$ such that (B.34)–(B.37) hold with $t\in(0,T]$ replaced by
$t\in(0,\infty)$.

###### Proof.

For ${\mathcal{F}}$ as in (B.2) one has

$\displaystyle\|{\mathcal{H}}(U)-{\mathcal{H}}(V)\|_{{L^{\frac{p}{2\gamma}}}(\Gamma)}\leq\|{\mathcal{F}}\big{(}U\big{)}(\partial_{x}u_{2}-\partial_{x}v_{2})\|_{L^{\frac{p}{2\gamma}}(\Gamma)}+\|\big{(}{\mathcal{F}}(U)-{\mathcal{F}}(V)\big{)}\partial_{x}v_{2}\|_{{L^{\frac{p}{2\gamma}}(\Gamma)}}.$
(B.38)

Using the Cauchy–Schwartz inequality we obtain

$\displaystyle\|\big{(}{\mathcal{F}}(U)-{\mathcal{F}}(V)\big{)}\partial_{x}v_{2}\|_{{L^{\frac{p}{2\gamma}}}(\Gamma)}^{\frac{p}{2\gamma}}$
$\displaystyle\leq\|\big{(}{\mathcal{F}}(U)-{\mathcal{F}}(V)\big{)}\|_{L^{{p}/{\gamma}}(\Gamma)}^{\frac{p}{2\gamma}}\|\partial_{x}v_{2}\|^{\frac{p}{2\gamma}}_{L^{{p}/{\gamma}}(\Gamma)}$
$\displaystyle\leq
C\|U-V\|_{(L^{p}(\Gamma))^{2}}^{\frac{p}{2\gamma}}\|v_{2}\|^{\frac{p}{2\gamma}}_{\widehat{W}^{1,p}(\Gamma)},$
(B.39)

where we used Proposition B.1 item (3) and $p/\gamma\leq p$. By a similar
argument one obtains

$\displaystyle\|{\mathcal{F}}\big{(}U\big{)}(\partial_{x}u_{2}-\partial_{x}v_{2})\|_{{L^{\frac{p}{2\gamma}}}(\Gamma)}^{\frac{p}{2\gamma}}$
$\displaystyle\leq\|{\mathcal{F}}(U)\|_{L^{{p}/{\gamma}}(\Gamma)}^{\frac{p}{2\gamma}}\|\partial_{x}u_{2}-\partial_{x}v_{2}\|^{\frac{p}{2\gamma}}_{L^{{p}/{\gamma}}(\Gamma)}$
$\displaystyle\leq
C\left(1+\|U\|_{(L^{p}(\Gamma))^{2}}^{\frac{p}{2\gamma}}\right)\|u_{2}-v_{2}\|^{\frac{p}{2\gamma}}_{\widehat{W}^{1,p}(\Gamma)},$
(B.40)

where we used Proposition B.1 item (2) and $p/\gamma\leq p$.

The inequality (B.34) follows from (B.33) and (1.25).

The inequality (B.35) follows from (B.33) and (1.23).

To prove (B.36) we write

$\displaystyle\|e^{(\Delta-\sigma
I)t}\partial_{x}{\mathcal{H}}(U)\|_{{\mathcal{X}}^{\beta}_{p}}=\|e^{\frac{1}{2}(\Delta-\sigma
I)t}e^{\frac{1}{2}(\Delta-\sigma
I)t}\partial_{x}{\mathcal{H}}(U)\|_{{\mathcal{X}}^{\beta}_{p}}$
$\displaystyle\quad\underset{\eqref{fracpow}}{\leq}Ce^{-\frac{\delta
t}{2}}t^{-\beta}\|e^{\frac{1}{2}(\Delta-\sigma
I)t}\partial_{x}{\mathcal{H}}(U)\|_{L^{p}(\Gamma)}\underset{\eqref{ehu}}{\leq}Ce^{-\delta
t}t^{-\beta-\frac{p+2\gamma-1}{2p}}\|U\|_{{L^{p}(\Gamma)\times\widehat{W}^{1,p}(\Gamma)}}.$
(B.41)

The inequality (B.37) is proved analogously by using (A.11).

The last statement follows from Corollary 2.1. ∎

##  Appendix C General theory of semilinear parabolic equations on graphs

In this section, we discuss local existence of classical solutions for

$\displaystyle u_{t}=u_{xx}+f(t,x,u,u_{x}),\quad x\in\Gamma,$ (C.1)
$\displaystyle\sum\limits_{\vartheta\sim e}\partial_{\nu}u_{e}(\vartheta)=0,\
u_{e}(\vartheta)=u_{e^{\prime}}(\vartheta),e\sim\vartheta,e^{\prime}\sim\vartheta,$
(C.2) $\displaystyle
u{(t_{0},\cdot)}=u_{0}(\cdot)\in{\mathcal{X}}^{\alpha}_{p},p\in[1,\infty).$
(C.3)

To that end, we first focus on local existence of strong solutions for

$u_{t}=\Delta u+F(t,u),u(t_{0})=u_{0}\in{\mathcal{X}}^{\alpha}_{p},$ (C.4)

where $\Delta={\mathcal{A}}_{p}$, cf. (LABEL:apnum), and $\alpha\geq 0$.

######  Definition C.1.

[25, Definition 3.3.1] A strong solution of (C.4) on $[t_{0},t_{1})$ with
initial condition $u(t_{0})=u_{0}\in{\mathcal{X}}_{p}^{\alpha}$ is a
continuous function $u:[t_{0},t_{1})\to L^{p}(\Gamma)$ such that
$u(t_{0})=u_{0}$, $u(t)\in\operatorname{dom}({\mathcal{A}}_{p})$ for
$t\in(t_{0},t_{1})$, $\frac{du}{dt}$ exists for $t\in(t_{0},t_{1})$,
$(t_{0},t_{1})\ni t\to F(t,u(t))\in L^{p}(\Gamma)$ is locally Hölder
continuous, and
$\int^{t_{0}+\sigma}_{t_{0}}\|F(t,u(t))\|_{L^{p}(\Gamma)}dt<\infty$ for some
$\sigma>0,$ and the differential equation $u_{t}+A_{p}u=F(t,u)$ is satisfied
on $(t_{0},t_{1})$ in $L^{p}(\Gamma)$.

In the following theorem we adopt classical results regarding strong solutions
of (C.4) to the setting of metric graphs.

######  Theorem C.1.

Let $p\in[1,\infty)$ and $\alpha\in{[}0,1)$. Assume that $F:[0,\infty)\times
X_{p}^{\alpha}\to L^{p}(\Gamma)$ is locally Hölder continuous in $t$ with
exponent $\nu\in(0,1)$ and locally Lipschitz continuous in $u$, that is, for
every $(t,u)\in[0,\infty)\times{\mathcal{X}}^{\alpha}_{p}$ there exists a
neighborhood $V\subset[0,\infty)\times{\mathcal{X}}^{\alpha}_{p}$ containing
$(t,u)$ and a constant $L=L(V)$ such that

$\|F(t_{1},u_{1})-F(t_{2},u_{2})\|_{L^{p}(\Gamma)}\leq
L(|t_{1}-t_{2}|^{\nu}+\|u_{1}-u_{2}\|_{{\mathcal{X}}^{\alpha}_{p}}),(t_{k},u_{k})\in
V,k=1,2.$ (C.5)

Then the following assertions hold.

  1. (1)

For any $t_{0}\in[0,\infty)$ and $u_{0}\in{\mathcal{X}}_{p}^{\alpha}$, there
exists $T_{\max}=T_{\max}(t_{0},u_{0})>0$ such that (C.4) has a unique strong
solution $u(t;t_{0},u_{0})$ on $[t_{0},t_{0}+T_{\max})$ with initial value
$u(t_{0};t_{0},u_{0})=u_{0}.$ Moreover,

$u(\cdot;t_{0},u_{0})\in
C([t_{0},t_{0}+T_{\max}),{\mathcal{X}}_{p}^{\alpha}),$ (C.6)

and if $T_{\max}<\infty,$ then

$\limsup_{t\nearrow
T_{\max}}\left\|u(t+t_{0};t_{0},u_{0})\right\|_{{X_{p}^{\alpha}}}=\infty.$
(C.7)

  2. (2)

For any $t_{0}\in[0,\infty)$ and $u_{0}\in{\mathcal{X}}_{p}^{\alpha}$, if
$u(t)$ is a strong solution of (C.4) on $[t_{0},t_{1})$ with initial condition
$u(t_{0})=u_{0}$, then $u$ satisfy the following integral equation

$u(t)=e^{\Delta(t-t_{0})}u_{0}+\int_{t_{0}}^{t}e^{\Delta(t-s)}F(s,u(s))ds.$
(C.8)

Conversely, if $t\mapsto u(t)$ is continuous as a mapping from $(t_{0},t_{1})$
into $X^{\alpha},$ one has
$\int^{t_{0}+\rho}_{t_{0}}\|F(t,u(t))\|_{L^{p}(\Gamma)}dt<\infty$ for some
$\rho>0,$ and if the integral equation (C.8) holds for $t_{0}<t<t_{1},$ then
$u(t)$ is a strong solution of the differential equation (C.4) on
$(t_{0},t_{1}).$ Furthermore,

$u\in
C^{\delta}((t_{0},T_{\max}),{\mathcal{X}}_{p}^{\alpha}),\delta\in(0,1-\alpha).$
(C.9)

  3. (3)

For any $t_{0}\in\mathbb{R}$ and $u_{0}\in X_{p}^{\alpha}$,

$u(\cdot;t_{0},u_{0})\in
C^{1}((t_{0},t_{0}+T_{\max}(t_{0},u_{0})),{\mathcal{X}}_{p}^{\beta}),\beta\in(0,\nu).$
(C.10)

  4. (4)

If $u_{n},u_{0}\in{\mathcal{X}}_{p}^{\alpha}$ and
$\|u_{n}-u_{0}\|_{{\mathcal{X}}_{p}^{\alpha}}\to 0$ as $n\to\infty$, then for
any $t_{0}\in\mathbb{R}$, one has

$\liminf_{n\to\infty}T_{\max}(t_{0},u_{n})\geq T_{\max}(t_{0},u_{0}),$ (C.11)

and

$\lim_{n\to\infty}\|u(t;t_{0},u_{n})-u(t;t_{0},u_{0})\|_{{\mathcal{X}}_{p}^{\alpha}}=0,$
(C.12)

uniformly in compact intervals of $[t_{0},t_{0}+T_{\max}(t_{0},u_{0}))$.

###### Proof.

(1) It follows from [25, Theorem 3.3.3 ] and [25, Theorem 3.3.4 ] together
with the fact that $e^{t\Delta}$ is analytic in $L^{p}(\Gamma)$.

(2) The first statement in part (2) follows from [25, Lemma 3.3.2], the fact
that $u\in C^{\delta}((t_{0},t_{1});X^{\alpha})$ is obtained in the course of
the proof of [25, Lemma 3.3.2].

