上世纪五十年代, Lehner和Wing在文献[1]中研究了无限平行板模型中最具一般的中子迁移方程

$ A \cdot = - \mu \frac{{\partial \cdot }}{{\partial x}} + \frac{c}{2}\int_{ - 1}^1 \cdot \;{\rm{d}}\mu ', {\rm{ }} $

(1.1)

其中$A$表示方程$(1.1)$所对应的迁移算子.之后, 迁移方程解的渐近稳定性态分析和相应迁移算子的谱分布情况已成为人们日益感兴趣的研究工作(部分见文献[2-19]).文献[2]研究了一类具各向异性、粒子单能、介质均匀带零边界条件的中子迁移方程, 采用半群等方法证明了这类迁移算子生成半群的紧性, 当初始条件$\psi_0 \in D(A^2)$时, 证明了该迁移方程解是渐近稳定的.文献[3]在$L_p(1\leq p < +\infty)$空间中研究了板模型中一类具周期和反射边界条件的各向异性、粒子单能、介质均匀的迁移方程, 证明了其相应的迁移算子产生$C_0$半群和其Dyson-Phillips展开式的二阶余项$R_2(t)$在$L_p(1 < p < +\infty)$空间紧以及在$L_1$空间弱紧, 得到了这类迁移算子的谱分析等结果.文献[4]研究了板模型中一类具周期和反射边界条件的各向异性、粒子单能、介质均匀的奇异迁移方程, 当初始条件$\psi_0 \in D(A^2)$时, 证明了该迁移方程解的渐近稳定性.至于更多奇异迁移方程解的构造性研究参阅文献[5-6].文献[7]采用豫解算子等方法研究了板模型中一类具抽象边界条件的各向异性、粒子单能、介质均匀的迁移方程, 证明了对任意的$r\in [0, 1)$, 有

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^r}\|K{(\lambda I- B)^{- 1}}K\| = 0, $

(1.2)

在某带域一致成立, 从而得到了这类迁移算子谱的存在性等结果.之后, 豫解算子方法在迁移方程中得到广泛应用^[8-13].文献[8]在$L_p(1 < p < +\infty)$空间中研究了种群细胞中一类具积分边界条件的增生扩散型迁移方程, 证明了相应迁移算子谱的存在性.文献[12]研究了种群细胞增生中一类具最大边界条件的迁移方程, 证明了这类迁移算子产生的$C_0$半群的Dyson-phillips展开式的第九阶余项的弱紧性, 得到了相应迁移算子谱的存在性等结果.受文献[12]的启发, 本文在$L_1$空间中研究了板模型中一类具抽象边界条件的各向异性、连续能量、非均匀介质的迁移方程, 采用半群理论、构造算子、比较算子和豫解算子等方法证明了这类迁移算子$A_H $产生的$C_0$半群的Dyson-phillips展开式的第九阶余项$R_9(t)$在$L_1$空间中是弱紧的, 得到了相应迁移算子的谱在区域$\Gamma_0$中仅由有限个具有限代数重数的离散本征值组成.当初始条件$\psi_0 \in D(A_H)$时, 证明了该迁移方程的解是渐近稳定的.

下面研究板几何中一类具抽象边界条件下的各向异性、连续能量、非均匀介质的迁移方程的初边值问题

$ \left\{ {\begin{array}{*{20}{l}} {\frac{{\partial \psi (x,v,\mu ,t)}}{{\partial t}} = - \mu \frac{{\partial \psi (x,v,\mu ,t)}}{{\partial x}} - \sigma (x,v)\psi (x,v,\mu ,t)}\\ {\qquad \qquad \qquad \quad + \int_E {\rm{d}} v'\int_{ - 1}^1 k (x,v,\mu ,v',\mu ')\psi (x,v',\mu ',t){\rm{d}}\mu '}\\ {\qquad \qquad \qquad = {B_H}\psi (x,v,\mu ,t) + K\psi (x,v,\mu ,t)}\\ {\qquad \qquad \qquad = {A_H}\psi (x,v,\mu ,t),}\\ {\psi (x,v,\mu ,0) = {\psi _0}(x,v,\mu ),{\psi ^i} = H({\psi ^0}),} \end{array}} \right. $

(1.3)

其中$B_H$表示有界的带有边界算子$H$和微分项的streaming算子, $K $表示带有积分项的扰动算子, $A_H$表示方程$(1.3)$所对应的迁移算子; $x \in [-a, a], 0 < a < + \infty $; $v, v'\in E=[v_m, v_M], 0 < v_m < v_M < +\infty $, 且$v_M$和$v_m$分别表示最大和最小速率; $\mu, \mu'\in [-1, 1] $, 且$\mu$表示粒子速度与$x$轴夹角的余弦值; 函数$k(x, v, \mu, v', \mu')$和$\sigma (x, v)$分别表示散射裂变核和总碰撞率, 且它们分别表示定义在$G=[-a, a]\times E$和$ D \times D_0=([-a, a]\times E\times [-1, 1])\times (E \times [-1, 1]) $上的有界可测函数; $H$为定义在边界空间上的线性算子.

令$X=L_1(D, {\rm d}x{\rm d}v{\rm d}\mu)$表相域$D$上有界可测函数全体按范数

$ \|f{{\|}_{X}}=\int_{-a}^{a}{\int_{E}{\int_{-1}^{1}{|}}}f(x,v,\mu )|\text{d}x\text{d}v\text{d}\mu $

构成的Banach空间.

