A spectral theory of linear operators on Gelfand triplets (New Developments in Geometric Mechanics)

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1 A spectral theory of linear operators on Gelfand triplets MI (Institute of Mathematics for Industry, Kyushu University) (Hayato CHIBA) chiba\copyright imikyushu-uacjp Dec 21, $\frac{du}{dt}=tu$ (11) $u$ $X$ $X$ $X$ $0$ $X$ $X$ $*1$ Landau [7] Gelfand 3 $*1$ sectorial operator $X$ Banach ( ) [10] sectorial operator [12] $S(t)$ Hilbert $e^{\sigma(t)t}=\sigma(s(t))$ $e^{\sigma(t)t}\subset\sigma(s(t))$ $S(t)$ $\sigma(t)$ $D(T)$ $X$ $S(t)$ $X$ $D(T)$ $D(T)$ [12]

2 79 Gelfand 3 $\searrow$ $L^{2}(R)$ $ix$ $\mathcal{m}$ : $\phi(x)\mapsto ix\phi(x)$ : $\sigma(\mathcal{m})=ir$ $(\lambda-\mathcal{m})^{-1}$ $(L^{2}(R)$ ) ( 0 ) $L^{2}(R)$ $\phi,$ $\psi\in L^{2}(R)$ $((\lambda-\mathcal{m})^{-1}\phi,\psi)$ $(( \lambda-\mathcal{m})^{-1}\phi, \psi)=\int\frac{1}{\lambda-ix}\phi(x)\psi(x)dx$ $1/(\lambda-ix)$ $\phi,\psi$ $\lim_{{\rm Re}(\lambda)arrow+0}\int\frac{1}{\lambda-ix}\phi(x)\psi(x)dx$ $*2$ $\phi,\psi$ $((\lambda-\mathcal{m})^{-1}\phi, \psi)$ $\int\frac{1}{\lambda-ix}\phi(x)\psi(x)dx+2\pi\phi(i\lambda)\psi(i\lambda)$ $\phi$ $\psi$ $R(\lambda;\phi,\psi)=\{\begin{array}{ll}\int_{R}\frac{1}{\lambda-ix}\psi(x)\phi(x)dx+2\pi\psi(i\lambda)\phi(i\lambda) ({\rm Re}(\lambda)<0),\lim_{{\rm Re}(\lambda)arrow+0}\int_{R}\frac{1}{\lambda-ix}\psi(x)\phi(x)dx (\lambda\in ir),\int_{r}\frac{1}{\lambda-ix}\psi(x)\phi(x)dx ({\rm Re}(\lambda)>0),\end{array}$ $*2$ $\lim_{{\rm Re}(\lambda)arrow+0}\int\frac{{\rm Re}(\lambda)}{{\rm Re}(\lambda)^{2}+({\rm Im}(\lambda)-x)^{2}}\phi(x)\psi(x)dx=\pi\phi({\rm Im}(\prime l))\psi({\rm Im}(\prime l))$

3 80 $R$ $\phi$ $\psi$ $R$ $X$ $L^{2}(R)$ $X$ $X$ $\phi\in X$ $R(\lambda;\phi, \psi)$ $\phi\mapsto R(\lambda;\phi, \psi)$ $X$ $R(\lambda;\bullet, \psi)$ $X$ $\psi\mapsto R(\lambda;\bullet, \psi)$ $X$ $X$ : $\langle A(\lambda)\psi \phi\rangle=\{\begin{array}{ll}\int_{r}\frac{1}{\lambda-ix}\psi(x)\phi(x)dx+2\pi\psi(i\lambda)\phi(i\lambda) ({\rm Re}(\lambda)<0),\lim_{{\rm Re}(\lambda)arrow+0}\int_{R}\frac{1}{\lambda-ix}\psi(x)\phi(x)dx (\lambda\in\iota R),\int_{R}\frac{1}{\prime t-ix}\psi(x)\phi(x)dx ({\rm Re}(\lambda)>0)\end{array}$ (12) $\langle\cdot \cdot\rangle$ $X $ $X$ Dirac $A(\lambda)=(\lambda-\mathcal{M})^{-1}$ $\mathcal{m}$ : $\mathcal{m}$ $(_{\ell}\lambda-\mathcal{m})^{-1}$ $L^{2}(R)$ $X$ $X $ (X - ) $L^{2}(R)$ $X$ $L^{2}(R)$ $L^{2}(R)$ $L^{2}(R)$ Hilbert $X\subset L^{2}(R)\subset X$ (13) 3 Gelfand 3 rigged Hilbert space Hilbert $(\lambda-t)^{-1}$ $X$ $X$ $X$ $(\lambda-t)^{-1}$ Riemann $R_{\lambda}$ Riemar

4 81 $X$ Banach $C^{0}$ $e^{tt}$ Laplace $e^{tt}= \lim_{yarrow\infty}\frac{1}{2\pi i}\int_{x-iy}^{x+iy}e^{\lambda t}(\lambda-t)^{-1}d\lambda$ (14) $x$ $($ $1(a))$ (a) (b) Fig 1 $\cross$ ( ) 1(b) : $\phi(x)\mapsto ix\phi(x)$ $e^{\prime tt}(\lambda-t)^{-1}$ $\mathcal{m}$ $X$ Laplace $e^{tt}= \lim_{yarrow\infty}\frac{1}{2\pi i}\int_{x-iy}^{x+iy}e^{\lambda t}a(\lambda)d\lambda$ (15) ( 2 Riemam ) $(T \phi)(x)=ix\phi(x)+k\int_{r}\phi(x)dx$ (16)