(3) The proof of part (3) stems from that of [25, Theorem 3.5.2]. For
completeness, we provide the details. It suffices to prove

$u(\cdot;t_{0},u_{0})\in
C^{1}((\tau,\tau+h),{\mathcal{X}}_{p}^{\beta}),t_{0}<\tau<\tau+h<t_{0}+T_{\max}(t_{0},u_{0}),\,\,0<\beta<\nu,$

for all $\tau\in(t_{0},t_{0}+T_{\max}(t_{0},u_{0}))$ and some $h=h(\tau)>0$.
For any fixed $\tau\in(t_{0},T_{\max}(t_{0},u_{0}))$, let
$u_{1}=u(\tau;t_{0},u_{0})$. Then one has $u(t;t_{0},u_{0})=u(t;\tau,u_{1}),\
\tau<t<T_{\max}(t_{0},u_{0})$ and for any $\sigma>0$,

$u(t;t_{0},u_{0})=u(t;\tau,u_{1})={e^{(\Delta-\sigma)(t-\tau)}u_{1}+\int_{\tau}^{t}e^{(\Delta-\sigma)(t-s)}g(s)ds,}$
(C.13)

where $g(t):={u(t;\tau,u_{1})}+F(t,u(t;\tau,u_{1}))$. By [25, Lemma 3.5.1], in
order to prove that the mapping

$(\tau,\tau+h)\ni t\mapsto
G(t):=\int_{\tau}^{t}e^{(\Delta-\sigma)(t-s)}g(s)ds$ (C.14)

is continuously differentiable into ${\mathcal{X}}^{\beta}_{p}$ for some
$h=h(\tau)>0$, hence, $u(\cdot,t_{0},u_{0})\in
C^{1}((\tau,\tau+h),{\mathcal{X}}_{p}^{\beta})$ it suffices to show that

$\|g(s)-g(t)\|_{L^{p}(\Gamma)}\leq K(t)(s-t)^{\nu},\ \tau<t<s<\tau+h,$ (C.15)

for some continuous $K(\cdot)$ and $\int_{\tau}^{\tau+h}K(t)dt<\infty$. Our
objective, therefore, is to derive (C.15). To that end, by (C.5), there exist
a neighborhood $V\subset[0,\infty)\times{\mathcal{X}}^{\alpha}_{p}$ containing
$(\tau,u_{1})$, and positive constants $L>0$, $h>0$ such that

$\displaystyle\|g(t)-g(s)\|_{L^{p}(\Gamma)}$ $\displaystyle\leq
L\big{(}|t-s|^{\nu}+\|u(t;\tau,u_{0})-u(s;\tau,u_{0})\|_{X_{p}^{\alpha}}\big{)},$
(C.16)

for $s,t\in(\tau,\tau+h)$ and $t<s$. Note that

$\displaystyle u(s;\tau,u_{1})-u(t;\tau,u_{1})=$
$\displaystyle\big{(}e^{(\Delta-\sigma)(s-t)}-I\big{)}e^{(\Delta-\sigma)(t-\tau)}u_{1}$
$\displaystyle+\int_{\tau}^{t}e^{(\Delta-\sigma)(t-\kappa)}\big{(}g(\kappa+s-t)-g(\kappa)\big{)}d\kappa+\int_{\tau}^{\tau+s-t}e^{(\sigma-\Delta)(s-\kappa)}g(\kappa)d\kappa.$

Then there exist constants $C_{1},C_{2}>0$ such that

$\displaystyle\|u(s;\tau,u_{1})-u(t;\tau,u_{1})\|_{{\mathcal{X}}_{p}^{\alpha}}$
$\displaystyle\leq
C_{1}\big{(}(s-t)\cdot\|(\sigma-\Delta)^{1+\alpha}e^{(\Delta-\sigma)(t-\tau)}{u_{1}}\|_{L^{p}(\Gamma)}+\int_{\tau}^{t}(t-\kappa)^{-\alpha}\|g(\kappa+s-t)-g(\kappa)\|_{L^{p}(\Gamma)}d\kappa$
$\displaystyle\qquad+\int_{\tau}^{\tau+s-t}\|(\sigma-\Delta)^{\alpha}e^{\Delta(s-\kappa)}g(\kappa)\|_{L^{p}(\Gamma)}d\kappa\big{)}$
$\displaystyle\leq
C_{2}\big{(}(s-t)(t-\tau)^{-\alpha}\|{(\sigma-\Delta)u_{1}}\|_{L^{p}(\Gamma)}+\int_{\tau}^{t}(t-\kappa)^{-\alpha}\|g(\kappa+s-t)-g(\kappa)\|_{L^{p}(\Gamma)}d\kappa$
$\displaystyle\qquad+(s-t)(t-\tau)^{-\alpha}\big{)}.$

This together with (C.16) imply

$\displaystyle\|g(s)-g(t)\|_{L^{p}(\Gamma)}\leq$ $\displaystyle
L(s-t)^{\nu}+C_{2}\big{(}(s-t)(t-\tau)^{-\alpha}\|{(\sigma-\Delta)u_{1}}\|_{L^{p}(\Gamma)}+(s-t)(t-\tau)^{-\alpha}\big{)}$
$\displaystyle+C_{2}\int_{\tau}^{t}(t-\kappa)^{-\alpha}\|g(k+s-t)-g(\kappa)\|_{L^{p}(\Gamma)}d\kappa.$
(C.17)

Then Gronwall’s inequality (see [25, Section 1.2.1] yields $L_{1}>0$ such that

$\displaystyle\|g(s)-g(t)\|_{L^{p}(\Gamma)}\leq(s-t)^{\nu}K(t),$ (C.18)

where

$K(t):=L_{1}\Big{(}1+{h}^{1-\nu}(t-\tau)^{-\alpha}\|{(\sigma-\Delta)u_{1}}\|_{L^{p}(\Gamma)}+{h}^{1-\nu}(t-\tau)^{-\alpha}\Big{)},$

as required in (C.15).

(4) Follows from [25, Theorem 3.4.1]. ∎

Equation (C.1) is related to (C.4) via the following elementary proposition.

######  Lemma C.1.

Let
$f:[0,\infty)\times\Gamma\times{\mathbb{R}}\times{\mathbb{R}}\rightarrow{\mathbb{R}}$.
Assume that there exists $\nu\in(0,1)$, such that for every triple
$(t,u,v)\in[0,\infty)\times{\mathbb{R}}\times{\mathbb{R}}$ there exists an
open neighborhood $U\subset[0,\infty)\times{\mathbb{R}}\times{\mathbb{R}}$ and
a constant $L=L(U)$ such that for all $(t_{k},u_{k},v_{k})\in U$, $k=1,2$ one
has

$|f(t_{1},x,u_{1},v_{1})-f(t_{2},x,u_{2},v_{2})|\leq
L(|t_{1}-t_{2}|^{\nu}+|u_{1}-u_{2}|+|v_{1}-v_{2}|),$ (C.19)

that is, $f=f(t,x,u,v)$ is locally Hölder continuous in $t$ with exponent
$\nu\in(0,1)$, locally Lipschitz continuous in $u$ and $v$, uniformly for
$x\in\Gamma$. Assume in addition that $f(t,\cdot,u,v)\in L^{p}(\Gamma)$. Let
$p\in[1,\infty)$ and $\alpha\in(0,1)$ be such that $2\alpha-p^{-1}>1$. For
$u\in{\mathcal{X}}^{\alpha}_{p}$ let $F(t,u)=f(t,\cdot,u,u_{x})$ then the
mapping $F:[0,\infty)\times{\mathcal{X}}_{p}^{\alpha}\to L^{p}(\Gamma)$ is
locally Hölder continuous in $t$ with exponent $\nu$ and Lipschitz continuous
in $u$.

###### Proof.

Fix $(t_{0},u_{0})\in[0,\infty)\times{\mathcal{X}}^{\alpha}_{p}$. Then due to
$2\alpha-\frac{1}{p}>1$, one has
$X_{p}^{\alpha}\hookrightarrow\widehat{C}^{1}(\widehat{\Gamma})$, see (A.9).
Therefore we may choose an open neighborhood
$V\subset[0,\infty)\times{\mathcal{X}}^{\alpha}_{p}$ containing
$(t_{0},u_{0})$ such that for $(t_{k},u_{k})\in V$, $k=1,2$, we have
$(t_{k},u_{k}(x),\partial_{x}u_{k}(x))\in U$ for all $x\in\Gamma$. Then by
(C.19) we have

$\displaystyle|f(t,x,u(x),u_{x}(x))-f(s,x,v(x),v_{x}(x))|\leq$
$\displaystyle\quad\leq L\big{(}|s-t|^{\nu}+|u(x)-v(x)|+|u_{x}(x)-v_{x}(x)|,\
x\in\Gamma.$ (C.20)

Hence, $f(t,\cdot,u(\cdot),u_{x}(\cdot))-f(s,\cdot,v(\cdot),v_{x}(\cdot))\in
L^{p}(\Gamma)$ and

$\displaystyle\|f(t,\cdot,u(\cdot),u_{x}(\cdot))-f(s,\cdot,v(\cdot),v_{x}(\cdot))\|_{L^{p}(\Gamma)}$
$\displaystyle\leq
L\big{(}|t-s|^{\nu}\sum_{e\in\mathcal{E}}|e|^{\frac{1}{p}}+\|u(\cdot)-v(\cdot)\|_{L^{p}(\Gamma)}+\|u_{x}(\cdot)-v_{x}(\cdot)\|_{L^{p}(\Gamma)}\big{)}$
$\displaystyle\leq\tilde{L}\big{(}|t-s|^{\nu}+\|u-v\|_{X_{p}^{\alpha}}\big{)},$

as required. ∎

Applying the above theorem, we have

######  Theorem C.2.

Assume that $f=f(t,x,u,v)$ is as in Lemma C.1 and suppose in addition that
$f(t,\cdot,u,v)\in\widehat{C}(\overline{\Gamma})$. Assume that $0<\alpha<1$
and $1<p<\infty$ are such that $2\alpha-p^{-1}>1$ and $2\nu-p^{-1}>0$. Then
the following assertions hold.