假设$O_1$:若$H$具有形式

$ H\left( {\begin{array}{*{20}{c}} {{u_1}}\\ {{u_2}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&{{H_{12}}}\\ {{H_{21}}}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{u_1}}\\ {{u_2}} \end{array}} \right), $

(1.4)

其中$H_{12}=\alpha J_1+\beta N_1, H_{21}=\alpha J_2+\beta N_2, $ $\alpha, \beta\in R^+, J_1$和$J_2$都是紧算子, 而且

$ \begin{array}{l} \begin{array}{*{20}{l}} {{N_1}:L_{1, 2}^0 \to L_{1, 1}^i, u(a, v, \mu ) \to {L_1}u( - a, v, \mu ) = u(a, v, \mu ), } \end{array}\\ \begin{array}{*{20}{l}} {{N_2}:L_{1, 1}^0 \to L_{1, 2}^i, u( - a, v, \mu ) \to {L_1}u(a, v, \mu ) = u( - a, v, \mu ).} \end{array} \end{array} $

令$\sigma={\rm ess}\inf\{\sigma (x, v)\}$.对任何Re$\lambda > -\sigma$, 考虑豫解方程$(\lambda I -B_H)\psi=\varphi$, 经过计算可以得到

$ {(\lambda I - {B_H})^{ - 1}} = {\chi _{(0, 1)}}(\mu )\sum\limits_{n \ge 0} {Q_\lambda ^ + } {H_{12}}{(P_\lambda ^ + {H_{12}})^n}D_\lambda ^ + + E_\lambda ^ + + {\chi _{( - 1, 0)}}(\mu )\sum\limits_{n \ge 0} {Q_\lambda ^ - } {H_{21}}{(P_\lambda ^ - {H_{21}})^n}D_\lambda ^ - + E_\lambda ^ -, $

其中算子$P^{-}_{\lambda}, P^{+}_{\lambda}, Q^{-}_{\lambda}, Q^{+}_{\lambda}, D^{-}_{\lambda}, D^{+}_{\lambda}, E^{-}_{\lambda}, E^{+}_{\lambda}$的定义见文献[10].

假设$O_2$:扰动算子$K$在$X$上是正则算子.故可由有限秩算子逼近, 即有

$ {k_n}(x, v, \mu, v', \mu ') = \sum\limits_{i \in I} {{\theta _i}} (x){f_i}(v, \mu ){g_i}(v', \mu '), $

从而得到

$ K\varphi (x, v, \mu ) = \sum\limits_{i \in I} {\int_E {\rm{d}} } v'\int_{ - 1}^1 {{\theta _i}} (x){f_i}(v, \mu ){g_i}(v', \mu ')\varphi (x, v', \mu '){\rm{d}}\mu ', $

(1.5)

其中$\theta_i(\cdot)\in L_{\infty}([-a, a]), f_i(\cdot, \cdot)\in L_{1}(D_0), g_{i}(\cdot, \cdot)\in L_{\infty}(D_0), I$为有限集.下面构造算子$\overline{B_H}$, $\overline{P}_{\lambda}$, $\overline{Q}_{\lambda}$, $\overline{D}_{\lambda}$和$ \overline{E}_{\lambda} $

$ \left\{ {\begin{array}{*{20}{l}} {\overline {{B_H}} :D(\overline {{B_H}} ) \subseteq X \to X, }\\ {\overline {{B_H}} \psi (x, v, \mu ) = - \mu \frac{{\partial \psi (x, v, \mu )}}{{\partial x}} - \sigma \psi (x, v, \mu ), }\\ {D(\overline {{B_H}} ) = \left\{ {\psi \in X\left| {\mu \frac{{\partial \psi }}{{\partial x}} \in X, \psi {|_{{D^i}}} = {\psi ^i} \in L_1^i, \psi {|_{{D^0}}} = {\psi ^0} \in L_1^0, \;{\psi ^i} = H{\psi ^0}} \right.} \right\};} \end{array}} \right. $

(1.6)

$ \left\{ {\begin{array}{*{20}{l}} {\overline {{P_\lambda }} :{X^i} \to {X^0};\overline {{P_\lambda }} \varphi = (\overline {P_\lambda ^ + } \varphi, \overline {P_\lambda ^ - } \varphi ), }\\ {\overline {P_\lambda ^ + } \varphi (a, v, \mu ) = \varphi ( - a, v, \mu )\exp (\frac{{ - 1}}{\mu }\int_{ - a}^a {(\lambda + \sigma )} {\rm{d}}\xi ), \mu \in (0, 1), }\\ {\overline {P_\lambda ^ - } \varphi ( - a, v, \mu ) = \varphi (a, v, \mu )\exp (\frac{1}{\mu }\int_{ - a}^a {(\lambda + \sigma )} {\rm{d}}\xi ), \mu \in ( - 1, 0);} \end{array}} \right. $

(1.7)

$ \left\{ {\begin{array}{*{20}{l}} {\overline {{Q_\lambda }} :{X^i} \to X;\overline {{Q_\lambda }} \varphi = (\overline {Q_\lambda ^ + } \varphi, \overline {Q_\lambda ^ - } \varphi ), }\\ {\overline {Q_\lambda ^ + } \varphi ( - a, v, \mu ) = \varphi ( - a, v, \mu )\exp (\frac{{ - 1}}{\mu }\int_{ - a}^x {(\lambda + \sigma )} {\rm{d}}\xi ), \mu \in (0, 1), }\\ {\overline {Q_\lambda ^ - } \varphi (a, v, \mu ) = \varphi (a, v, \mu )\exp (\frac{1}{\mu }\int_x^a {(\lambda + \sigma )} {\rm{d}}\xi ), \mu \in ( - 1, 0);} \end{array}} \right. $