5 82 $L^{2}(R,g(x)dx)$ (11) ( $K>0$ $g$ ) [3,4] $K_{c}=\Subset/\pi$ 4 $\mathcal{m}$ $K$ $K$ $K>K_{c}$ $\lambda=\lambda$( $K=K_{c}$ $0<K<K_{c}$ $\phi,$ $\psi$ ( $((\lambda-t)^{-1}\phi, \psi)$ ), 2 2 Riemam Laplace 2 Riemam (11) $u(t)$ $X$ [3, 4] Fig 2 $K$ $K>K_{c}$ 0 K K $<$ $<$ Riemalm D$()$ R$()$ 2 Gelfand 3 $X$ $C$ Hausdorff $X $ $X $ $X$ $\mu\in$ $\mu(\phi)$ $\phi\in X$ $a,$ $b\in C,$ $\phi,$ $\psi\in X$ $\mu,$ $\xi\in X $ $\langle\mu \phi\rangle$ Dirac

6 83 $\langle\mu a\phi+b\psi\rangle=\overline{a}\langle\mu \phi\rangle+\overline{b}\langle\mu \psi\rangle$, (21) $\langle a\mu+b\xi \phi\rangle=a\langle\rho r \phi\rangle+b\langle\xi \phi\rangle$, (22) $X $ ( ) ( ) $*$ $*$ $\phi\in X$ $\langle\mu J \phi\ranglearrow\langle\mu \phi\rangle$ $\{\mu j\}\subset X $ $\mu\in$ $X$ $\langle\mu j \phi\ranglearrow\langle/l \phi\rangle$ $\{\mu J\}\subset$ $\mu\in X $ $(\cdot,$ $\cdot)$ Hilbert $X$ $H$ $X$ Hilbert $\subset X $ 21 Hausdorff $X$ Hilbert $X$ 3 $X\subset H\subset X $ (23) rigged Hilbert space, Gelfand 3 $i:^{c}harrow X $ ; $\psi\in$ $i(\psi)$ $\langle\psi $ $i(\psi)(\phi)=\langle\psi \phi\rangle=(\psi, \phi)$, $\phi\in X$ (24) $i$ $\prime Harrow X $ : $X$ $i$ ( ) $X$ $i$ (Tr\ eves [18]) 21 Gelfand 3 Schwartz Gelfand [8] $X=C_{0}^{\infty}(R^{m}),$ $H=L^{2}(R^{m})$ Gelfand 3 Schwartz 3 Gelfand 3 Chiba [5] 31 $C$ Hilbert $H$ $\{E(B)\}_{B\in B}$ $H$ $H= \int_{r}\omega de(\omega)$ :

7 84 Fig $E[\psi, \phi](\omega)$ 3 $\Omega$ $K$ $T:=H+K$ Schr\"odinger $H$ $K$ $\Omega\subset C$ $\tilde{i}$ $\tilde{i}$ $I$ ( 3) $T=H+K$ $C$ $\Omega$) (Xl) $X(\Omega)$ (X2) $X(\Omega)$ (X3) $X(\Omega)$ (Xl), (X2) Gelfand 3 $X(\Omega)\subset H\subset X(\Omega) $ (31) ( Treves [18] ), Frechet Banach Hilbert $C^{\infty}$ Frechet $*3$ Montel $*4$ Banach-Steinhaus, $*3$ $*4$ Montel Montel $C^{\infty}$ $C^{\infty}$ $C^{\infty}$ Montel Schwartz Montel Montel [9,13] Banach-Steinhaus $X$ $\chi$ $\chi$ 4 $A$ (i) $A$ (ii) $A$ (iii) $A$

8 85 [5] $H$ $E(B)$ (X4) $\phi\in X(\Omega)$ $E[\phi, \phi](\omega)$ $\lambda\in I\cup\Omega$ (X5) $(E(B)\phi, \phi)$ $I$ $\Omega\cup I$ $E[\cdot,$ $\cdot](\lambda)$ : $X(\Omega)\cross X(\Omega)arrow C$ (X4) $\phi,$ $\psi\in X(\Omega)$ $E[\phi, \psi](\omega)$ : $(E(B)\phi, \psi)$ $I$ $d(e(\omega)\phi, \psi)=e[\phi, \psi](\omega)d\omega$, $\omega\in I$ (32) $E[\phi, \psi](\omega)$ $\omega\in I\cup\Omega$ $I$ $\omega\in R$ $ix(\omega)$ $X(\Omega)$ $X(\Omega) $ : $ix(\omega)arrow X(\Omega) $ $\langle A(\lambda)\psi \phi\rangle=\{\begin{array}{ll}\int_{r}\frac{1}{\lambda-\omega}e[\psi,\phi](\omega)d\omega+2\pi\sqrt{-1}e[\psi,\phi](\lambda) (\lambda\in\omega),\lim_{yarrow-0}\int_{r}\frac{1}{x+\sqrt{-1}y-\omega}e[\psi,\phi](\omega)d\omega (\lambda=x\in I),\int_{R}\frac{1}{\lambda-\omega}E[\psi,\phi](\omega)d\omega ({\rm Im}(\lambda)<0),\end{array}$ (33) ${\rm Im}(\lambda)<0$ $\langle A(\lambda)\psi \phi\rangle$ $\{{\rm Im}(\lambda)<0\}\cup\Omega\cup I$ $\langle A(\lambda)\psi \phi\rangle=((\lambda-h)^{-1}\psi,\phi)$ $H$ $A(,l)$ $X(\Omega) $ $\Omega$ $(\lambda-h)^{-1}$ $*5$ $H$ 1 $(\Omega$ $)$ $\Omega$) $X(\Omega) $ $A(\lambda)\circ i$ : $X(\Omega)arrow X(\Omega) $ $*6$ $Q$ $Q $ : $D(Q )arrow X(\Omega) $ ; $Q $ $D(Q )$ $X(\Omega)$ $C$ $\phi\mapsto\langle\mu Q\phi\rangle$ $\Omega$) $\mu\in X(\Omega) $ $*5$ (iv) $A$ Treves[18] $X$ Banach $\Omega$ $\Omega$ Riemann 4 Banach [1] $*6X(\Omega)$ Banach