(1) $($Existence of classical solutions$)$ For given $t_{0}\in{\mathbb{R}}$
and $u_{0}\in{\mathcal{X}}_{p}^{\alpha}$, (C.1) has a unique classical
solution $u(t,x;t_{0},u_{0})$ on some interval $[t_{0},t_{0}+T)$ with initial
condition $u(t_{0},x;t_{0},u_{0})=u_{0}(x)$ in ${\mathcal{X}}_{p}^{\alpha}$,
and

$u(\cdot,\cdot;t_{0},u_{0})\in
C([t_{0},t_{0}+T_{\max}),{\mathcal{X}}_{p}^{\alpha})\cap
C^{\delta}((t_{0},t_{0}+T_{\max}),{\mathcal{X}}_{p}^{\alpha})\cap
C^{1}((t_{0},t_{0}+T_{\max},{\mathcal{X}}_{p}^{\beta})$ (C.21)

for all $\delta\in(0,1-\alpha)$ and $\beta\in(0,\nu)$. Moreover if
$T_{\max}<\infty,$ then

$\limsup_{t\rightarrow
T^{-}_{\max}}\left\|u(t+t_{0},\cdot;t_{0},u_{0})\right\|_{{\mathcal{X}}_{p}^{\alpha}}=\infty.$
(C.22)

(2) $($Continuity with respect to initial conditions$)$ If
$u_{n},u_{0}\in{\mathcal{X}}_{p}^{\alpha}$ and
$\|u_{n}-u_{0}\|_{{\mathcal{X}}_{p}^{\alpha}}\to 0$ as $n\to\infty$, then for
any $t_{0}\in\mathbb{R}$, $\liminf_{n\to\infty}T_{\max}(t_{0},u_{n})\geq
T_{\max}(t_{0},u_{0})$ and

$\lim_{n\to\infty}\|u(t,\cdot;t_{0},u_{n})-u(t,\cdot;t_{0},u_{0})\|_{{\mathcal{X}}_{p}^{\alpha}}=0$

uniformly in compact intervals of $[t_{0},t_{0}+T_{\max}(t_{0},u_{0}))$.

(3) $($Comparison principle$)$ If
$u_{1},u_{2}\in\widehat{C}(\overline{\Gamma})$ and $u_{1}(x)\leq u_{2}(x)$,
$x\in\Gamma$ then $u(t,x;t_{0},u_{1})\leq u(t,x;t_{0},u_{2})$, $x\in\Gamma$
for $t>t_{0}$ such that both $u(t,x;t_{0},u_{1})$ and $u(t,x;t_{0},u_{2})$
exist.

###### Proof of Theorem C.2.

(1) First, let us define a mapping
$F:[0,\infty)\times{\mathcal{X}}^{\alpha}_{p}\rightarrow L^{p}(\Gamma)$ as
follows

$F(t,u)(x)=f(t,x,u(x),u_{x}(x)),u\in{\mathcal{X}}^{\alpha}_{p}.$ (C.23)

Then by Lemma C.1 $F=F(t,u)$ satisfies the hypotheses of Theorem C.1.
Therefore for every $u_{0}\in{\mathcal{X}}^{\alpha}_{p}$ there exists a unique
strong solution $u(t;t_{0},u_{0})$ of (C.4) defined on the maximal time
interval $(t_{0},T(t_{0},u_{0}))$ and satisfying the initial condition
$u(t_{0};t_{0},u_{0})=u_{0}$. Let us denote

$u(t,x;t_{0},u_{0}):=u(t;t_{0},u_{0})(x),x\in\Gamma.$ (C.24)

Then (C.21) follows from (C.6), (C.9) and (C.10), while (C.22) follows from
(C.16).

Next, we prove that $u(t,x;t_{0},u_{0})$ is a classical solution of (C.1). To
that end we note that $2\alpha-p^{-1}>1$ and $2\nu-p^{-1}>0$. Then there is
$\beta\in(0,\nu)$ such that $2\beta-p^{-1}>0$, hence, by (A.9) one has

${\mathcal{X}}^{\alpha}_{p}\hookrightarrow\widehat{C}^{1}(\overline{\Gamma}),\quad{\mathcal{X}}_{p}^{\beta}\hookrightarrow\widehat{C}(\overline{\Gamma}).$

This together with (C.6) and (C.10) yield

$u(\cdot,\cdot;t_{0},u_{0})\in\widehat{C}^{0,1}([t_{0},t_{0}+T_{\max}(t_{0},u_{0}))\times\overline{\Gamma})\cap\widehat{C}^{1,0}((t_{0},t_{0}+T_{\max}(t_{0},u_{0}))\times\overline{\Gamma}).$
(C.25)

Next, defining

$\tilde{f}(t,x)=f(t,x,u(t,x;t_{0},u_{0}),u_{x}(t,x;t_{0},u_{0})).$

and using (C.25) one obtains

$\tilde{f}\in\widehat{C}^{0,0}((t_{0},t_{0}+T_{\max}(t_{0},u_{0}))\times\overline{\Gamma}),$
(C.26)

Re-writing (C.1) we obtain

$-u_{xx}(t,x;t_{0},u_{0})=-u_{t}(t,x;t_{0},u_{0})+\tilde{f}(t,x).$

This together with (C.25) and (C.26) yield

$u(\cdot,\cdot;t_{0},u_{0})\in\widehat{C}^{1,2}((t_{0},t_{0}+T_{\max}(t_{0},u_{0}))\times\overline{\Gamma}).$

Therefore, $u(t,x;t_{0},u_{0})$ is a classical solution of (C.1) with initial
condition $u(t_{0},x;t_{0},u_{0})=u_{0}(x)\in{\mathcal{X}}_{p}^{\alpha}$.

Now, we claim that (C.1) has a unique classical solution $u(t,x;t_{0},u_{0})$
satisfying $u(t_{0},x;t_{0},u_{0})=u_{0}(x)$ and (C.21). In fact, if
$u(t,x;t_{0},u_{0})$ is a classical solution of (C.1) satisfying
$u(t_{0},x;t_{0},u_{0})=u_{0}(x)$ and (C.21), then $u(t,\cdot;t_{0},u_{0})$
satisfies

$u(t,\cdot;t_{0},u_{0})=e^{\Delta(t-t_{0})}u_{0}(\cdot)+\int_{t_{0}}^{t}e^{\Delta(t-s)}F(s,u(s,\cdot;t_{0},u_{0}))ds,$

and hence is a strong solution of (C.4) with initial condition
$u(t_{0},\cdot;t_{0},u_{0})=u_{0}(\cdot)$. The claim then follows from the
uniqueness of strong solutions of (C.4).

(2) Follows from Theorem C.1 (4).

(3) We divide the proof into 3 steps.

Step 1. Let us introduce the following notation

$\displaystyle
w_{0}:=u_{2}-u_{1},w(t,x):=u(t,x;t_{0},u_{2})-u(t,x;t_{0},u_{1}),$ (C.27)
$\displaystyle U_{k}:=u(t,x;t_{0},u_{k}),V_{k}:=\partial_{x}U_{k},k=1,2.$
(C.28)

Then by assumption $w_{0}\geq 0$ and one has

$\displaystyle-w_{t}+w_{xx}+f(t,x,U_{1},V_{1})-f(t,x,U_{2},V_{2})=0,$ (C.29)
$\displaystyle\sum\limits_{\vartheta\sim e}\partial_{\nu}w_{e}(\vartheta)=0,$
(C.30)

For given $U_{1}=u(t,x;t_{0},u_{1})$ and $U_{2}=u(t,x;t_{0},u_{2})$ we will
rewrite (C.29) as a linear parabolic equation for $w(x,t)$. To that end, let

$\Phi(x,t,s):=f\left(t,x,sU_{1}+(1-s)U_{2},sV_{1}+(1-s)V_{2}\right).$

Then one has

$\Phi(x,t,1)-\Phi(x,t,0)=f(t,x,U_{1},(U_{1})_{x})-f(t,x,U_{2},(U_{2})_{x}),$
(C.31)

and

$\displaystyle\Phi(x,t,1)-\Phi(x,t,0)$
$\displaystyle=\int_{0}^{1}\frac{\partial\Phi}{\partial s}(t,x,s)ds$
$\displaystyle=\int_{0}^{1}\frac{\partial f}{\partial
u}(t,x,sU_{1}+(1-s)U_{2},sV_{1}+(1-s)V_{2})(U_{1}-U_{2})ds$
$\displaystyle\quad+\int_{0}^{1}\frac{\partial f}{\partial
v}(t,x,sU_{1}+(1-s)U_{2},sV_{1}+(1-s)V_{2})(V_{1}-V_{2})ds$
$\displaystyle=B(x,t)(U_{1}-U_{2})+C(x,t)(V_{1}-V_{2}),$

where we denoted

$\displaystyle B(x,t)=\int_{0}^{1}\frac{\partial f}{\partial
u}(t,x,sU_{1}+(1-s)U_{2},sV_{1}+(1-s)V_{2})ds\quad\text{ and }$ $\displaystyle
C(x,t)=\int_{0}^{1}\frac{\partial f}{\partial
v}(t,x,sU_{1}+(1-s)U_{2},sV_{1}+(1-s)V_{2})ds$

Therefore (C.29) can be rewritten as follows

$L(w):=-w_{t}+w_{xx}+B(x,t)w_{x}+C(x,t)w=0.$

Step 2. Let us define $\widehat{w}(x,t)$via $w(x,t)=e^{-\lambda
t}\hat{w}(x,t)$, where $\lambda=\sup_{x,t}|C(x,t)|$. Then one has

$\displaystyle L(w)$ $\displaystyle=-(\widehat{w}_{t}e^{-\lambda
t}-\lambda\widehat{w}e^{-\lambda t})+\widehat{w}_{xx}e^{-\lambda
t}+B(x,t)\widehat{w}_{x}e^{-\lambda t}+\widehat{w}e^{-\lambda t}$
$\displaystyle=e^{-\lambda
t}\left(-\widehat{w}_{t}+\widehat{w}_{xx}+B(x,t)\widehat{w}_{x}+(C(x,t)+\lambda)\widehat{w}\right)=0.$

Thus $\hat{w}$ satisfies $\hat{L}(\widehat{w})=0$ where $\widehat{L}$ is
defined as

$\widehat{L}(\widehat{w})=-\widehat{w}_{t}+\widehat{w}_{xx}+B(x,t)\widehat{w}_{x}+\widehat{C}(x,t)\widehat{w},\text{
where }\widehat{C}(x,t)=C(x,t)+\lambda\geq 0$ (C.32)

Step 3. In this step we show that for any
$u_{1},u_{2}\in{\mathcal{X}}_{p}^{\alpha}$ with $u_{2}-u_{1}\geq 0$ we have
$u(t,x;t_{0},u_{2})-u(t,x;t_{0},u_{1})\geq 0$, $t\geq t_{0}$

First, assume that $u_{1},u_{2}\in
C(\overline{\Gamma})\cap{\mathcal{X}}_{p}^{\alpha}$ and
$\inf_{x\in\Gamma}(u_{2}-u_{1})(x)>0$. Then by definition $\widehat{w}_{0}\in
C(\overline{\Gamma})\cap{\mathcal{X}}_{p}^{\alpha}$ and
$\inf_{x\in\Gamma}\widehat{w}_{0}(x)=\inf_{x\in\Gamma}e^{\lambda
t}(u_{2}-u_{1})(x)>0$. We claim that $\widehat{w}(t,x;\hat{w}_{0})>0$ for all
$t>t_{0}$ and $x\in\Gamma$. Assuming the contrary, there exist $t_{1}>t_{0}$
and $x_{1}\in\Gamma$ such that
$\widehat{w}(t_{1},x_{1};t_{0},\widehat{w}_{0})=0$ and
$\widehat{w}(t,x;t_{0},\widehat{w}_{0})>0$ for all $t_{0}<t<t_{1}$ and
$x\in\Gamma$. Let $e_{1}\in\mathcal{E}$ be such that $x_{1}\in\bar{e}_{1}$.
Using the maximum principle for parabolic equations (on bounded intervals) we
infer that $x_{1}$ is an end point of edge $e_{1}$, that is, $x_{1}\in\partial
e_{1}$. Then Hopf’s Lemma gives

$\partial_{\nu}\widehat{w_{e}}(t_{1},x_{1};t_{0},\hat{w}_{0})<0,\text{\ for
every edge\ }\,\,e\in\mathcal{E}\,{\rm with}\,\,x_{1}\in\partial e,$

where $\frac{\partial\widehat{w}}{\partial\nu}$, as usual, denotes the
derivative of $\widehat{w}$ with respect to the unit outer normal. This
implies, in particular, that

$\sum_{e\sim x_{1}}{\partial_{\nu}\widehat{w}_{e}(x_{1})}<0.$

This contradicts the vertex condition (C.30). Thus
$u(t,x;t_{0},u_{2})-u(t,x;t_{0},u_{1})\geq 0$ as asserted.