(1.8)

$ \left\{ {\begin{array}{*{20}{l}} {\overline {{D_\lambda }} :X \to {X^0};\overline {{D_\lambda }} \varphi = (\overline {D_\lambda ^ + } \varphi, \overline {D_\lambda ^ - } \varphi ), }\\ {\overline {D_\lambda ^ + } \varphi (x, v, \mu ) = \frac{1}{\mu }\int_{ - a}^a {\exp } (\frac{{ - 1}}{\mu }\int_{x'}^a {(\lambda + \sigma )} {\rm{d}}\xi )\varphi (x', v, \mu ){\rm{d}}x', \mu \in (0, 1), }\\ {\overline {D_\lambda ^ - } \varphi ( - a, v, \mu ) = \frac{1}{\mu }\int_{ - a}^a {\exp } (\frac{1}{\mu }\int_{ - a}^{x'} {(\lambda + \sigma )} {\rm{d}}\xi )\varphi (x', v, \mu ){\rm{d}}x', \mu \in ( - 1, 0);} \end{array}} \right. $

(1.9)

$ \left\{ {\begin{array}{*{20}{l}} {\overline {{E_\lambda }} :X \to X;\overline {{E_\lambda }} \varphi = (\overline {E_\lambda ^ + } \varphi, \overline {E_\lambda ^ - } \varphi ), }\\ {E_\lambda ^ + \varphi (x, v, \mu ) = \frac{1}{\mu }\int_{ - a}^x {\exp } (\frac{{ - 1}}{\mu }\int_{x'}^x {(\lambda + \sigma )} {\rm{d}}\xi )\varphi (x', v, \mu ){\rm{d}}x', \mu \in (0, 1), }\\ {E_\lambda ^ - \varphi ( - a, v, \mu ) = \frac{1}{\mu }\int_x^a {\exp } (\frac{1}{\mu }\int_x^{x'} {(\lambda + \sigma )} {\rm{d}}\xi )\varphi (x', v, \mu ){\rm{d}}x', \mu \in ( - 1, 0).} \end{array}} \right. $

(1.10)

令$ P(A_H)=\sigma(A_H)\cap \Gamma_0 $为迁移算子$A_H$的谱点集, 其中

$ {\Gamma _0} = \{ \lambda \in {\bf{C}}|{\rm{Re}}\lambda \ge {\lambda _0} + \varepsilon \} (\varepsilon > 0), \\{{\lambda }_{0}}=\left\{ \begin{array}{*{35}{l}} -\sigma , & \|H\|\le 1, \\ -\sigma +\frac{1}{2a}\log (\|H\|), & \|H\|>1. \\ \end{array} \right. $

引理2.1^[11] 设$B_H$是Banach空间$X$上一强连续半群的生成元, 其型为$\omega$, 若$K$是有界算子, 且存在$m\in N, \eta > \omega$, 满足

(1) 对任意的$ {\rm Re}\lambda > \eta, (\lambda I-B_H)^{-1}[K(\lambda I-B_H)^{-1}]^m $是空间$X$上弱紧算子,

(2) 在区域$ \{\lambda \in {\bf C}| {\rm Re}\lambda \geq \eta \} $上, 一致成立

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } \,|\text{Im}\lambda |\|{{(\lambda I-{{B}_{H}})}^{-1}}{{[K{{(\lambda I-{{B}_{H}})}^{-1}}]}^{m}}\|=0, $

则对任意的$ t > 0 $, 可得$ R_{2m+1} $是$X$上的弱紧算子.

定理2.1 若假设$O_1$和$O_2$被满足, 则对任意的$ t > 0 $, 可得$ R_{9}(t) $是空间$X$上的弱紧算子, 从而得到$ P(A_H)$仅由有限个具有限代数重数的离散本征值组成.

证 由于本定理的证明较长, 故分以下5步证明.

第1步 证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } \,|\text{Im}\lambda {{|}^{\frac{1}{2}}}\|K\overline{E_{\lambda }^{+}}K\|=0 $

(2.1)

在$ \Gamma_0$上一致地成立.事实上, 对任意的$\varphi\in X$, 有

$ \overline {E_\lambda ^ + } \varphi (x, v, \mu ) = \frac{1}{\mu }\int_{ - a}^x {\exp } (\frac{{ - 1}}{\mu }[(x-x')\lambda + \int_{x'}^x {\lambda + \sigma } {\rm{d}}\xi])\varphi (x', v, \mu ){\rm{d}}x'. $

(2.2)

做变换$s=\frac{x-x'}{\mu}, t=x-\mu's$, 可以得到做变换$s=\frac{x-x'}{\mu}, t=x-\mu's$, 可以得到

$ K\overline {E_{\lambda, \varepsilon }^ + } K\varphi (x, v, \mu ) = \theta (x)f(v, \mu )\int_E {\rm{d}} v'\int_{ - a}^x {\rm{d}} \mu 'h(v', \frac{{x - t}}{s})\int_\varepsilon ^{ + \infty } {\frac{{{\rm{d}}s}}{s}} \\\times \theta (t)\exp ( - \lambda s - \int_t^x \sigma {\rm{d}}\xi ){\chi _{(x - t, + \infty )}}(s) \\\times \int_E {\rm{d}} v''\int_{ - 1}^1 {\rm{d}} \mu ''g(v'', \mu '')\varphi (t', v'', \mu ''). $