9 86 $\langle Q \mu \phi\rangle=\langle\mu Q\phi\rangle$ Hilbert $Q$ $H$ $Q$ $Q^{*}$ $(Q\phi, \psi)=(\phi, Q^{*}\psi)$ $\Omega$ $Q^{*}$ $)$ $(Q^{*}) $ $Q^{\cross}$ $Q^{\cross}=(Q^{*}) $ $i\circ Q=Q^{\cross}\circ i _{D(Q)}$ $Q$ $Q^{\cross}$ $H$ $K$ (X6) $H$ $X(\Omega)$ $HY\subset X(\Omega)$ (X7) $K$ $X(\Omega)$ $Y$ $H$ - $K^{*}$ $X(\Omega)$ (X8) $\lambda\in\{{\rm Im}(\lambda)<0\}\cup I\cup\Omega$ (X6) (X7) $D(H^{\cross})$ id( $K,$ $H^{\cross},K^{\cross}$ $K$ $H^{\cross},K^{\cross},$ $T^{\cross}$ $K^{\cross}A(\lambda)iX(\Omega)\subset ix(\omega)$ $\Omega$ $)$ $H,K$ $\Omega$) $T^{\cross}$ $X(\Omega) $ $H$- $K(\lambda-H)^{-1}$ (X7) $(\lambda-h)^{-1}$ (X8) $(\lambda-t)v=0$ $T=H+K$ $X(\Omega) $ $(\lambda-h)^{-1}$ $(id-(\lambda-h)^{-1}k)v=0$ 31 $\lambda\in\omega UIU\{\lambda {\rm Im}(\lambda)<0\}$ $(id-a(\lambda)k^{\cross})\mu=0$ (34) $\mu\in X(\Omega)$ $\mu$ $K^{\cross}$ $(id-k^{\cross}a(\lambda))k^{\cross}\mu=0$ (35) $K^{\cross}\mu=0$ (34) $\mu=0$ $id-k^{\cross}a(\lambda)$ $K^{\cross}A(\lambda)$ $ix(\omega)$ $(\Omega$ $)$ (X8) well-defined 32 $\mu$ $T^{\cross}\mu=\lambda\mu$ (36)

10 $\blacksquare$ 87 $D(\lambda-H^{\cross})\supset R(A(\lambda))$ $ix(\omega)$ $(\lambda-h^{\cross})a(\lambda)=id:ix(\omega)arrow$ $(\lambda-h^{\cross})(id-a(\lambda)k^{\cross})\mu=(\lambda-h^{\cross}-k^{\cross})\mu=(\lambda-t^{\cross})\mu=0$ $T^{\cross}$ $T^{\cross}$ $X(\Omega) $ $T^{\cross}$ $C$ $T^{\cross}$ 32 $n=1,2,$ $\cdots$ $A^{(n)}(\lambda)$ : $ix(\omega)arrow X(\Omega) $ $\langle A^{(n)}(\lambda)\psi \phi\rangle=\{\begin{array}{l}\int_{r}\frac{1}{(\lambda-\omega)^{n}}e[\psi, \phi](w)d\omega+2\pi\sqrt{}=\text{ } \frac{(-1)^{n-1}}{(n-1)!}\frac{d^{n-1}}{dz^{n-1}} _{z=\lambda}e[\psi, \phi](z), (\lambda\in\omega),\lim_{yarrow-0}\int_{r}\frac{1}{(x+\sqrt{-1}y-\omega)^{n}}e[\psi, \phi](\omega)d\omega, (\lambda=x\in I),\int_{R}\frac{1}{(\lambda-\omega)^{n}}E[\psi, \phi](\omega)d\omega, ({\rm Im}(\lambda)<0)\end{array}$ $\langle A^{(n)}(\lambda)\psi \phi\rangle$ $\Omega$ $((\lambda-h)^{-n}\psi, \phi)$ $A^{(1)}(\lambda)$ 33 (i) $(\lambda-\mathscr{x})^{n}a^{(f)}(\lambda)=a^{0-n)}(\lambda)$, (ii) $A^{(J)}(\lambda)(\lambda-H^{\cross})^{n}=A^{0-n)}(\lambda)$ $(\lambda-h^{\cross})\mu\in ix(\omega)$ $j\geq n\geq 0$ $A^{0)}(\lambda)$ $A^{(0)}(\lambda):=id$ $A(\lambda)(\lambda-H^{\cross})\mu=\mu$ (iii) $\frac{d^{j}}{d\lambda^{j}}\langle A(\lambda)\psi \phi\rangle=(-1)^{j}j!\langle A^{0+1)}(\lambda)\psi \phi\rangle,$ $j=0,1,$ $\cdots$ (iv) $\psi\in X(\Omega)$ $A(\lambda)\psi$ (37) $A( \lambda)\psi=\sum_{j=0}^{\infty}(\lambda_{0}-\lambda ya^{(i+1)}(\lambda_{0})\psi,$ (38) $\Omega$)