Next, for any $u_{1},u_{2}\in{\mathcal{X}}_{p}^{\alpha}$, due to
$2\alpha-p^{-1}>0$ and (A.9), there exist $u_{1,n},u_{2,n}\in
C(\overline{\Gamma})\cap{\mathcal{X}}_{p}^{\alpha}$ such that

$\lim\limits_{n\to\infty}(u_{2,n}-u_{1,n})=u_{2}-u_{1}\quad{\rm in}\quad
L^{p}(\Gamma).$

Moreover, since $\inf_{x\in\Gamma}(u_{2}-u_{1})\geq 0$ we may assume that
$\inf_{x\in\Gamma}(u_{2,n}-u_{1,n})>0$. Therefore one has

$\widehat{w}_{n}:=e^{\lambda t}(u_{2,n}-u_{1,n})\in
C(\overline{\Gamma})\cap{\mathcal{X}}_{p}^{\alpha},\
\inf_{x\in\Gamma}\widehat{w}_{n}(x)>0,\widehat{w}_{n}\to\widehat{w}\text{ in
}L^{p}(\Gamma),$ (C.33)

and, by the previous argument,

$\widehat{w}(t,x;t_{0},\hat{w}_{n})>0,t>t_{0},x\in\Gamma,n\geq 1.$ (C.34)

Finally, using

$\widehat{w}(t,\cdot;t_{0},\hat{w}_{n})\to\widehat{w}(t,\cdot;t_{0},\hat{w}_{n})\quad{\rm
as}\quad n\to\infty\quad{\rm in}\quad L^{p}(\Gamma),$ (C.35)

we obtain $u(t,x;t_{0},u_{2})-u(t,x;t_{0},u_{1})\geq 0$, $t\geq t_{0}$ as
required.

∎

######  Proposition C.1.

[25, p. 190]. Let $\alpha\in[0,1)$ $a_{1},a_{2}>0$, $T>0$. Assume that an
integrable function $u:[0,T]\rightarrow[0,\infty)$ satisfies

$0\leq u(t)\leq a_{1}t^{-\alpha}+a_{2}\int_{0}^{t}(t-s)^{-\alpha}u(s)ds,$
(C.36)

for $t\in[0,T]$. Then there exists $C=C(\alpha,a_{2},T)>0$ such that

$0\leq u(t)\leq\frac{a_{1}C}{1-\alpha}t^{-\alpha},\ t\in(0,T).$ (C.37)

## References

  * [1] H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), pp. 593–676.
  * [2] G. Arumugam and J. Tyagi, Keller-Segel chemotaxis models: a review, Acta Appl. Math., 171 (2021), pp. Paper No. 6, 82.
  * [3] S. Becker, F. Gregorio, and D. Mugnolo, Schrödinger and polyharmonic operators on infinite graphs: parabolic well-posedness and $p$-independence of spectra, J. Math. Anal. Appl., 495 (2021), pp. Paper No. 124748, 44.
  * [4] N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), pp. 1663–1763.
  * [5] G. Berkolaiko, R. Carlson, S. A. Fulling, and P. Kuchment, eds., Quantum graphs and their applications, vol. 415 of Contemporary Mathematics, American Mathematical Society, Providence, RI, 2006.
  * [6] G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, vol. 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013.
  * [7] R. Borsche, S. Göttlich, A. Klar, and P. Schillen, The scalar Keller-Segel model on networks, Math. Models Methods Appl. Sci., 24 (2014), pp. 221–247.
  * [8] D. Borthwick, E. M. Harrell, II, and K. Jones, The heat kernel on the diagonal for a compact metric graph, Ann. Henri Poincaré, 24 (2023), pp. 1661–1680.
  * [9] G. Bretti, R. Natalini, and M. Ribot, A hyperbolic model of chemotaxis on a network: a numerical study, ESAIM: M2AN, 48 (2014), pp. 231–258.
  * [10] V. I. Burenkov, Sobolev spaces on domains, vol. 137 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998.
  * [11] F. Camilli and L. Corrias, Parabolic models for chemotaxis on weighted networks, J. Math. Pures Appl. (9), 108 (2017), pp. 459–480.
  * [12] C. Cattaneo, The spread of the potential on a homogeneous tree, Ann. Mat. Pura Appl. (4), 175 (1998), pp. 29–57.
  * [13] M. Efendiev, E. Nakaguchi, and W. L. Wendland, Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system, Discrete Contin. Dyn. Syst., (2007), pp. 334–343.
  * [14] P. Exner, J. P. Keating, P. Kuchment, T. Sunada, and A. Teplyaev, eds., Analysis on graphs and its applications, vol. 77 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, RI, 2008\.  Papers from the program held in Cambridge, January 8–June 29, 2007.
  * [15] P. Exner and P. Šeba, Free quantum motion on a branching graph, Rep. Math. Phys., 28 (1989), pp. 7–26.
  * [16] L. Friedlander, Genericity of simple eigenvalues for a metric graph, Israel J. Math., 146 (2005), pp. 149–156.
  * [17] E. Galakhov, O. Salieva, and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), pp. 4631–4647.
  * [18] Y. Gong, S. He, and A. Kiselev, Random search in fluid flow aided by chemotaxis, Bull. Math. Biol., 84 (2022), pp. Paper No. 71, 46.
  * [19] Y. Gong and A. Kiselev, A simple reaction-diffusion system as a possible model for the origin of chemotaxis, J. Biol. Dyn., 17 (2023), pp. Paper No. 2260833, 16.
  * [20] A. Grigoryan and A. Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, Duke Math. J., 109 (2001), pp. 451–510.
  * [21] F. R. Guarguaglini and R. Natalini, Global smooth solutions for a hyperbolic chemotaxis model on a network, SIAM J. Math. Anal., 47 (2015), pp. 4652–4671.
  * [22] B. Harley, H. D. Kim, M. Zaman, I. Yannas, D. Lauffenburger, and L. Gibson, Microarchitecture of three-dimensional scaffolds influences cell migration behavior via junction interactions, Biophys J., 95 (2008).
  * [23] E. M. Harrell, II, Spectral theory on combinatorial and quantum graphs, in Spectral theory of graphs and of manifolds, CIMPA 2016, Kairouan, Tunisia, vol. 32 of Sémin. Congr., Soc. Math. France, Paris, 2018, pp. 1–37.
  * [24] Z. Hassan, Y. P. Shen, and Y. P. Zhang, Global existence of classical solutions of chemotaxis systems with logistic source and consumption or linear signal production on $\mathbb{R}^{n}$, preprint, arXiv:2310.16001v1, (2023).
  * [25] D. Henry, Geometric theory of semilinear parabolic equations, vol. 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981\.
  * [26] D. Henry, Perturbation of the boundary in boundary-value problems of partial differential equations, vol. 318 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2005.  With editorial assistance from Jack Hale and Antônio Luiz Pereira.
  * [27] T. Hillen and K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), pp. 183–217.
  * [28] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), pp. 103–165.
  * [29] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), pp. 52–107.
  * [30] T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 926–973.
  * [31] , Pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments, J. Math. Anal. Appl., 490 (2020), pp. 124204, 30\.
  * [32] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), pp. 399–415.
  * [33] , A model for chemotaxis, J. Theoret. Biol., 30 (1971), pp. 225–234.
  * [34] A. Kiselev, F. Nazarov, L. Ryzhik, and Y. Yao, Chemotaxis and reactions in biology, J. Eur. Math. Soc. (JEMS), 25 (2023), pp. 2641–2696.
  * [35] A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case, J. Math. Phys., 53 (2012), pp. 115609, 9.
  * [36] A. Kiselev and X. Xu, Suppression of chemotactic explosion by mixing, Arch. Ration. Mech. Anal., 222 (2016), pp. 1077–1112.
  * [37] V. Kostrykin, J. Potthoff, and R. Schrader, Heat kernels on metric graphs and a trace formula, in Adventures in mathematical physics, vol. 447 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, pp. 175–198.
  * [38] T. Kottos and U. Smilansky, Quantum graphs: a simple model for chaotic scattering, Journal of Physics A: Mathematical and General, 36 (2003), p. 3501.
  * [39] P. Kuchment, Differential and pseudo-differential operators on graphs as models of mesoscopic systems, in Analysis and applications—ISAAC 2001 (Berlin), vol. 10 of Int. Soc. Anal. Appl. Comput., Kluwer Acad. Publ., Dordrecht, 2003, pp. 7–30.
  * [40] P. Kuchment, Quantum graphs: I. some basic structures, Waves in Random Media, 14 (2004), pp. S107–S128.
  * [41] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), pp. 1158–1191.
  * [42] B. B. Mandal and S. C. Kundu, Cell proliferation and migration in silk fibroin 3d scaffolds, Biomaterials, 30 (2009), pp. 2956–2965.
  * [43] D. Mugnolo, Semigroup methods for evolution equations on networks, Understanding Complex Systems, Springer, Cham, 2014.
  * [44] K. J. Painter, Mathematical models for chemotaxis and their applications in self-organisation phenomena, Journal of Theoretical Biology, 481 (2019), pp. 162–182.  Celebrating the 60th Birthday of Professor Philip Maini.
  * [45] N. Rodríguez and M. Winkler, Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation, Math. Models Methods Appl. Sci., 30 (2020), pp. 2105–2137.
  * [46] N. Rodriguez and M. Winkler, On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime, European Journal of Applied Mathematics, 33 (2022), p. 919–959.
  * [47] J.-P. Roth, Le spectre du laplacien sur un graphe, in: G. Mokobodzki, D. Pinchon (Eds.), Théorie du Potentiel, Proc. Orsay 1983, 1096 (1984), pp. 521–539.
  * [48] R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$, J. Differential Equations, 262 (2017), pp. 5635–5690.
  * [49] R. B. Salako, W. Shen, and S. Xue, Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?, J. Math. Biol., 79 (2019), pp. 1455–1490.
  * [50] W. Shen, Z. Shen, S. Xue, and D. Zhou, Population dynamics under climate change: persistence criterion and effects of fluctuations, J. Math. Biol., 84 (2022), pp. Paper No. 30, 42.
  * [51] W. Shen and S. Xue, Persistence and spreading speeds of parabolic-elliptic Keller-Segel models in shifting environments, J. Differential Equations, 269 (2020), pp. 6236–6268.
  * [52] , Forced waves of parabolic-elliptic Keller-Segel models in shifting environments, J. Dynam. Differential Equations, 34 (2022), pp. 3057–3088.
  * [53] , Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst., 42 (2022), pp. 2893–2925.
  * [54] , Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 2981–3002.
  * [55] C. Spadaccio, A. Rainer, S. De Porcellinis, M. Centola, F. De Marco, M. Chello, M. Trombetta, and J. Genovese, A g-csf functionalized plla scaffold for wound repair: An in vitro preliminary study., Annu Int Conf IEEE Eng Med Biol Soc. 2010, (2010).
  * [56] Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), pp. 6142–6161.
  * [57] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), pp. 849–877.
  * [58] H. Triebel, Interpolation theory, function spaces, differential operators, vol. 18 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978.
  * [59] T. Tsujikawa, Stability of aggregating patterns in a chemotaxis model including growth, in China-Japan Symposium on Reaction-Diffusion Equations and their Applications and Computational Aspects (Shanghai, 1994), World Sci. Publ., River Edge, NJ, 1997, pp. 159–165.
  * [60] J. von Below, A characteristic equation associated to an eigenvalue problem on c2-networks, Linear Algebra and its Applications, 71 (1985), pp. 309–325.
  * [61] J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), pp. 316–337.
  * [62] , A maximum principle for semilinear parabolic network equations, in Differential equations with applications in biology, physics, and engineering (Leibnitz, 1989), vol. 133 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1991, pp. 37–45.
  * [63] L. Wang, C. Mu, and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), pp. 1847–1872.
  * [64] Q. Wang, J. Yan, and C. Gai, Qualitative analysis of stationary Keller-Segel chemotaxis models with logistic growth, Z. Angew. Math. Phys., 67 (2016), pp. Art. 51, 25.
  * [65] M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), pp. 708–729.
  * [66] , Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), pp. 2889–2905.
  * [67] , Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), pp. 1516–1537.
  * [68] , Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), pp. 261–272.
  * [69] , Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. (9), 100 (2013), pp. 748–767.
  * [70] , Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), pp. 1056–1077.
  * [71] T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst., (2015), pp. 1125–1133.
  * [72] J. Zhao, C. Mu, L. Wang, and K. Lin, A quasilinear parabolic-elliptic chemotaxis-growth system with nonlinear secretion, Appl. Anal., 99 (2020), pp. 86–102.
  * [73] J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), pp. 120–140.
  * [74] P. Zheng, C. Mu, X. Hu, and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), pp. 509–522.