(2.3)

令$K \overline{E^{+}_{\lambda, \varepsilon}}K=A_1A_{\varepsilon}A_2$, 其中

$ {A_1}:{L_1}( - a, a) \to X, {A_1}\varphi (x) = \theta (x)f(v, \mu )\varphi (x), \\{A_2}:X \to {L_1}( - a, a), {A_2}\varphi (x, v, \mu ) = \int_E {\rm{d}} v\int_{ - 1}^1 \theta (x)g(v, \mu )\varphi (x, v, \mu ){\rm{d}}\mu, \\\left\{ {\begin{array}{*{20}{l}} {\overline {{A_\varepsilon }} :{L_1}( - a, a) \to {L_1}( - a, a), }\\ {\overline {{A_\varepsilon }} \varphi (x) = \int_{ - a}^x {\rm{d}} t\int_E {\rm{d}} v\int_\varepsilon ^{ + \infty } {\frac{{ds}}{s}} \exp ( - \lambda s - \int_t^x \sigma {\rm{d}}\xi )h(v, \frac{{x - t}}{s})\varphi (t){\chi _{(x - t, + \infty )}}(s).} \end{array}} \right. $

因为$A_1$, $A_2$一致有界, 所以只需要证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|\overline {{A_\varepsilon }}\| = 0 $

(2.4)

在$\Gamma_0$上一致地成立.由于对任意$\varphi \in L_1(-a, a)$, 利用H\"{o}lder不等式可以得到

$ \|\overline {{A_\varepsilon }}\| \le \int_{ - 2a}^{2a} {\rm{d}} x\int_E {\rm{d}} v|\int_\varepsilon ^{ + \infty } {\exp } [-(\lambda + \sigma-\frac{\varepsilon }{2})s]{l_{x, v, n}}(s)h(v, \frac{x}{s})\varphi (t){\chi _{(x, + \infty )}}(s)|, $

(2.5)

所以由(2.4)-(2.5)式, 只需证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\int_{ - 2a}^{2a} {\rm{d}} x\int_E {\rm{d}} v|\int_\varepsilon ^{ + \infty } {\exp } [-(\lambda + \sigma-\frac{\varepsilon }{2})s] \\\times {l_{x, v, n}}(s)h(v, \frac{x}{s})\varphi (t){\chi _{(x, + \infty )}}(s)| = 0 $

(2.6)

在$\Gamma_0$一致地成立, 其中$l_{x, v, n}(\cdot)$一致收敛于$\varphi_{x, v}(\cdot)$.

下面, 对任意的$ n\in{\bf N} $, 若$x\in(-2a, 2a)$固定且$v\in E$固定, 设$(s_i)_{1\leq i\leq m}$是其支撑集上的一个划分, 并且满足对任意的$ i\in \{ 1, 2, \cdot\cdot\cdot, m-1 \}$, 当$s\in [s_i, s_{i+1})$时有$G_{x, v}(s)=G_{x, v}(s_i)$, 所以可得

$ \int_\varepsilon ^{ + \infty } {\exp } ([-\lambda-(\sigma-t)]s){G_{x, v}}{\rm{d}}s \\= \frac{1}{{\lambda + \sigma }}\sum\limits_{i = 1}^{m - 1} {{G_{x, v}}} ({s_i})\exp ([-\lambda-(\sigma-t)]({s_i} - {s_{i + 1}})). $

(2.7)

从而可得

$ |\int_\varepsilon ^{ + \infty } {\exp } ([-\lambda-(\sigma-t)]s){G_{x, v}}{\rm{d}}s| \le \frac{{2(m - 1)\sup |h( \cdot, \cdot )|}}{{\varepsilon |{\rm{Im}}\lambda {|^{\frac{1}{2}}}}}. $

(2.8)

于是

$ |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\int_{ - 2a}^{2a} {\rm{d}} x\int_E {\rm{d}} v|\int_\varepsilon ^{ + \infty } {\exp } [-(\lambda-\sigma (x-t)s]{\rm{d}}s \cdot {l_{x, v, n}}(s)h(v, \frac{x}{s})\varphi (t){\chi _{(x, + \infty )}}(s)| \\{\rm{ }} \le {\rm{ }}\frac{{8aM(m - 1)\sup |h( \cdot, \cdot )|}}{{\varepsilon |{\rm{Im}}\lambda {|^{\frac{1}{2}}}}}, $

(2.9)

其中$M=v_M -v_m$.因为

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } \frac{{8aM(m - 1)\sup |h( \cdot, \cdot )|}}{{\varepsilon |{\rm{Im}}\lambda {|^{\frac{1}{2}}}}} = 0, $

(2.10)

故(2.1)式获证.