11 -Hx)kBO) $\blacksquare$ 88 (i),(ii) (iii) $\langle A(\lambda)\psi \phi\rangle$ (iii) $\langle A(\lambda)\psi \phi\rangle=\sum_{j=0}^{\infty}(\lambda_{0}-\lambda\dot{y}\langle A^{0+1)}(\lambda_{0})\psi \phi\rangle,$ (39) $A(\lambda)\psi$ $X(\Omega)$ (iv) Banach-Steinhaus $(\lambda-t)^{n}v=0$ $n=2$ $(\lambda-h-k)(\lambda-h-k)v=(\lambda-h)^{2}(id-(\lambda-h)^{-2}k(\lambda-h))\circ(id-(\lambda-h)^{-1}k)v=0$ $(\lambda-h)^{2}$ $(\lambda-h)^{-n}$ $(id-(\lambda-h)^{-2}k(\lambda-h))\circ(id-(\lambda-h)^{-1}k)v=0$ $A^{(n)}(\lambda)$ $(id-a^{(2)}(\lambda)k^{\cross}(\lambda-h^{\cross}))\circ(id-a(\lambda)k^{\cross})\mu=0$ $B^{(n)}(\lambda)$ $D(B^{(n)}(\lambda))\subset X(\Omega) arrow X(\Omega) $ : $B^{(n)}(\lambda)=id-A^{(n)}(\lambda)K^{\cross}(\lambda-H^{\cross})^{n-1}$ (310) $2$) $(\lambda)b^{(1)}(\lambda)\mu=0$ $B^{(n)}(\lambda)$ $A^{(n)}(\lambda)K^{\cross}(\lambda-H^{\cross})^{n-1}$ ( $(\lambda$$)=$ l -0 $(\lambda$ $\cross$ $)$( -H )k, $j>k$ (311) 34 $V_{\lambda}= \bigcup_{m\geq 1}Ker\mathscr{A}^{m)}(\lambda)\circ B^{(m-1)}(\lambda)\circ\cdots\circ B^{(1)}(\lambda)$ (312) $\dim V_{\lambda}$ $KerB^{(1)}(\lambda)$ 35 $\mu\in V_{\lambda}$ $M$ $(\lambda-t^{\cross})^{m}\mu=0$ $\bigcup_{m\geq 1}Ker(\lambda-T^{\cross})^{m}$ $(\Omega$ $)$ $\bigcup_{m\geq 1}Ker(\lambda-T^{\cross})^{m}$

12 89 33 $R_{\lambda}=(\lambda-T)^{-1}$ $R_{\lambda}\psi=(\lambda-H)^{-1}(id-K(\lambda-H)^{-1})^{-1}\psi$ (313) $X(\Omega) $ $(\lambda-h)^{-1}$ $\hat{\omega}=\omega\cup I\cup\{_{1}l {\rm Im}(\lambda)<0\}$ 36 $(id-k^{\cross}a(\lambda))^{-1}$ $X(\Omega) $ $R_{\lambda}=A(\lambda)\circ(id-K^{\cross}A(\lambda))^{-1}=(id-A(A)K^{\cross})^{-1}\circ A(\lambda)$, 2 $id-k^{\cross}a(\lambda)$ $(\Omega$$)$ $(id-a(\lambda)k^{\cross})a(\lambda)=a(\lambda)(id-k^{\cross}a(\lambda))$ $ix(\omega)arrow$ R : $\lambda\in\hat{\omega}$ (314) $id-a(\lambda)k^{\cross}$ $R(A(\lambda))$ $X(\Omega) $ $A(\lambda)\circ i$ $\text{ _{}\lambda}\circ i:x(\omega)arrow$ $\lambda\in\hat{\omega}$ 37 2 $\hat{\rho}(t)$ $V_{\lambda}\subset\hat{\Omega}$ $\lambda \in V_{\lambda}$ (i) $\Omega$) $\Omega$) $R_{\lambda }\circ i$ $X(\Omega) $ (ii) $\psi\in X(\Omega)$ $id-k^{\cross}a(\lambda)$ $)$ $\{\text{ _{}\lambda }\circ i(\psi)\}_{\lambda }$ $\in$ $\Omega$) $c\hat{\tau}(t):=\hat{\omega}\backslash \hat{\rho}(t)$ $(\hat{\tau}_{p}(t)$ $\lambda\in\hat{\sigma}(t)$ ( $\hat{\sigma}_{r}(t)$ $\circ i$ $\Omega$) $\lambda\in\hat{\sigma}(t)$ $\hat{\sigma}_{c}(t)=\hat{\sigma}(t)\backslash (\hat{\sigma}_{p}(t)\cup\hat{\sigma}_{r}(t))$ $*7$ ; $\hat{\rho}(t)$ $\hat{\rho}(t)$ $\Omega$) Banach Banach $i^{-1}k^{\cross}a(\lambda)i$ [19, 14] $\Omega$) Banach $X(\Omega)$ $\lambda\in\hat{\rho}(t)$ ( $id-i^{-1}k^{\cross}a(\lambda)i$ $\Omega$ $)$ 315) $\hat{\rho}(t)$ $*7$ Banach-Steinhaus