in this estimation scheme.
Therefore, we have

\begin{eqnarray}
\CR (\diag (g_1,g_2),{\cal M}_{xp,n})
\leq
\frac{\hbar g_{x_0}}{2(n+1/2)}
+\frac{\hbar g_{p_0}}{2(n+1/2)},
\nonumber
\end{eqnarray}

which implies that arbitrarily precise estimate is possible
if ${\cal M}_{xp,n}$ with large enough $n$ is fortunately given.

The efficiency $(\ref{eqn:effmle})$
is achievable by the following maximum likelihood estimator
up to the first order of $1/N$:

\begin{eqnarray}
\hat{x}_0 &=&
\argmax_{x_0} \sum_{j=1}^{N/2} \ln p(x_j - x_0),\nonumber\\
\hat{p}_0 &=&
\argmax_{p_0} \sum_{j=1}^{N/2} \ln \tilde{p}(p_j - p_0),
\label{eqn:mlexp}
\end{eqnarray}

$x_1,x_2,..., x_{N/2}$
$p_1,p_2,..., p_{N/2},$ be
data produced by the measurement of
the position and the momentum of the given states,
and  their probability distribution is denoted by
$p(x)$ and $\tilde{p}(p)$, respectively.

§ PLANCK'S CONSTANT AND UNCERTAINTY

In this section, we focus on Planck's constant.

\begin{eqnarray}
P'=\hbar^{-1/2} P,\, X'=\hbar^{-1/2} X,\nonumber\\
p'_0=\hbar^{-1/2} p_0,\, x'_0=\hbar^{-1/2} x_0,
\nonumber
\end{eqnarray}

and define

\begin{eqnarray}
{\cal M}'_{xp}=\{\rho(p'_0, x'_0)\,|\,
\rho(p'_0, x'_0)=\pi(\exp i(p'_0 X'-x'_0 P')|\phi_0\rgl\,)\}.
\nonumber
\end{eqnarray}

Since Planck's constant does not appear
in the commutation relation $[P', X']= -i$,
the attainable CR type bound of ${\cal M}_{xp}'$ is
not dependent on $\hbar$
if definition of $|\phi_0\rgl$ does not include $\hbar$.

If $\CR({\cal M}'_{xp})$ has some finite value,
the identity

\begin{eqnarray}
\CR({\cal M}_{xp})=\hbar\CR({\cal M}'_{xp})
\nonumber
\end{eqnarray}

\begin{eqnarray}
\lim_{\hbar\rightarrow 0} \CR({\cal M}_{xp})=0.
\nonumber
\end{eqnarray}

Therefore, in the limit of $\hbar\rightarrow 0$,
position and momentum can be simultaneously
measured as precisely as needed.

However, it must be noticed that
the attainable CR type bound of the position shifted model

\begin{eqnarray}
{\cal M}_x=\{\rho(x_0)\,|\,
\rho(x_0)=\pi(\exp (-ix_0 P)|\phi_0\rgl\,)\}
\nonumber
\end{eqnarray}

and of the momentum  shifted model

\begin{eqnarray}
{\cal M}_p=\{\rho(p_0)\,|\,
\rho(p_0)=\pi(\exp (ip_0 P)|\phi_0\rgl\,)\}
\nonumber
\end{eqnarray}

also tends to zero as $\hbar\rightarrow 0$,
and that the ratio

\begin{eqnarray}
\frac{\CR({\cal M}_{xp})}{\sqrt{\CR({\cal M}_x)\:\CR({\cal M}_p)}}
\frac{\CR({\cal M}_{xp}')}{\sqrt{\CR({\cal M}'_x)\:\CR({\cal M}'_p)}}
\nonumber
\end{eqnarray}

is independent of $\hbar$, where
${\cal M}'_x$ and ${\cal M}'_p$ are defined
in the same manner as ${\cal M}'_{xp}$.
Therefore, noncommutative nature of the model is unchanged
even if $\hbar$ tends to $0$.
Actually, as in $(\ref{eqn:sfnb})$the index $\beta$ of the noncommutative nature of the model
is independent of $\hbar$.

Notice the discussion in this section is essentially valid for
the mixed position-momentum shifted model,

\begin{eqnarray}
\{\rho(p_0, x_0)\,|\,
\rho(p_0, x_0)
\rho_0 D^*(x_0,\,p_0\,),\,(x_0,\,p_0)\in\R^2\},
\nonumber
\end{eqnarray}

where the state $\rho_0$ is mixed,
and $D(x_0,\,p_0\,)$ is the operator defined by $(\ref{eqn:weilop})$.

§ SEMIPARAMETRIC ESTIMATION OF THE SHIFT PARAMETERS

In section $\ref{sec:estshift}$, our conclusion is that
if we are fortunate enough, we can estimate
the average of the position and the momentum
with arbitrary accuracy at the same time.

One may argue that this is because we make
full use of knowledge about the shape of the wave function
of the given state. However, this argument is not thoroughly true.

In the classical estimation theory, we have the following very strong result.
Suppose that we are intersected in the mean value $\theta\in\R$
of the probability distribution,
and that  the shape of the probability distribution is unknown
except it is  symmetric around $\theta$.
In other words, we set up the semiparametric model such that,

\begin{eqnarray}
\{ p(x\, |\, \theta, g)\: |\:
p(x\, |\, \theta, g)=g(x-\theta),\: \theta\in\R,\:
\mbox{$g(x)$ is symmetric around $0$}\},\nonumber\\
\end{eqnarray}

and estimate the parameter $\theta\in\R$ from
the data $x_1,x_2,...,x_N\in\R$.

If $g(x)$ is known, the variance of  the best consistent estimator
is given by

\begin{eqnarray}
\frac{1}{N J}+o\left(\frac{1}{N}\right)
\label{eqn:semicr}
\end{eqnarray}

where $J$ is the Fisher information,

\begin{eqnarray}
J=\int \left(\frac{d}{dx}\ln g(x)\right)^2 g(x)dx.
\end{eqnarray}

In the case where  $g(x)$ is not known,
the theorem 2.2 in the  Ref. $[5]$ insists
that the bound $(\ref{eqn:semicr})$ is attainable:

If $g(x)$ is absolutely continuous,
the bound $(\ref{eqn:semicr})$ is attainable
by some consistent estimate
(see pp. 649-650 in th Ref $[5]$).

By the use of this theorem,
if $p(x)$ and $\tilde{p}(p)$ defined
in the end of section $\ref{sec:estshift}$ are
symmetric about $x_0$ and $p_0$ respectively,
we can use the semiparametric estimates, instead
of the maximum likelihood estimates $(\ref{eqn:mlexp})$,
and can achieve the same efficiency as  $(\ref{eqn:mlexp})$.
Then, if we are so fortunate that the $|\phi_0\rgl$
is happen to be $|n\rgl$ with very large $n$,
our estimate is quite accurate.

CHAPTER: TIME-ENERGY UNCERTAINTY IN VIEW OF HYPOTHESIS TEST

§ CONVENTIONAL DISCUSSION ABOUT TIME-ENERGY UNCERTAINTY

This chapter is the result of the joint research with Mr. S. Osawa of
Tokyo Institute of Technology,
which aimed at a consistent and physically meaningful formulation of
the time-energy uncertainty.

There are various formulation of the time-energy uncertainty relation,
corresponding to the variety of the interpretation of
uncertainty $\Delta t$ of time $t$.
Some authors introduce `time operator' which forms canonical pair with
the Hamiltonian $H$ of the system.
In this chapter, it is shown that the formulation of
the time-energy uncertainty is quite reasonable, consistent, and physically

An acceptable interpretation of  $\Delta t$  is the
time interval during which
the state of a system can hardly
be distinguished from the initial state.
For example, it is derived in the explanation
of the sudden approximation in Messiah $[16]$.
The outline is as follows.