第2步 证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\sum\limits_{n \ge 0} \|{K\overline {Q_\lambda ^ + } {H_{12}}(} \overline {P_\lambda ^ + } {H_{12}})D_\lambda ^ + K\| = 0 $

(2.11)

在$ \Gamma_0$一致地成立.显然, 对任意的$n\in N\setminus \{0\}$, $\overline{P_\lambda^+} J_1$和$\overline{P_\lambda^+} N_1$是不可交换的算子, 而且$(\overline{P_\lambda^+} H_{12})^n$具有如下形式

$ {(\overline {P_\lambda ^ + } {H_{12}})^n} = \sum\limits_{j = 1}^{{2^n}} {{P_j}} . $

(2.12)

又由于

$ \|K\overline {Q_\lambda ^ + } {H_{12}}{P_j}\overline {D_\lambda ^ + } K\| \le \|K\overline {Q_\lambda ^ + } {H_{12}}\| \cdot \|{P_j}\overline {D_\lambda ^ + } K\|. $

(2.13)

所以要证明(2.11)式在$ \Gamma_0$一致地成立, 即证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|{P_j}\overline {D_\lambda ^ + } K\| = 0, $

(2.14)

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|K\overline {Q_\lambda ^ + } {H_{12}}{P_{{2^n}}}\overline {D_\lambda ^ + } K\| = 0 $

(2.15)

同时在$ \Gamma_0$上一致成立, 其中$j=1, 2, \cdot\cdot\cdot, {2^n-1}$.事实上, 因为$J_1$是紧算子的, 所以不妨设$J_1$是秩一算子, 即

$ {J_1}\varphi ( - a, v, \mu ) = \theta (x)f(v, \mu )\int_E {\rm{d}} v\int_0^1 g (v', \mu ')\varphi (a, v', \mu ')|\mu '|{\rm{d}}\mu ', $

(2.16)

其中$f(v, \mu)$和$g(v', \mu')$都是简单可测函数, 所以(2.14)式转化为证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|{J_1}{(\overline {P_\lambda ^ + } {N_1})^k}\overline {D_\lambda ^ + } K\| = 0 $

(2.17)

在$ \Gamma_0$上一致成立.设$\varphi\in X$, 则

$ {J_1}{(\overline {P_\lambda ^ + } {N_1})^k}\overline {D_\lambda ^ + } K\varphi (x, v, \mu ) = \theta (x)f(v, \mu )\int_E {\rm{d}} v'\int_0^1 {\rm{d}} \mu 'g(v', \mu ')f(v', \mu ') \\\times \int_{ - a}^a {\rm{d}} x \cdot \exp [\frac{{-1}}{{\mu '}}\int_{x'}^x {(\lambda-\sigma {\rm{d}}\xi )((2k + 1)a-x)}] \\\times \theta (x')\int_E {\rm{d}} v''\int_{ - 1}^1 g (v'', \mu '')\varphi (x - s\mu ', v'', \mu ''){\rm{d}}\mu ''. $

(2.18)

定义算子

$ {B_1}:\varphi \in X \to \theta (x)f(v, \mu )\int_E {\rm{d}} v\int_0^1 g (v, \mu )\varphi (x, v, \mu ){\rm{d}}\mu \in {L_1}( - a, a), \\{B_2}:\gamma \in \mathbb{R}\to \gamma \eta (v, \mu ) \in L_{1, 1}^i, \\{B_k}:\varphi \in {L_1}( - a, a) \to \int_E {\rm{d}} v\int_0^1 {\rm{d}} \mu f(v, \mu )g(v, \mu )\theta (x) \\\times \int_{ - a}^a {\exp } [\frac{{-1}}{\mu }\int_{x'}^x {(\lambda-\sigma {\rm{d}}\xi )((2k + 1)a-x)}]\varphi (x), $

显然可得

$ {J_1}{(\overline {P_\lambda ^ + } {N_1})^k}\overline {D_\lambda ^ + } K = {B_2}{B_k}{B_1}. $

(2.19)

由于$ \| J_1(\overline{P_\lambda^+} L_1)^k \overline{D^{+}_{\lambda}}K \| \leq\| B_2 \| \cdot \| B_k \| \cdot\| B_1 \|, $而且算子$B_1, B_2$, 和$B_k$都有界, $B_1$和$B_2$与$\lambda$无关, 所以只需证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|{B_k}\| = 0 $

(2.20)

在$ \Gamma_0$上一致成立.设$\varphi\in L_1((-a, a);{\rm d}x), \bar{\varphi}$, 是$\varphi$在$R$上的平凡拓展, 于是$B_k\varphi$可写为

$ {B_k}\varphi = \int_R {{F_\lambda }} ((2k + 1)a - x)\bar \varphi {\rm{d}}x = ({F_\lambda } * \bar \varphi )((2k + 1)a). $

(2.21)

利用Young不等式可得

$ |{B_k}\varphi | \le \|{F_\lambda }\|{_{{L_\infty }(R)}}\|\bar \varphi \|{_{{L_1}( - a, a)}}, $

从而得到

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\theta (x)\int_\varepsilon ^{ + \frac{1}{\varepsilon }} | \int_E {\rm{d}} v\int_0^1 {\rm{d}} \mu f(v, \mu )g(v, \mu )\exp [\frac{{-1}}{\mu }\int_{x'}^x {(\lambda-\sigma {\rm{d}}\xi )}]{\rm{d}}\mu |{\rm{d}}t = 0. $

(2.22)

所以(2.14)式在$\Gamma_0$上一致成立.下面证明(2.15)式也在$\Gamma_0$上一致成立.事实上, 由于$H_{12}=\alpha J_1+\beta N_1, J_1$是$X$上的弱紧算子, 则只需证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|K\overline {Q_\lambda ^ + } {N_1}{P_{{2^n}}}\overline {D_\lambda ^ + } K\| = 0 $

(2.23)