13 $R_{\lambda\ovalbox{\tt\small REJECT} h}i(\psi)$ i $\blacksquare$ (i) $\psi\in X(\Omega)$ (ii) ${\rm Im}(\lambda)<0$ $R_{\lambda}i\psi$ (ii) $\hat{\rho}(t)$ $R_{\lambda}\circ i=i\circ(\lambda-t)^{-1}$ ${\rm Im}(\lambda)<0$ $\Omega$) - $\psi,$ $\phi\in X(\Omega)$ $\langle R_{\lambda}\psi \phi\rangle=((\lambda-t)^{-1}\psi, \phi)$ $\langle R_{\lambda}\psi \phi\rangle$ $((\lambda-t)^{-1}\psi,\phi)$ $\psi_{\lambda}=i^{-1}(id-k^{\cross}a(\lambda))^{-1}i(\psi)$ $R_{\lambda+h}i(\psi)-R_{\lambda}i(\psi)=(A(\lambda+h)-A(\lambda))i(\psi_{\lambda})+R_{\lambda+h}i\circ i^{-1}k^{\cross}(a(\lambda+h)-a(\lambda))i(\psi_{\lambda})$ $harrow 0$ $X(\Omega) $ $0$ $A(\lambda)\circ i$ 1 2 $\lambda+$4 $i^{-1}k^{\cross}a(\lambda)i$ $\{R_{\lambda}, \circ i\}_{\lambda \in V_{\lambda}}$ (ii) Banach-Steinhaus $R_{\lambda+h}i$ $harrow 0$ $i^{-1}k^{\cross}a(\lambda)i$ $\Omega$ $)$ $(\psi$ $)$ $h$ $R_{\lambda}i(\psi)$ $\Omega$) 39 (i) $(\lambda-t^{\cross})\circ R_{\lambda}=id _{ix(\omega)}$ $R_{\lambda}\circ(\lambda-T^{\cross})\mu=\mu$ (ii) $\mu\in X(\Omega) $ $(\lambda-t^{\cross})\mu\in ix(\omega)$ (iii) $T^{\cross}\circ R_{\lambda}=R_{\lambda}\circ T^{\cross}$ 33 (iii) well-defined 34 $\Sigma\subset\hat{\sigma}(T)$ $\{\lambda {\rm Im}(\lambda)<0\}$ $\Pi_{\Sigma}:iX(\Omega)arrow X(\Omega) $ $\gamma\subset\omega\cup I\cup$ $\Pi_{\Sigma}\phi=\frac{1}{2\pi\sqrt{-1}}\int_{\gamma}R_{\lambda}\phi d\lambda$, $\phi\in ix(\omega)$, (315)

14 91 $\Pi_{\Sigma}\circ\Pi_{\Sigma}$ $*8$ Pettis $\Pi_{\Sigma}$ $\Pi_{\Sigma}$ $\Sigma$ 310 $\Pi_{\Sigma}(iX(\Omega))\cap(id-\Pi_{\Sigma})(iX(\Omega))=\{0\}$ $ix(\omega)\subset\pi_{\sigma}(ix(\omega))\oplus(id-\pi_{\sigma})(ix(\omega))\subset X(\Omega) $ (317) $\phi\in X(\Omega)$ $\mu_{1},\mu_{2}\in X(\Omega) $ $\phi$ $i(\phi)=\langle\phi =\mu_{1}+\mu_{2}$, $\mu_{1}\in\pi_{\sigma}(ix(\omega)),$ $\mu_{2}\in(id-\pi_{\sigma})(ix(\omega))$ (318) $\Pi_{\Sigma}$ 311 $T^{\cross}$- : $\Pi_{\Sigma}\circ T^{\cross}=T^{\cross}\circ\Pi_{\Sigma}$ $\lambda_{0}$ 312 $\Pi_{0}$ $\lambda_{0}$ $V_{0}=$ $\bigcup_{m\geq 1}KerB^{(m)}(\lambda_{0})\circ\cdot\cdot\cdot$ $\circ B^{(1)}(\lambda_{0})$ $\lambda_{0}$ $\Pi_{0}iX(\Omega)$ $\Pi_{0}iX(\Omega)=\nabla_{0}$ $\Pi\circ\Pi=\Pi$ $\Pi$ ( ) 312 $\lambda_{0}$ $E_{-1}=-\Pi_{0}$ $id=(\lambda-t^{\cross})\circ \text{ _{}\lambda}$ $= \sum_{j}^{\infty}=-\infty(\lambda_{0}-\lambda)^{j}e_{j}$ $\{E_{j}\}_{j}$ $E_{-1}$ 35 $\hat{\sigma}_{p}(t)\subset\sigma_{p}(t^{\cross})$ $C_{-}=\{{\rm Im}(\lambda)<0\}$ $\hat{\sigma}(t)$ $\sigma(t)$ (i) $\hat{\sigma}(t)\cap C_{-}\subset\sigma(T)\cap C_{-}$, $\hat{\sigma}_{p}(t)\cap C_{-}\subset\sigma_{p}(T)\cap C_{-}$ $\sigma_{p}(t)$ $\sigma(t)$ $*8$ $x$ $X $ $s$ Hausdorff Borel $f:sarrow X $ $\phi\in X$ $\mu$ $s$ $\langle I(]) \phi\rangle=\int_{s}\langle f \phi\rangle d\mu$ (316) $I( \int)\in$ $l \omega=\int_{s}f^{d\mu}$ $f$ Pettis $x$ $f$ Pettis [5] $f$ Pettis