We suppose the Hamiltonian to change-over in a continuous way
from a certain initial time $t_0$ to a certain final time $t_1$. We put

\begin{eqnarray}
\Delta t \equiv t_1-t_0
\nonumber
\end{eqnarray}

and denote by $H(t)$ the Hamiltonian at time $t$.

Let $|0 \rangle $ denote the state vector of the system at time $t_0$ ,
and $U(t_1, t_0)$ the time evolution operator
from $t_0$ to $t_1$.

The sudden approximation consists in writing

\begin{eqnarray}
U(t_1,t_0)|0\rangle \approx |0\rangle .
\nonumber
\end{eqnarray}

Messiah regarded

\begin{eqnarray}
\lgl 0 | U^ {\dag}(t_1, t_0)\,(I-|0\rgl\lgl 0|)\,U(t_1, t_0)|0\rgl
\label{eq:w},
\end{eqnarray}

as  a `probability of finding
the system in a state other than the initial state'.
One obtains the expansion of $w$ in powers of $\Delta t$
by the perturbation method. Put

\begin{eqnarray}
\overline {H} = \frac{1}{ \Delta t} \int_{t_0}^{t_1} H(t) dt.
\nonumber
\end{eqnarray}

We then have

\begin{eqnarray}
\frac {\Delta t^2}{\hbar^2}\lgl 0 |\overline{H}
(I-|0\rgl\lgl 0|) \overline{H} |0 \rgl
+ O(\Delta t)^3.
\nonumber
\end{eqnarray}

\begin{eqnarray}
\langle 0 |\overline{H}Q_0 \overline{H} |0 \rangle
=\langle 0|\overline{H}^2|0 \rangle - \langle 0|\overline{H}|0 \rangle ^2
=\lgl(\Delta \overline{H})^2\rgl,
\nonumber
\end{eqnarray}

we have

\begin{eqnarray}
w= \frac{\Delta t^2 \lgl(\Delta \overline{H})^2\rgl }{\hbar^2}+ O(\Delta t)^3.
\nonumber
\end{eqnarray}

Thus the condition for the validity of the sudden approximation,
$w\ll1$, requires that

\begin{equation}
\Delta t \ll \frac{\hbar}{\sqrt{\lgl(\Delta \overline{H})^2\rgl}}
\label{eqn:messTE}
\end{equation}

We can point out a  defect in  this discussion.
In Messiah's  discussion,
the following testing scheme is
implicitly assumed;
Let $M_{ms}$ the measurement which takes value
on the set $\Omega=\{0,\, 1\}$ such that,

\begin{eqnarray}
M_{ms}(\{ 0 \})=|0\rgl\lgl 0|,\;\;
M_{ms}(\{ 1 \})=I-|0\rgl\lgl 0|\;\; ;
\nonumber
\end{eqnarray}

If  the outcome of the measurement
is  $0$, we accept that the system is in the initial state.
However, if there is a testing scheme which works better than this,
$\Delta t$ should be smaller than  $(\ref{eqn:messTE})$ implies.
Hence, for the Messiah's discussion to be valid,
the optimality of the testing scheme must be shown.
In this study, we investigate this point
from the viewpoint of hypothesis testing.

§ TIME-ENERGY UNCERTAINTY AS A HYPOTHESIS TEST

Here, we formulate the problem as a hypothesis testing
(see Ref. [18]).
Consider the model

\begin{eqnarray}
{\cal M}=\{\rho(t)\,|\,
\rho(t)=U(t_1,t_0)\,\rho(t_0)\,U^{\dagger}(t_1,t_0),\, t\in\R\}
\nonumber
\end{eqnarray}

and the hypothesises

\begin{eqnarray}
\begin{array}{ccc}
H_0: & \rho(t)=\rho(t_0) & \mbox{( null hypothesis)}, \\
H_1: & \rho(t)=\rho(t_1) & \mbox{ (alternative hypothesis)}.
\end{array}
\label{eqn:TEqutest}
\end{eqnarray}

Then, we choose a test
which maximize the probability $\gamma$
when $H_1$ is really true.
The maximization is to be done
under the restriction
that the probability of rejecting   $H_0$
when $H_0$ holds true is smaller than the significance level $\alpha$,
for, otherwise, the test which always reject $H_0$ is chosen.
We call $\gamma$ the power of the test.

Two steps are needed to maximize
the power $\gamma$ of  the test.
The first step is to find the most powerful tests
of the following classical hypothesis testing of the parameter family

\begin{eqnarray}
{\cal M}(M)
=\{ p_M(x\,|\, t)\, |\,
p_M(x\,|\, t)dx=\tr \rho(t) M(dx),\;\; t\in\R\}
\nonumber
\end{eqnarray}

of the probability distributions which is deduced from ${\cal M}$
by the measurement $M$; The hypothesis are set to be

\begin{eqnarray}
\begin{array}{ccc}
H_0: &  p_M (x | t)= p_M (x | t_0) & \mbox{( null hypothesis)}, \\
H_1: &  p_M (x | t)=p_M (x | t_1) &\mbox{ (alternative hypothesis)},
\end{array}
\label{eq:TEctest}
\end{eqnarray}

corresponding to $(\ref{eqn:TEqutest})$.
We denote by $\gamma_M$ the power of the most powerful tests
in this hypothesis testing.
In the second step, we adjust the measurement $M$ to maximize $\gamma_M$,
and obtain the optimal tests and its power.

§ POWER OF THE TEST

Let us consider the power of test and the optimum measurement
when we are given $N$ copies of the state.
To begin with, consider the first step.

Whatever the significance levle of the test is,
Stein's lemma in the  classical statistics
gives the maximum power $\gamma_M$ in the classical
hypothesis testing $(\ref{eq:TEctest})$ as

\begin{eqnarray}
\lim_{N\rightarrow\infty}\frac{1}{N}\ln(1-\gamma_M(N)\,)
=-D(\,p_M(x | t_0\,)\,||\,p_M(x | t_1)\,),
\label{eqn:gM}
\end{eqnarray}

where $D(\,p(x)\,||\,q(x)\,)$
is Kullback Divergence defined by

\begin{eqnarray}
\int p(x)\ln\frac{p(x)}{q(x)}dx.
\nonumber
\end{eqnarray}

When $N$ is very large, $(\ref{eqn:gM})$ roughly writes

\begin{equation}
\gamma_M  \approx 1- \exp[-N D(\,p_M(x | t_0\,)\,||\,p_M(x | t_1)\,)\,],
\label{eqn:gM2}
\end{equation}

where the argument $N$ in $\gamma_M(N)$ is dropped for notational simplicity.

Expansion of $D(\,p_M(x | t_0\,)\,||\,p_M(x | t_1)\,)$ in powers of
$\Delta t=t_1 -t_0$ gives

\begin{eqnarray}
D(\,p_M(x | t_0\,)\,||\,p_M(x | t_1)\,)
=\frac{1}{2}J_M(\theta_0)\,(\Delta t)^2 + o(\Delta t)^2,
\nonumber
\end{eqnarray}

where $J_M(\theta_0)$ is the classical Fisher information
of the classical model ${\cal M}(M)$.
Hence, when $\delta t$ is very small and $N$ is very large,
$\gamma_M$ can be written roughly as

\begin{equation}
\gamma_M
\approx 1-\exp\left(-\frac{N}{2}J_M(t_0)(\Delta t)^2\,\right)
+o(\Delta t)^2
\qquad(\Delta t \ll 1),
\label{eqn:gM3}
\end{equation}

Let us move to the second step.
By virtue of $(\ref{eqn:gM2})$,
when $\Delta t$ is very small, the maximization of
$\gamma_M$ is equivalent to that of $J^M(\theta_0)$,
the answer of which is given as

\begin{eqnarray}
\max_M J_M(\theta_0)=J^S(\theta_0),
\nonumber
\end{eqnarray}

where $J^S(\theta_0)$ is the SLD Fisher information of
the model ${\cal M}$ (see Theorem 1 in Ref. [20]).
$J^S(\theta_0)$ is easily calculated as

\begin{eqnarray}
J^S(\theta_0)=\frac{4}{\hbar^2}\lgl\Delta H^2\rgl,
\label{eqn:TEjs}
\end{eqnarray}

and finally we obtain the power $\gamma_max$ of the optimum test
when $N$ is very large and $\Delta t$ is very small;

\begin{eqnarray}
\gamma_{max}
\approx
1-\exp\left(-\frac{2N}{\hbar^2}(\Delta t)^2\lgl(\Delta H)^2\rgl\right)
+o(\Delta t)^2
\qquad(\Delta t \ll 1,\,N\gg 1).
\nonumber
\end{eqnarray}

Now we can show the condition that $\rho(t)\,(t\ge t_1)$
can hardly be distinguished from $\rho(t_0)$  using $N$ data
when $\Delta t \ll 1$ and $N \gg 1$ are satisfied;
the condition writes

\begin{eqnarray}
\gamma_{max}\approx
\exp\left(-\frac{2N}{\hbar^2}(\Delta t)^2\lgl(\Delta H)^2\rgl\right)
+o(\Delta t)^2
\ll 1,
\nonumber
\end{eqnarray}

or equivalently,

\begin{eqnarray}
\frac{2N \Delta t^2\lgl(\Delta H)^2\rgl}{\hbar^2}\ll 1
\quad(N\gg 1,\,\Delta t\ll 1).
\label{eqn:htestTE}
\end{eqnarray}

Notice that Messiah's condition $(\ref{eqn:messTE})$
is identical to $(\ref{eqn:htestTE})$ when $\Delta t$ is very small.

§ THE OPTIMAL MEASUREMENT

In this section, it is shown that the test
based on the measurement $M_{ms}$ is one of the optimal tests.

The classical Fisher information $J_{M_{ms}}(t_0)$
of the model ${\cal M}(M_{ms})$
is given by

\begin{eqnarray}
J_{M_{ms}}(t_0)=\lim_{t \to t_0}
\left(\frac{\dot{p}(0|t)^2}{p(0|t)}
\nonumber
\end{eqnarray}

\begin{eqnarray}
p(0|t)&=&\tr \rho(t) M_{ms}({0})\nonumber\\
\nonumber
\end{eqnarray}

Expansion of $p(0|t)$ in powers of $t-t_0$ gives

\begin{eqnarray}
+ o(t-t_0)^2\nonumber\\
&=&-\frac{1}{\hbar^2}(\lgl(\Delta H)^2\rgl)\,(t-t_0)^2 +o(t-t_0)^2,
\nonumber
\end{eqnarray}

which leads to

\begin{eqnarray}
J_{M_{ms}}(t_0)=\frac{4}{\hbar^2}\Delta H^2.
\label{eqn:TEjm}
\end{eqnarray}

From $(\ref{eqn:TEjs})$ and $(\ref{eqn:TEjm})$,
$M_{ms}$ is one of the optimum measurements.