在$\Gamma_0$上一致成立.由于对任意的$\varphi\in X$, 则

$ K\overline {Q_\lambda ^ + } {N_1}(\beta \overline {P_\lambda ^ + } {N_1})\overline {D_\lambda ^ + } K = \theta (x){\beta ^n}f(v, \mu )\int_{ - a}^a {\rm{d}} x\int_E {\rm{d}} v'\int_0^1 g (v', \mu ')f(v', \mu ') \\\times \theta (x')\exp [\frac{{-1}}{{\mu '}}\int_{x'}^x {(\lambda-\sigma )} {\rm{d}}\xi ((2n + 1)a-x)]{\rm{d}}\mu ' \\\times \int_E {\rm{d}} v''\int_{ - 1}^1 g (v'', \mu '')\varphi (x - s\mu ', v'', \mu ''){\rm{d}}\mu ''. $

(2.24)

令$K \overline{Q_\lambda^+} N_1 (\beta \overline{P_\lambda^+} N_1) \overline{D^{+}_{\lambda}}K=E_2 E_n E_1$, 其中

$ {E_1}:\varphi \in X \to \theta (x)\int_E {\rm{d}} v\int_{ - 1}^1 g (v, \mu )\varphi (x, v, \mu ){\rm{d}}\mu \in {L_1}(( - a, a);{\rm{d}}x), \\{E_2}:\varphi \in {L_1}(( - a, a);{\rm{d}}x) \to \theta (x){\beta ^n}f(v, \mu )\varphi (x) \in X, \\{E_n}:\varphi \in {L_1}(( - a, a);{\rm{d}}x) \to \int_{ - a}^a {\rm{d}} x\int_E {\rm{d}} v\int_0^1 f (v, \mu )g(v, \mu ) \\\times \exp [\frac{{-1}}{\mu }\int_{x'}^x {(\lambda-\sigma )} {\rm{d}}\xi ((2n + 1)a-x)]{\rm{d}}\mu \in {L_1}(( - a, a);{\rm{d}}x), $

因为算子$E_1, E_2$, 和$E_n$都有界, 而且$E_1$和$E_2$与$\lambda$无关, 类似于(2.20)式的证明, 同理可以得到

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|{E_n}\| = 0 $

(2.25)

在$\Gamma_0$上一致成立, 即(2.15)式成立, 故(2.11)式在$\Gamma_0$上一致成立.

第3步 证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|K{(\lambda I - \overline {{B_H}} )^{ - 1}}K\| = 0 $

(2.26)

在$\Gamma_0$上一致成立.由于

$ \|K{(\lambda I - \overline {{B_H}} )^{ - 1}}K\|{\rm{ }} \le {\rm{ }}\sum\limits_{n \ge 0} \|{K\overline {Q_\lambda ^ + } {H_{12}}(} \overline {P_\lambda ^ + } {H_{12}}{)^n}\overline {D_\lambda ^ + } K\| + \|K\overline {E_\lambda ^ + } K\| \\+ \sum\limits_{n \ge 0} \|{K\overline {Q_\lambda ^ - } {H_{21}}(} \overline {P_\lambda ^ - } {H_{21}}{)^n}\overline {D_\lambda ^ + } K\| + \|K\overline {E_\lambda ^ - } K\|. $

(2.27)

故要证明(2.26)式在$\Gamma_0$上一致成立, 即证明(2.1)式和(2.11)式以及方程

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|K\overline {E_\lambda ^ - } K\| = 0, $

(2.28)

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\sum\limits_{n \ge 0} \|{K\overline {Q_\lambda ^ - } {H_{21}}(} \overline {P_\lambda ^ - } {H_{21}}{)^n}\overline {D_\lambda ^ - } K\| = 0 $

(2.29)

同时在$\Gamma_0$上一致成立.由于(2.28)式与(2.1)式以及(2.29)和(2.11)式具有相同的结构, 故同理可证明(2.28)和(2.29)式在$\Gamma_0$上一致成立.故(2.26)式在$\Gamma_0$上一致成立.

第4步 证明

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda {|^{\frac{1}{2}}}\|K{(\lambda I - {B_H})^{ - 1}}K\| = 0 $

(2.30)

在$\Gamma_0$上一致成立.由于

$ \|K{(\lambda I - {B_H})^{ - 1}}K\| \le \|K{(\lambda I - \overline {{B_H}} )^{ - 1}}K\|, $

(2.31)

于是由第3步可得到(2.30)式在$\Gamma_0$上一致成立.

第5步 证明对任意的$t > 0$, $R_9(t)$是$X$上的弱紧算子.事实上, 对任意的${\rm Re}\lambda > \lambda_0$, 可得$(\lambda I-B_H)^{-1}K$在$X$空间上的弱紧算子.由于$(\lambda I-B_H)^{-1}$在$\Gamma_0$上一致有界, 故$(\lambda I-B_H)^{-1}[K(\lambda I-B_H)^{-1}]^{m}$在$X$空间上是弱紧算子.令

$ W(\lambda ) = {(\lambda I - {B_H})^{ - 1}}{[K{(\lambda I-{B_H})^{-1}}]^4}, $

(2.32)

可得

$ \|W(\lambda )\| = \|{(\lambda I - {B_H})^{ - 1}}\|{^3}\|{[K(\lambda I-{B_H})K]^2}\|. $

(2.33)

又由于

$ \|{(\lambda I - {B_H})^{ - 1}}\| \le \frac{1}{{{\rm{Re}}\lambda + \sigma }} \le \frac{1}{\varepsilon }, $

(2.34)

所以可得

$ |{\rm{Im}}\lambda | \cdot \|W(\lambda )\| \le \frac{1}{{{\varepsilon ^3}}}[|{\rm{Im}}\lambda {|^{\frac{1}{2}}} \cdot \|K(\lambda I-{B_H})K\|{^2}]. $

(2.35)

于是由第4步知, 存在$m=4$使得

$ \mathop {\lim }\limits_{|{\rm{Im}}\lambda | \to + \infty } |{\rm{Im}}\lambda | \cdot \|W(\lambda )\| = 0. $

(2.36)

最后, 由引理2.1可得$R_9(t)$是$X$上的弱紧算子, 从而得到$P(A_H)$仅由有限个具有限代数重数的离散本征值组成.即此定理获证.