15 $\gamma$ $\gamma$ $\gamma$ $\gamma$ $\blacksquare$ 92 (ii) $\Sigma\subset C_{-}$ $\sigma(t)$ $\sigma(t)$ $\hat{\sigma}(t)$ $\lambda\in C_{-}$ $\sigma(t)$ $\prime 1\in\hat{\sigma}(T)$ $\lambda\in C_{-}$ $\circ i=i\circ(\lambda-t)^{-1}$ $(\lambda-t)^{-1}$ ( 38) (i) $X(\Omega)$ $\hat{\sigma}(t)$ $\hat{\sigma}(t;x(\omega))$ $X_{1}(\Omega)$ $X_{2}(\Omega)$ $(X1)\sim(X8)$ 2 $\hat{\sigma}(t;x_{1}(\omega)),\hat{\sigma}(t;x_{2}(\omega))$ 314 $X_{2}(\Omega)$ $X_{1}(\Omega)$ $X_{2}(\Omega)$ $X_{1}(\Omega)$ (i) $\hat{\sigma}(t;x_{2}(\omega))\subset\hat{\sigma}(t;x_{1}(\omega))$ $\Sigma$ (ii) $\hat{\sigma}(t;x_{1}(\omega))$ $\hat{\sigma}(t;x_{1}(\omega))$ $\hat{\sigma}(t;x_{2}(\omega))$ $\hat{\sigma}(t;x_{1}(\omega))$ $\lambda\in\hat{\sigma}(t;x_{2}(\omega))$ $R_{\lambda}$ $X_{1}(\Omega)$ $X_{1}(\Omega) $ $X_{2}(\Omega)$ $X_{2}(\Omega) $ $\Pi_{\Sigma}$ (i) (ii) $\Pi_{\Sigma}iX_{1}(\Omega)\neq\{0\}$ $X_{2}(\Omega)$ $X_{1}(\Omega)$ $\blacksquare$ $\Pi_{\Sigma}iX_{2}(\Omega)\neq\{0\}$ 2 $\Omega$) $*$9, $X_{1}$ $X_{2}$ $L$ $*$ 10, $U\subset X_{1}$ $LU\subset X_{2}$ $L=L(\lambda)$ $L(\lambda)$ $U$ $X_{1}$ $L(\lambda)$ Banach $L(\lambda)$ ( $U$ ) $L$ $U\subset X_{1}$ $LU\subset X_{2}$ $*9$ $*10$ Schr\"odinger [15] complex deformation [11] Gelfand 3 Banach (resonance pole)

16 $\cross$ $\blacksquare$ 93 $L=L(\lambda)$ $L(\lambda)$ $U$ Banach $X_{1}$ $L(\lambda)$ ( $(\lambda$ $)$ )L $X_{2}$ Montel Montel ( ) ( ) $i^{-1}k^{\cross}a(\lambda)i$ $\lambda\in\hat{\omega}$ 315 $(\lambda$ $U_{\lambda}\subset\hat{\Omega}$ $i^{-1}$ K $)$ i : $X(\Omega)arrow$ $X(\Omega)$ $id-$ $\lambda \in U_{\lambda}$ $i^{-1}k^{\cross}a(\lambda)i$ $\Omega$ $)$ $\lambda\not\in\hat{\sigma}(t)$ $R_{\lambda}\circ i=a(\lambda)\circ i\circ(id-i^{-1}k^{\cross}a(\lambda)i)^{-1}$ $\{(id-i^{-1}k^{\cross}a(\lambda )i)^{-1}\psi\}_{\lambda \in V_{1}}$ $A(\lambda)\circ i$ $\Omega$) $\lambda \mapsto(id-i^{-1}k^{\cross}a(\lambda )i)^{-1}\psi$ $\lambda \in V_{\lambda}$ $id-i^{-1}k^{\cross}a(\lambda)i$ $X(\Omega)$ Banach Neumam Banach Neumann Bmyn [2] $X(\Omega)$ $\lambda\in\hat{\rho}(t)$ $i^{-1}k^{\cross}a(\lambda)i$ Banach $X(\Omega)$ $\Omega$ $id-i^{-1}k^{\cross}a(\lambda)i$ $)$ $i^{-1}k^{\cross}a(\lambda)i$ 316 $i^{-1}k^{\cross}a(\lambda)i:x(\omega)arrow X(\Omega)$ $\lambda\in\hat{\omega}$ $D\subset\hat{\Omega}$ (i) $D$ $\hat{\sigma}_{p}(t)$ $\hat{\omega}$ (ii) 312 (iii) $\hat{\sigma}_{c}(t)=\hat{\sigma}_{r}(t)=\emptyset$ $X(\Omega)$ Banach Riesz-Schauder $X(\Omega)$ Banach Riesz-Schauder (Ringrose [16]),