*Part III

The general  model theory
Part III :
The general  model theory

CHAPTER: GEOMETRICAL STRUCTURE

§ $W$-CONNECTION AND $E$-CONNECTION IN THE WIDER SENSE

This chapter presents the results obtained in the joint research with
Dr. A. Fujiwara of Osaka University.
In this section, we treat with the generalization of
the $w$-connection in the faithful model theory
to the general case.

To define the generalized $w$-connection, we need to specify
the logarithmic derivative ${\bf L}(X)$ among
the various roots of the matrix equation $(\ref{eqn:deflogd})$.
Moreover, we need to check whether
a  connection can be defined by $\ref{eqn:defpttw}$ or not.
For the system of vectors

\begin{eqnarray}
\{
{\bf L}(\partial_{\zeta^i}|_{W_1})W_2
{\bf L}(\partial_{\zeta^i}|_{W_1})\,) I\:
|\: i=1,2,..., 2dr-1
\}
\label{eqn:spantrans}
\end{eqnarray}

might not be a linearly independent.
However, for any choice
of the logarithmic derivative,
if $W_2$ is  near enough to $W_1$,
$(\ref{eqn:spantrans})$ is linearly independent,
because the mapping

\begin{eqnarray}
L\longrightarrow LW-\tr(\pi(W)L)I
\nonumber
\end{eqnarray}

is continuous.

Therefore, we define

\begin{eqnarray}
{\cal N}(W_1, \varepsilon)
=\{ W\: |\:  W\in{\cal N}, W\in B(W_1, \varepsilon)\},
\nonumber
\end{eqnarray}

\begin{eqnarray}
B(W_1, \varepsilon)
\equiv
\left\{ W\:
\left|\: \sqrt{\lgll W-W_1, W-W_1\rgll} \leq\varepsilon
\right.
\right\},
\nonumber
\end{eqnarray}

and we restrict ourselves to the consideration of
${\cal N}(W_1, \varepsilon)$ with enough small $\varepsilon$.
Then, a choice of the logarithmic derivative
defines a generalization of the $w$-connection.

In the almost same way, we can define generalized $e$-connection
in the manifold

\begin{eqnarray}
{\cal M}(\theta, \varepsilon)
=\{ \rho\: |\:  \rho\in{\cal M}, W\in B(\rho_1, \varepsilon)\},
\nonumber
\end{eqnarray}

with small $\varepsilon$.
Or, we can deduce a generalized $e$-connection
from a $w$-connection as in $(\ref{eqn:ptwte})$.
Obviously, the equation$(\ref{eqn:etortion2})$
hold true in any generalization
of the $w$-connection and $e$-connection.
The generalized $w$-connection and the generalized $e$-connection
deduced from that $w$- connection satisfies theorem $\ref{theorem:lslsdecom}$.

§ VANISHING CONDITIONS

In this section, we examine
the conditions  that $e$-tortion,
$w$-tortion, and/or  Uhlmann curvature would vanish.

We consider following conditions
for the model ${\cal M}={\cal P}_r$ and
for the manifold ${\cal N}={\cal W}_r$.
Here, $W$ denotes a member of ${\cal W}_r$.

(A) Algebraic conditions

* The SLD can be chosen to satisfy
$W^*[L^S_{X}, L^S_{Y}]W=0$.

* The SLD can be chosen to satisfy
$[L^S_{X}, L^S_{Y}]W=0$.

* The SLD can be chosen to satisfy
$[L^S_{X}, L^S_{Y}]=0$.

(G) Geometric conditions

* Uhlmann curvature vanishes at $\pi(W)$.

* There is a generalized $w$-tortion which vanishes at $W$
for horizontal vectors.

(1)  G1 is equivalent to  A1

(2)  G2 is equivalent to  A2

(1) : Since Uhlmann curvature is the vertical component of
${\bf M}\Ttw(\hlift(X),\hlift(Y))=(1/4)[L_X^S,L_Y^S]W$,
Uhlmann curvature vanishes iff
$(1/4)[L_X^S,L_Y^S]W$ is orthogonal to
the vertical subspace,
or equivalently, iff
for any skew Hermitian matrix $A$,

\begin{eqnarray}
{\rm Re}\,\tr \left\{(WA)^*(1/4)[L_X^S,L_Y^S]W\right\}
=(-1/4){\rm Re}\,\tr \left\{AW^*[L_X^S,L_Y^S]W\right\}
\nonumber
\end{eqnarray}

holds true,
which is equivalent to $W^*[L_X^S,L_Y^S]W=0$.

(2) : The statement is derived directly from $(\ref{eqn:mttw})$.

As for the Algebraic conditions,

\begin{eqnarray}
A1\,\Leftarrow\, A2\Leftarrow\, A3,\nonumber
\nonumber
\end{eqnarray}

but non of the converses do not hold true.

The former half of the assertion is trivial.
As for the $A1\Rightarrow A2$,
we have the following counter-example:

\begin{eqnarray}
\begin{array}{cc}
\sigma\\
\end{array}
\right),\:\:\:
\begin{array}{cc}
{\bf a}^*\sigma
\end{array}
\right),\:\:\:
\begin{array}{cc}
{\bf 0}^*
\end{array}
\right),
\nonumber
\end{eqnarray}

where $\sigma$ is a $r\times r$ reversible matrix,
${\bf a}$ a member of $R^m$ which is not eigenvector of
$r\times r$ matrix $B$, and $0$ the vector all of whose components are zeros.
The following is the counter-example of $A2\Rightarrow A3$:

\begin{eqnarray}
\begin{array}{cc}
\sigma\\
\end{array}
\right),\:\:\:
\begin{array}{cc}
\end{array}
\right),\:\:\:
\begin{array}{cc}
\end{array}
\right),
\label{eqn:gence23}
\end{eqnarray}

where $d$ is larger than or equal to $r+2$,
$B$ is a $(d-r)\times r$ full rank matrix,
$C$ is a $(d-r)\times (d-r)$ matrix,
and $BB^*$ does not commute with $C$.

If the model is faithful or pure, the conditions A1-A3 are equivalent.
Therefore, not only G1$\Leftarrow$G2,
but also G1$\Rightarrow$G2 holds true.
However, as is understood by the above two lemmas,
this is not the case generally:

G2 implies G1, but not vice versa.

§ PURE STATE MODEL REVISITED

In the pure state model, we can also introduce
the generalized $e$- and $w$- connections.
One remarkable fact about the pure state model is
that the generalized $e$-connection in  the total space model ${\cal P}_1$
can be chosen so that the $e$-tortion vanishes.

Actually, by taking $L^S_X=2X\rho$, we have

\begin{eqnarray}
\frac{1}{4}[[L^S_X, L^S_Y],\rho]
&=&[[X\rho, Y\rho],\rho]\nonumber\\
&=& \frac{1}{4}
[[(|l_X\rgl\lgl\phi|+|\phi\rgl\lgl l_X|),
(|l_Y\rgl\lgl\phi|+|\phi\rgl\lgl l_Y|) ], |\phi\rgl\lgl\phi|]\nonumber\\
\nonumber
\end{eqnarray}

if we  choose  SLD and the logarithmic derivative
for horizontal vectors this way,
Uhlamnn curvature is equal to the $w$-tortion for the horizontal vectors,
which means that
the pure state model is locally quasi-classical
iff there is a generalized $w$-connection whose tortion vanishes.

CHAPTER: ATTAINABILITY OF SLD CR BOUND

§ COMMUTATIVE SLD AND ATTAINABILITY OF THE BOUND

In the faithful model,
a necessary and sufficient condition for SLD CR bound
to be attained
is that the SLD's are commutative,
and in the pure state model, that condition
is the existence of the commutative SLD's.
one might come up with the conjecture that
in general, attainability of SLD CR bound is
equivalent to the existence of commutative SLD's,
which is the algebraic condition A1 in section $\ref{sec:genvanish}$.

Actually, it is easily shown that SLD CR bound can be achieved
if SLD's commute. The optimal measurement
is the one in the theorem $\ref{theorem:mpCR}$,
that is, the simultaneous spectral decomposition of
commutative SLD's.
However, in the followings, the case of $(\ref{eqn:gence23})$
is shown to be a counter-example of
the converse of the statement. Therefore,
the algebraic condition A1 is only a sufficient condition
for the attainable SLD CR bound.

${\cal K}$ be a $d+\max\{r,d-r\}$-dimensional complex vector space
such that ${\cal H}$ is a its subspace,
$P$ the  projection from
${\cal K}$ onto ${\cal H}$,
$L^j\:(j=1,2)$ the matrices which satisfy

\begin{eqnarray}
L^j W=
\sum_{k=1}^2[J^{S-1}]^{j,k}
{\bf M}\left(\hlift_W \left(\frac{\partial}{\partial\theta^k}\right)\right),
\label{eqn:defL^j}
\end{eqnarray}

$\tilde{L}^j\:(j=1,2)$ the matrices in ${\cal K}$
such that

\begin{eqnarray}
\tilde{L}^1\equiv\left(
\begin{array}{ccc}
0 & B^*C_1 & 0\\
C_1B &0 & C_1D \\
0 & D^*C_1 &0
\end{array}
\right),\:
\tilde{L}^2 \equiv \left(
\begin{array}{ccc}
0 & B^*C_2 & 0\\
C_2B &0 & C_2D \\
0 & D^*C_2 &0
\end{array}
\right),
\nonumber
\end{eqnarray}

\begin{eqnarray}
\label{eqn:c1c2}
\end{eqnarray}

and $D$ will be defined soon.
$\tilde{L}^j\:(j=1,2)$ are defined so that
they satisfy

\begin{eqnarray}
P\tilde{L}^j W=L^j W
\: (j=1,2),
\nonumber
\end{eqnarray}

where $W$ in the left hand side of the equation means

\begin{eqnarray}
\left(
\begin{array}{c}
\end{array}
\right).
\label{eqn:w=wo}
\end{eqnarray}

By virtue of

\begin{eqnarray}
\tilde{L}^1\tilde{L}^2=
\left(
\begin{array}{ccc}
B^*C_1C_2B & 0 & B^*C_1C_2D\\
0 & C_1(BB^*+DD^*)C_2  & 0\\
D^*C_1C_2B & 0 & D^*C_1C_2D
\end{array}
\right)
\nonumber
\end{eqnarray}

and  $(\ref{eqn:c1c2})$,
if we choose $D$ such that

\begin{eqnarray}
D=(\alpha I - BB^*)^{1/2},
\nonumber
\end{eqnarray}

where $\alpha$ is the maximum eigenvalue of the matrix $BB^*$,
$\tilde{L}^1$ and $\tilde{L}^2$ commute.