由定理2.1可设

$ P({A_H}) = \{ {\lambda _1}, {\lambda _2}, {\lambda _3}, \cdot \cdot \cdot, {\lambda _n}, \cdot \cdot \cdot \}, $

(2.37)

并假设这些本证值按照实部递减排列, 从而可得

$ {\rm Re}\lambda_1>{\rm Re}\lambda_2>{\rm Re}\lambda_3>\cdot\cdot\cdot>{\rm Re}\lambda_n>\cdot\cdot\cdot>-\lambda_0. $

引理2.2^[17] 假设$O_1$和$O_2$被满足, 则对任意的$\varepsilon > 0 $, 有

$ \|{(\lambda I - {A_H})^{ - 1}}(I - P)\|{_X} $

在区域$ \{\lambda \in {\bf C}| {\rm Re}\lambda \geq {\rm Re}\lambda_{n+1}+\varepsilon \} $上一致有界, 其中$P=P_1+P_2+P_3+\cdot\cdot\cdot+P_n$, $P_i$表示由$\lambda_i (1\leq i \leq n)$对应的射影算子.

定理2.2 若假设$O_1$和$O_2$被满足, 则对任意的$\varepsilon > 0, t > 0 $, 存在$M > 0$使得

$ \|{{V}_{H}}(t)-\sum\limits_{i=1}^{n}{\exp }({{\lambda }_{i}}t)\exp ({{D}_{i}}t){{P}_{i}}\|\le M\exp (\varepsilon +\text{Re}{{\lambda }_{n+1}})t, $

(2.38)

其中$V_H(t)$表示由迁移算子$A_H$生成的$C_0$半群, $D_i$表示由$\lambda_i (1\leq i \leq n)$对应的幂零算子.

证 由假设$O_1$和半群理论知, $U_H(t)$是$X$上的$C_0$正半群, 而且半群的谱型为$-\sigma$, 所以对任意的$\varepsilon > 0, t > 0 $, 存在$M_1 > 0$使得

$ \| U_H(t)\| \leq M_1 \exp(-\sigma+\varepsilon)t. $

(2.39)

由半群生成定理知, $A_H$生成的$C_0$半群$V_H(t)$, 而且其渐近表达式为

$ V_H(t)=\sum_{j\geq 0}U_j(t), $

(2.40)

其中$U_0(t)=U(t), $

$ U_j(t)\psi=\int_0^t U(s)K U_{j-1}(t-s)\psi {\rm d}s. $

(2.41)

于是由文献[15]知, 对任意的$\psi\in X, j\in {\bf N}$, 当${\rm Re}\lambda > -\lambda_0$时, 可以得到

$ \int_0^\infty \exp(-\lambda t)U_j(t)\psi {\rm d}t= (\lambda I-B_H)^{-1}[(\lambda I-B_H)^{-1}]^j \psi, $

(2.42)

$ \| U_j(t)\| \leq \exp((-\sigma+\varepsilon)t) M_1^{j+1}\frac{\| K \| ^j t^j}{j!} . $

(2.43)

令

$ F(t) = {V_H}(t)(I - P) - \sum\limits_{j = 0}^3 {{U_j}} (t), $

(2.44)

则对任意的$\psi\in X, F(t)$在$ [0, +\infty)$上是强连续的, 于是可以得到

$ F(t)(I - P)\psi = {V_H}(t)(I - P)\psi - \sum\limits_{j = 0}^3 {{U_j}} (t)(I - P)\psi . $

(2.45)

由(2.42)式可得

$ \int_0^\infty {\exp } (- \lambda t)F(t)(I- P)\psi {\rm{d}}t \\= {\rm{ }}{(\lambda I- {A_H})^{ - 1}}(I - P)\psi - \sum\limits_{j = 0}^3 {{{(\lambda I - {B_H})}^{ - 1}}} {[K{(\lambda I-{B_H})^{-1}}]^j}(I - P)\psi . $

(2.46)

又因为

$ {(\lambda I - {A_H})^{ - 1}} = \sum\limits_{j = 0}^\infty {{{(\lambda I - {B_H})}^{ - 1}}} {[K{(\lambda I - {B_H})^{ - 1}}]^j}, $

(2.47)

所以

$ \int_0^\infty {\exp } ( - \lambda t)F(t)(I - P)\psi {\rm{d}}t \\= \sum\limits_{j = 4}^\infty {{{(\lambda I - {B_H})}^{ - 1}}} {[K{(\lambda I-{B_H})^{-1}}]^j}(I - P)\psi = \Xi (\lambda ), $

(2.48)

其中$ \Xi(\lambda)=(\lambda I-B_H)^{-1}[K(\lambda I-B_H)^{-1}]^3 K(\lambda I-A_H)^{-1}(I-P)\psi. $显然

$ \| \Xi(\lambda)\| \leq\| (\lambda I-B_H)^{-1}\| ^2 \| [K(\lambda I-B_H)^{-1}K \| ^2 \| (\lambda I-A_H)^{-1}(I-P)\| \cdot \| \psi\| . $