17 $I$ ( $I$ ) $\lambda_{0}\in I$ ( ) $\lambda_{0}$ $\mathcal{p}_{0}\phi=\lim_{\epsilonarrow-0}\sqrt{-1}\epsilon\cdot(\lambda_{0}+\sqrt{-1}\epsilon-t)^{-1}\phi$, $\phi\in H$, (319) $\Pi_{0}$ $i\circ \mathcal{p}_{0}=\pi_{0}\circ i$ $\Omega$) $\mathcal{p}_{0}h\neq\emptyset$ $\Pi_{0}iX(\Omega)\neq\emptyset$ $\sigma_{p}(t)\subset\hat{\sigma}_{p}(t)$ $\blacksquare$ $\hat{\sigma}_{p}(t)$ $C^{0}$- $e^{\sqrt{-1}tt}$ $\sqrt{-1}t=\sqrt{-1}(h+k)$ Laplace $(e^{\sqrt{-1}tt} \psi,\phi)=\frac{1}{2\pi\sqrt{-1}}\lim_{xarrow\infty}\int_{-x-\sqrt{-1}y}^{x-\sqrt{-1}}\mathcal{y}e^{\sqrt{-1}\lambda t}((\lambda-t)^{-1}\psi, \phi)d\lambda$, $x,y\in R$, (320) $tarrow\infty$ 38 $\phi,$ $\psi\in X(\Omega)$ $(e^{\sqrt{-1}tt} \psi,\phi)=\frac{1}{2\pi\sqrt{-1}}\lim_{xarrow\infty}\int_{-x-\sqrt{-1}}^{x-\sqrt{-1}}ye^{\sqrt{-1}\lambda t}\langle R_{\lambda}\psi y \phi\rangle d\lambda$, (321) 316 $\langle R_{\lambda}\psi \phi\rangle$ $\Pi_{0}$ $\lambda_{0}$ $M$ $\frac{1}{2\pi\sqrt{-1}}\int_{\gamma_{0}}e^{\sqrt{-1}\lambda t}\langle R_{\lambda}\psi \phi\rangle d\lambda=\sum_{k=0}^{m-1}e^{\sqrt{-1}\lambda_{0}t}\frac{(-\sqrt{-1}t)^{k}}{k!}\langle(\lambda_{0}-t^{\cross})^{k}\pi_{0}\psi \phi\rangle$, 0 $(\sqrt{-1}t$

18 95 $(e^{\sqrt{-1}tt}\psi,\phi)$ ), $e^{\sqrt{-1}tt}\psi$ Landau [7], Schr\"odinger [11, 15] $\lambda_{0}$ $\mu_{0}\in X(\Omega) $ $(e^{\sqrt{-1}tt})^{\cross}=((e^{\sqrt{-1}tt})^{*}) $ $(e^{\sqrt{-1}\tau t})^{\cross}\mu_{0}=e^{\sqrt{-1}\lambda_{0}t}\mu_{0}$ $\mu_{0}$ $\mu_{0}$ $\Omega$) $\Omega$) $T>0$ $\epsilon>0$ $\phi_{0}\in X(\Omega)$ $0\leq t\leq T$ $ \langle(e\sqrt{}=$ $Tt)^{\cross}\phi_{0} \psi\rangle-\langle(e^{\sqrt{-1}\tau t})^{\cross}\mu_{0} \psi\rangle <\epsilon$, $0\leq t\leq T$ $(e^{\sqrt{-1}tt}\phi_{0}, \psi)\sim e^{\sqrt{-1}\lambda_{0^{f}}}\langle\mu_{0} \psi\rangle$, (322) $tarrow\infty$ 4 41 [3,4] $g_{1}(z)$ $-1<\omega<1$ $g_{1}(\omega)>0$ $g(\omega)$ $g(\omega)=\{\begin{array}{ll}0 (\omega<-1),g_{1}(\omega) (-1<\omega<1),0 (\omega>1),\end{array}$ (41) ${}^{t}h=l^{2}(r,g(\omega)d\omega)$ $L^{2}$ $H$ $\sigma$( $H$ $(H\phi)(\omega)=\omega\phi(\omega)$ $supp(g)=[-1,1]$ $(E(\omega)\psi, \phi):=e[\psi,\phi](\omega)=\{\begin{array}{ll}0 (\omega<-1),\psi(\omega)\overline{\phi(\omega)}g_{1}(\omega) (-1<\omega<1),0 (\omega>1),\end{array}$ (42) $X$ $L^{2}(R,g(\omega)d\omega)$

19 96 $g(\omega)$ $\psi,\phi\in X$ $((\lambda-h)^{-1}\psi, \phi)$ $\omega<-1$, $\omega>1$ $E[\psi, \phi](\omega)$ $((\lambda-h)^{-1}\psi, \phi)$ $((\lambda-h)^{-1}\psi, \phi)$ $((\lambda-h)^{-1}\psi, \phi)$ $0$ $-1<\omega<1$ $\langle A(\lambda)\psi \phi\rangle=\int_{r}\frac{1}{\lambda-\omega}\psi(\omega)\overline{\phi(\omega)}g_{1}(\omega)d\omega+2\pi\sqrt{-1}\cdot\psi(\lambda)\overline{\phi(\lambda)}g_{1}(\lambda)$, (43) $-1<\omega<1$ $+1$ $n$ $((\lambda-h)^{-1}\psi, \phi)$ $\int_{r}\frac{1}{\lambda-\omega}\psi(\omega)\overline{\phi(\omega)}g_{1}(\omega)\pi n\cdot\psi(\lambda)\overline{\phi(\lambda)}g_{1}(\lambda)$, $-1$ $(\lambda-h)^{-1}$ $X$ $X $ $\pm 1$ Riemann $P_{0}(\omega)\equiv 1\in L^{2}(R,g(\omega)d\omega)$ $K$ $(K\phi)(\omega)=-\sqrt{-1}\kappa(\phi, P_{0})P_{0}(\omega)$ $\kappa>0$ $T:=H+K$ $\sqrt{-1}$ $g$ (16) $K$ $H$ $\sigma_{c}(t)=[-1,1]$ $\int_{r}\frac{1}{\lambda-\omega}\mathscr{a}\omega)d\omega-\frac{\sqrt{-1}}{k}=0$, (44) $\lambda=x+\sqrt{-1}y,$ $x,y\in R$ $\int_{r}\frac{x-\omega}{(x-\omega)^{2}+y^{2}}\mathscr{a}\omega)d\omega=0$, $\int_{r}\frac{y}{(x-\omega)^{2}+y^{2}}g(\omega)d\omega=-\frac{1}{\kappa}$ (45) $g(\omega)$ $x=0$ $\kappa>0$ $K$ $x=0$ $K$ $\kappa$ $yarrow-o$ $\lim_{yarrow-0}\int_{r}\frac{y}{\omega^{2}+f}g(\omega)d\omega=-\pi g(0)=-\frac{1}{k}$ (46) $Karrow(\pi g(0))^{-1}$ $\sigma_{c}(t)=[-1,1]$ $\lambda(\kappa)arrow 0$ $\kappa=(\pi g(0))^{-1}$