Let $E$ be a projection valued measurement such that

\begin{eqnarray}
\tilde{L}^i=\int \hat\theta^i E(d\hat\theta)\: (i=1,2),
\nonumber
\end{eqnarray}

and $M$ a measurement deduced from $E$ as

\begin{eqnarray}
\nonumber
\end{eqnarray}

Then, we have

\begin{eqnarray}
&=&[{\rm Re}\,\tr \pi(\tilde{W}) \tilde{L}^j \tilde{L}^k]\nonumber\\
&=&[{\rm Re}\,\tr \pi(W) L^j L^k]\nonumber\\
\nonumber
\end{eqnarray}

Because, as is shown in the previous section,
any  matrices $A_1, A_2$ which satisfy

\begin{eqnarray}
A_j W=
{\bf M}\left(\hlift_W \left(\frac{\partial}{\partial\theta^j}\right)\right)\:
\end{eqnarray}

do not commute with each other,
$(\ref{eqn:gence23})$ is a counter-example of the conjecture.

§ A NECESSARY CONDITION AND THE MAIN CONJECTURES

If SLD CR bound is achieved,
G2($\Leftrightarrow$A1) in section $\ref{sec:genvanish}$
holds true.

Let $M$ be a measurement which satisfies $(\ref{eqn:genxx=J})$,
$E$ the Naimark dilation of $M$,
${\cal K}\supset{\cal H}$ the Hilbert space which $E$ lives in,
$P$ the projection from ${\cal H}$ onto ${\cal K}$,
$W$ a member of ${\cal W}_d$ such that $\rho(\theta)=\pi(W)$,
$L^j\,(j=1,...,m)$ the operators in ${\cal H}$
which satisfy $(\ref{eqn:defL^j})$,
$\tilde{L}^j\,(j=1,...,m)$ the operators ${\cal K}$ which satisfy

\begin{eqnarray}
{\bf M}^j(\hat\theta, E, W)=\tilde{L}^j W,
\nonumber
\end{eqnarray}

where $W$ in the equation means $(\ref{eqn:w=wo})$.
Then, for SLD CR bound to be attained,

\begin{eqnarray}
[V[E]]_{jj}(=[V[M]]_{jj})=tr \rho(\theta)(L^j)^2(=[J^{S-1}]^{jj}),
\nonumber
\end{eqnarray}

or, its equivalence,

\begin{eqnarray}
\tr \rho(\theta)(\tilde{L}^j)^2=\tr \rho(\theta)(L^j)^2
\label{eqn:extL=L}
\end{eqnarray}

must hold true.

\begin{eqnarray}
P\tilde{L}^j W= L^j W
\nonumber
\end{eqnarray}

follows directly from their definitions,
we have

\begin{eqnarray}
\tr W^*(\tilde{L}^j)^2 W
&=&\tr W^*(\tilde{L}^j)P(\tilde{L}^j)+\tr W^*(\tilde{L}^j)(1-P)(\tilde{L}^j)
\nonumber\\
&=&\tr W^*(L^j)^2W=\tr \rho(\theta)(L^j)^2,
\nonumber
\end{eqnarray}

which, combined with $(\ref{eqn:extL=L})$, leads to

\begin{eqnarray}
P\tilde{L}^j W=L^j W.
\nonumber
\end{eqnarray}

Hence, we have

\begin{eqnarray}
&&P(\tilde{L}^j \tilde{L}^k-\tilde{L}^k \tilde{L}^j)W
\nonumber\\
&=&P(\tilde{L}^j P\tilde{L}^k-\tilde{L}^k P\tilde{L}^j)W
\nonumber\\
&=&P(\tilde{L}^j P\tilde{L}^k P-\tilde{L}^k P\tilde{L}^j P)W.
\nonumber
\end{eqnarray}

$[\tilde{L}^j, \tilde{L}^k]=0$ follows from the definition,
this means

\begin{eqnarray}
[P\tilde{L}^j P, P\tilde{L}^k P]W=0.
\nonumber
\end{eqnarray}

Because we can take the SLD's such that

\begin{eqnarray}
P\tilde{L}^j P =\sum_k [J^{S-1}]^{j,k}L^S_k,
\nonumber
\end{eqnarray}

we have the theorem.

If G2($\Leftrightarrow$A1) in section $\ref{sec:genvanish}$
holds true, the equality in CR inequality can be achieved.

In the pure state model,
this conjecture is true as is mentioned
in the end of section $\ref{sec:gentwt}$.
Because of $G1\Leftrightarrow G2$ and theorem $\ref{th:estuhl1}$,
the conjecture is valid also in the faithful model.

$w$-tortion is a good index of noncommutative nature of the model.

This statement is proved to be true in the 2-parameter pure state model.

CHAPTER: CONCLUSIONS

As for a geometrical side of the thesis, $w$-connection is proposed
as a medium to unify Nagaoka's information geometry
and Uhlmann's parallelism.

Our conjecture is that the tortion of
$w$-connection is a good measure of noncommutative nature of the model.
This conjecture is proved
for the 2-parameter pure state model.
The following seem to support the conjecture:

(1) The attainable CR type bound of
the general pure state model
with the weight matrix $J^S$.

(2) The condition for the pure state model and the faithful model
to be locally quasi-classical.

The proof (or disproof) of the conjecture is an open problem.

As for the global property of the model,
the faithful model is quasi-classical iff the model is parallel.
However, in the pure state model,
being parallel is sufficient condition but not necessary condition.
Therefore, Uhlmann's RPF might not characterize the global
property of the model in general.
However, the condition for being parallel seems to have
intrinsic relation with some kind of symmetry.

As for the determination of the attainable CR type bound,
we succeeded in the case of the 2-parameter pure state model
and the coherent model.
The CR type bound with weight matrix $J^S$ is also
calculated for arbitrary pure state model.

We successfully applied the result in the estimation theory
to the analysis of the position-momentum uncertainty.
The main points are that the mean value of the position and the momentum
can be estimated up to arbitrary efficiency and
that Planck's constant has nothing
to do with noncommutative nature of the position-momentum shifted model.
As for the analysis of the time-energy uncertainty,
we succeeded in the formulation of the problem
in a good shape without the help of `time operator'.

[1]
Y. Aharonov and J. Anandan,
“Phase change during a cyclic quantum evolution,”
Phys. Rev. Lett. 58, 1593–1596 (1987).
[2]
S. Abe,
“Quantized geometry associated with uncertainty and correlation,”
Phys. Rev. A 48, pp. 4102-4106 (1993).
[3]
S. Amari,
Differential-Geometrical Methods in Statistics,
Lecture Notes in Statistics, Vol. 28
(Springer, Berlin, 1985).
[4]
M. V. Berry,
“Quantal phase factors accompanying adiabatic changes,”
Proc. Roy. Soc. London A392, 45–57 (1984).
[5]
P. J. Bickel,
“The 1980 Wald Memorial Lectures  On Adaptive Estimation,”
The Annals of Statistics, Vol.10, No.3, pp. 647-671 (1982).
[6]
A. Fujiwara,
“A Geometrical Study in Quantum Information Systems,”
doctoral thesis (1995).
[7]
A. Fujiwara, private communication.
[8]
A. Fujiwara and H. Nagaoka,
“Quantum Fisher metric and estimation for pure state models,”
Phys. Lett. 201A,119-124 (1995).
[9]
A. Fujiwara and H. Nagaoka,
“Coherency in view of quantum estimation theory,”
in Quantum coherence and decoherence,
edited by K. Fujikawa and Y. A. Ono,
(Elsevier, Amsterdam, 1996), pp.303-306.
[10]
C. W. Helstrom,
“Minimum Mean-Square Error Estimation in Quantum Statistics,”
Phys. Lett., 25A, 101-102(1967).
[11]
C. W. Helstrom,
Quantum Detection and Estimation Theory
(Academic Press, New York, 1976).
[12]
A. S. Holevo,
Probabilistic and Statistical Aspects of Quantum Theory
(North-Holland, Amsterdam, 1982) (in Russian, 1980).
[13]
M. Hübner,
“Explicit computation of the Bures distance for density matrices,”
Phys. Lett. A163, 239–242 (1992).
[14]
M. Hübner,
“Computation of Uhlmann's parallel transport for density matrices
and the Bures distance on three-dimensional Hilbert space,”
Phys. Lett. A179, 226–230 (1993).
H. Kuratuji and S. Iida,
”Effective Action for Adiabatic Process”,
Prog. Theo. Phys.74,439-445 (1985).
[15]
S. Kobayashi and K. Nomizu,
Foundations of Differential Geometry, I, II
(John Wiley, New York, 1963, 1969).
[16]
A. Messiah,
“MECANIQUE QUANTIQUE,”
Dunod, Paris (1959).
[17]
E. L. Lehmann,
“Theory of Point Estimation,”
Jhon Wiley (1983).
[18]
E. L. Lehmann,
“Testing Statistical Hypothesis,”
2nd ed., Jhon Wiley (1986).
[19]
H. Nagaoka,
“On Fisher Information of Quantum Statistical Models,"
SITA'87, 19-21, Nov., (1987)(in Japanese).
[20]
H. Nagaoka,
“A New Approach to Cramer-Rao Bounds for Quantum State Estimation,”
IEICE Technical Report, IT89-42, 9-14(1989).
[21]
H. Nagaoka,
“On the Parameter Estimation Problem for
Quantum Statistical Models,"
SITA'89, 577-582 Dec. (1989).
[22]
“Differential Geometrical Aspects of Quantum State Estimation
and Relative Entropy,”
METR 94-14 (1994).
[23]
H. Nagaoka,
private communication No. 1 (1996).
[24]
M. Ozawa,
“Quantum measuring processes of continuous observables,”
J. Math. Phys. 25, 79-87 (1984).
[25]
H. P. Robertson,
Phys. Rev., 34(1), 163-164, (1924).
[26]
J. J. Sakurai,
“Modern Quantum Mechanics,”
Benjamin/Cummings Publishing Company,Inc,(1985).
[27]
J .Samuel and R.Bhandari,
“General Setting for Berry's Phase,”
2239-2342, (1988).
[28]
“GEOMETRIC PHASES IN PHYSICS,”
Advanced Series in Mathematical Physics,vol. 5,
World Scientific (1989).
[29]
B. Simon,
“Holonomy, the quantum adiabatic theorem, and Berry's phase,”
Phys. Rev. Lett. 51, 2167–2170 (1983).
[30]
“Positive functions on $C^*$-algebras,”
Proc. Am. Math. Soc. 6, 211-216(1955).
[31]
A. Uhlmann,
“Parallel transport and `Quantum holonomy' along density operators,”
Rep. Math. Phys., vol.24, pp.229-240(1986).
[32]
A. Uhlmann,
“An energy dispersion estimate,”
Phys. Lett. A, 161, pp.329–331(1992).
[33]
A. Uhlmann,
“Density operators as an arena for differential geometry,”
Rep. Math. Phys., vol.33, pp.253-263(1993).
[34]
H. Yuen and M. Lax,
“Multiple-Parameter Quantum Estimation and Measurement of
Nnnselfadjoint Observables,”
IEEE Trans. on Information Theory, Vol. IT-19, No. 6, 740-750(1973).
[35]
T. Y. Yung,
“Asymptotic Efficient Approaches to
Quantum-Mechanical Parameter Estimation,”
Information Sciences, 9, pp.25-42 (1975).