(2.49)

令$\Gamma_{n, \varepsilon}=\{\lambda \in {\bf C}| {\rm Re}\lambda > \beta_{n, \varepsilon}-\frac{\varepsilon}{2}\} $, 其中$\beta_{n, \varepsilon}={\rm Re}\lambda_{n+1}+\varepsilon $.由定理2.1知, 存在$ M_2 > 0$使得对充分大的$|{\rm Im}\lambda |$有

$ \| [K(\lambda I-B_H)^{-1}K \| \leq \frac{M_2}{|{\rm Im}\lambda |^{\frac{2}{3}}} $

(2.50)

在$\Gamma_{n, \varepsilon}$上一致成立.由引理2.2知$ \| (\lambda I-A_H)^{-1}(I-P)\| $在$\Gamma_{n, \varepsilon}$上一致有界.于是由(2.49)式知, 存在$ M_3 > 0$使得对充分大的$|{\rm Im}\lambda |$有

$ \|\Xi (\lambda )\| \le \frac{{{M_3}}}{{|{\rm{Im}}\lambda {|^{\frac{4}{3}}}}} $

(2.51)

在$\Gamma_{n, \varepsilon}$上一致成立.因为$ \Xi(\lambda) $在$\Gamma_{n, \varepsilon}$上是解析的, 所以由文献[15]知, 对任意的$ t > 0 $, 存在连续函数

$ \alpha(t)=\frac{1}{2\pi{\rm i}}\int_{\tau -{\rm i}\infty}^{\tau +{\rm i}\infty} \exp(\lambda t) \Xi(\lambda){\rm d}\lambda, $

(2.52)

其中$\tau > \max \{0, \beta_{n, \varepsilon} \}$.由(2.48)式和Laplace变换可得$F(t)(I-P)=\alpha(t)$.于是(2.52)式可化为

$ \alpha (t) = \frac{1}{{2\pi {\rm{i}}}}\mathop {\lim }\limits_{y \to + \infty } (\int_{{\beta _{n, \varepsilon }} - {\rm{i}}y}^{{\beta _{n, \varepsilon }} + {\rm{i}}y} {\exp } ( - \lambda t)\Xi (\lambda ){\rm{d}}\lambda \\+ \int_{{\beta _{n, \varepsilon }}}^\tau {\exp } ((x + {\rm{i}}y)t)\Xi (x + {\rm{i}}y){\rm{d}}x - \int_{{\beta _{n, \varepsilon }}}^\tau {\exp } ((x - {\rm{i}}y)t)\Xi (x - {\rm{i}}y){\rm{d}}x). $

(2.53)

由Lebesgue控制收敛定理和(2.51)式可得

$ \mathop {\lim }\limits_{y \to + \infty } \int_{{\beta _{n, \varepsilon }}}^\tau {\exp } ((x + {\rm{i}}y)t)\Xi (x + {\rm{i}}y){\rm{d}}x - \int_{{\beta _{n, \varepsilon }}}^\tau {\exp } ((x - {\rm{i}}y)t)\Xi (x - {\rm{i}}y){\rm{d}}x = 0. $

(2.54)

从而得到

$ F(t)(I - P)\psi = \frac{1}{{2\pi {\rm{i}}}}\int_{{\beta _{n, \varepsilon }} - {\rm{i}}\infty }^{{\beta _{n, \varepsilon }} + {\rm{i}}\infty } {\exp } (\lambda t)\Xi (\lambda ){\rm{d}}\lambda . $

(2.55)

于是由Fubini定理得到

$ \|F(t)(I - P)\psi\| \le \frac{1}{{2\pi {\rm{i}}}}\exp ({\beta _{n, \varepsilon }}t)\int_{ - \infty }^{ + \infty } \|{\Xi (} {\beta _{n, \varepsilon }} + \zeta {\rm{i}})\|{\rm{d}}\zeta . $

(2.56)

由(2.49)-(2.50)式可得

$ \int_{ - \infty}^{+ \infty} \| \Xi( \beta_{n, \varepsilon}+\zeta{\rm i} )\| {\rm d}\zeta \leq M_4\| \psi \|, $

(2.57)

其中$M_4 > 0$, 由(2.56)式可得

$ \| F(t)(I-P)\psi \| \leq \frac{M_4}{2\pi{\rm i}}\exp(\beta_{n, \varepsilon}t)\| \psi \| . $

(2.58)

又由(2.43)-(2.45)式和(2.58)式可得

$ \|{V_H}(t)(I - P)\|{\rm{ }} \le {\rm{ }}\|F(t)(I - P)\| + {U_k}(t)(I - P)\| \\\le \frac{{{M_4}}}{{2\pi {\rm{i}}}}\exp (({\rm{Re}}{\lambda _{n + 1}} + \varepsilon )t) + \sum\limits_{k = 0}^3 {\exp } (( - {\lambda _0} + \varepsilon )t)M_1^{k + 1}\frac{{\|K\|{^k}{t^k}}}{{k!}} \\\le {\rm{ }}M\exp (({\rm{Re}}{\lambda _{n + 1}} + \varepsilon )t), $

(2.59)

其中$ M=\frac{M_4}{2\pi{\rm i}}+ \sup\limits_{t\geq 0 } \Big(\exp(({\rm Re}\lambda_{n+1}-\lambda_0)t) \sum\limits _{k=0}^3 M_1^{k+1} \frac{\| K\| ^k t^k}{k!} \Big), $从而此定理获证.