20 97 $X$ 316 $\hat{\sigma}_{c}(t)=\hat{\sigma}_{r}(t)=\emptyset$ 2 Riemann 2 Riemann $\int_{r}\frac{1}{\lambda-\omega}g_{1}(\omega)d\omega+2\pi\sqrt{-1}\cdot g_{1}(\omega)-\frac{\sqrt{-1}}{k}=0$ (47) (44) $\kappaarrow(\pi \mathscr{a}0))^{-1}$ (44) (47) $\partial u/\partial t=\sqrt{-1}$tu $e^{\sqrt{-1}tt}$ l 36 Laplace 4 Fig 4 $\gamma $ $\gamma$ Laplace [-1, 1] Riemam Riemann 2 $\pm 1$ $\gamma$ 36 $g(\omega)$ $g(\omega)$ $\omega\in R$ ( (42) 2 ), $\gamma$ $u(t)$ $tarrow\infty$ $0$ [3,4]

21 98 42 Schr\"odinger [6] Schr\"odinger $R^{m}$ $T=-\Delta+V$ $\Delta$ ( ) ${}^{t}h=l^{2}(r^{m})$ $V$ $V:R^{m}arrow C$ $-\Delta,$ $V$ $H,$ $K$ $H$ $( \lambda-h)^{-1}\psi(x)=\frac{1}{(2\pi)^{m/2}}\int_{r^{m}}\frac{1}{\lambda- \xi ^{2}}e^{\sqrt{-1}x\cdot\xi}F[\psi](\xi)d\xi$, $F$ $S^{m-1}\subset R^{m}$ Fourier $m-1$ $\xi\in R^{m}$ $\xi=r\omega,$ $r\geq 0,$ $\omega\in S^{m-1}$ $(\lambda-h)^{-1}\psi(x)$ $( \lambda-h)^{-1}\psi(x)=\frac{1}{(2\pi)^{m/2}}\int_{0}^{\infty}\frac{1}{\lambda-r}(\int_{s^{m-1}}\frac{\sqrt{}\mu^{-2}}{2}e^{\sqrt{-1}\sqrt{r}x\cdot\omega}\mathcal{f} [\psi](\sqrt{\gamma}\omega)d\omega)dr$, (48) $\arg(\lambda)=0$ $H$ $\{\lambda -2\pi<\arg(\lambda)<0\}$ $L^{2}$(Rm)- it $f(z):=f[\psi](\sqrt{z}\omega)$ $\frac{1}{(2\pi)^{m/2}}\int_{0}^{\infty}\frac{1}{\lambda-r}(\int_{s^{m-1}}\frac{\sqrt{r}^{m-2}}{2}e^{\sqrt{-1}\sqrt{r}x\cdot\omega}\mathcal{f}[\psi](\sqrt{r}\omega)d\omega)dr$ $+ \frac{\pi\sqrt{-1}}{(2\pi)^{m/2}}\sqrt{\lambda}m-2\int_{s^{m-1}}e^{\sqrt{-1}\sqrt{\lambda}x\cdot\omega}\mathcal{f} [\psi](\sqrt{\lambda}\omega)d\omega$, (49) $V$ $X(\Omega)$ $a>0$ $V$ $e^{2a x }V(x)\in L^{2}(R^{m})$ (410) $a>0$ $X(\Omega):=L^{2}(R^{m}, e^{2a x }dx)$ $X(\Omega) $ $L^{2}(R^{m}, e^{-2a x }dx)$ $T=-\Delta+K$ $(X1)\sim(X8)$, $\psi\in L^{2}(R^{m}, e^{2a x }dx)$ $F[\psi](\sqrt{\lambda}\omega)$ $\psi\in L^{2}(R^{m}, e^{2a x }dx)$ Gelfand 3 $L^{2}(R^{m},e^{2a x }dx)\subset L^{2}(R^{m})\subset L^{2}(R^{m}, e^{-2a x }dx)$ (411) 316 $r$ $F[\psi](r\omega)$ $\{r\in C -a<{\rm Im}(r)<a\}$ Riemam $P(a)=\{\lambda -a<{\rm Im}(\sqrt{\lambda})<a\}$ Riemann $z=0$ ) $m$ [6] (49) ( [6]

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