Notes on spectral clusters for semiclassical Schrodinger operators (Spectral and Scattering Theory and Related Topics)

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(1)29. 数理解析研究所講究録第2023巻 2017年 29-34. Notes. on. clusters for semiclassical. spectral. Yoshihisa Center for Mathematical. Modeling. Toyonaka. Miyanishi and Data. 560. \leftrightarrow. January. Schrödinger operators. Science, Osaka University. 8531, Japan. 20 2016. Abstract In this short note, we discuss the relationships between eigenvalues of Schrödinger operators and periodic trajectories of classical mechanics. For a Hamiltonian function H(x,p) : T^{*}\mathrm{R}^{n}\rightarrow \mathrm{R}\cup\{\pm\infty\} , let \hat{H}\equiv. self‐adjoint Weyl type pseudo‐differential operator and \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(H) be the spectrum of \hat{H}. consists of only eigenvalues, we define the (semiclassical) essential spectrum by. Op_{h}^{W}(H(x,p)). Ue. a. $\Lambda$_{h}(E, c)=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\hat{H})\cap[E-c, E+\mathrm{c}]. If. difference. D $\sigma$(\hat{H})\equiv\overline{\{\frac{E_{i}(h)-E_{J}(h)}{h}|E_{\mathrm{t} (h),E_{j}(h)\in$\Lambda$_{h}(E,c)\} \subset \mathrm{R} where. \displaystle\frac{\wedg }{\ cdot\}^{ ve }\mathrm{s}\mathrm{s}. mems. the set of. including Hamiltonians near. E. or. with. D $\sigma$(\hat{H})=\mathrm{R}.. accumulating points as h\rightarrow 0 We prove the srcalled Helton type theorem singular potentials, that is, either every classical Hamiltonian flow is periodic .. Introduction. 1 Let. first recall the Helton theorem. For. us. (M,g). ,. we. set the classical mechanics and. (QP). (CP) where \triangle denotes the and. [8]. Laplacian and. a compact oriented, smooth Riemanman quantum mechanics by. \left{begin{ary}l -\triangle\primeu_{j}(x)=$\lambd$_{j}u (x),\ {u_j}(x),$\lambd$_{j}\:mathr{E}\mathr{i}\mathr{g}\mathr{e}\mathr{n}\mathr{f}\mathr{u}\mathr{n}\mathr{c}\mathr{}\mathr{i}\mathr{o}\mathr{n}\mathr{e}\mathr{x}\mathr{p}\mathr{}\mathr{n}\mathr{s}\mathr{i}\mathr{o}\mathr{n}, \end{ary}\ight. \left{\begin{ar y}{l X_{H}=(\frac{\partilH}{\partilp},-\frac{\partilH}{\partilx}),\ exptX_{H}:S^{*}M\rightarowS^{*}M:\mathrm{G}\mathrm{e}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}, \end{ar y}\right. H(x,p)=\sqrt{g_{s\mathrm{t}}(p,p)}\in C^{\infty}(T^{*}M). H(xp)=\sqrt{g_{st}(pp)}=\sqrt{g^{l}Jp_{i}p_{J}}\in C^{\infty}(T^{*}\mathrm{R}^{n}). that either. \mathrm{a}\mathrm{n}\mathrm{d}\rightar ow three. D $\sigma$(\sqrt{-\triangle})=\mathrm{R}. ss means. examples. (Example 1). the set of. or. every. geodesic. accumulating points. on. curve. .. We note that. n‐dimensional. manifold. \displaystyle \triangle=\frac{1}{\sqrt{g} \frac{ $\theta$}{\partial x}\{\sqrt{g}g^{ $\iota$}J\frac{\partial}{\partial x_{j} \}. local charts. Under these circumstances, Helton of (CP) is closed. Here. in R. To understand the. meaning. of this. theorem,. we. proved. shall. see. :. Let. (S^{2}, g_{s\mathrm{t} ). be. a. standard 2-\dim sphere in \mathrm{R}^{3}. .. Then. $\lambda$_{j}=j(j+1). with. multiplicity 2j+1. and. D $\sigma$(\sqrt{-\triangle})=\mathrm{Z}. (Example 2). Let. (M, g). be. a. \displaystyle \sqrt{$\lambda$_{j} =j+\frac{1}{2}+O(\mathrm{j}^{-1})j\in \mathrm{N} (Example 3). Let. 2-\dim Zoll surface with. period. 2 $\pi$. .. and. (\mathrm{R}^{2}/(2 $\pi$ \mathrm{Z})^{2}g_{f^{la\mathrm{t} }). It is known. (See. e.g.. [9,. Lemma. D $\sigma$(\sqrt{-\triangle})=\mathrm{Z}. be. a. 2-\dim flat torus. Then. D $\sigma$(\sqrt{-\triangle})=\mathrm{R}.. $\lambda$_{\mathcal{J}^{k}},=j^{2}+k^{2}jk\in \mathrm{N}. and. so. 29.2.1]). that.

(2) 30. Every geodesic flow is periodic in Example 1 and 2, however, we find some geodesic curves are not closed in Example 3. These faithfully reflect properties of the difference specturm D $\sigma$(\sqrt{-\mathrm{A} ) We also would like to mention some recent results for compact manifold cases. For the case of magnetie Schrödinger operators (which is called Bochner Laplacian), the analogous result is given by R. Kuwabara [11], and if the periodic points of Hamiltonian with a smooth scalar potential have measure 0 on the energy surface H^{-1}(e) T. Tate [15] proved .. ,. D_{e} $\sigma$(\sqrt{-\triangle})=\mathrm{R}. It is. our. emphasized that manifolds. are. compact and the potentials. allows. us. to treat. Hydrogen. atoms. (SP). (CP) H(xp). where. including singular potentials and celestial mechanics. To do \mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}_{\rangle}. on we. in the above theorems. So. Euclidean spaces. Such set (SP) and (CP) by. situation. a. \left{\begin{ar y}{l \hat{H}u_j(x,h)=E_{j}(h)u_{J}(x,h)\ {u_j}(x,h)E_{\mathrm{J}(h)\: mathrm{E}\mathrm{i}\ athrm{g}\ athrm{e}\ athrm{n}\mathrm{f}\ athrm{u}\mathrm{n}\mathrm{c}\ athrm{}\ athrm{i}\ athrm{o}\ athrm{n}\mathrm{s}\mathrm{a}\ thrm{n}\mathrm{d}\mathrm{e}\ athrm{i}\ athrm{g}\ athrm{e}\ athrm{n}\mathrm{v}\ athrm{a}\ thrm{l}\ athrm{u}\mathrm{e}\ athrm{s}, \end{ar y}\right. \left{\begin{ar y}{l X_{H}=(\frac{\partilH}{\partilp}-\frac{$\thea$H}{\partilx}),\ exptX_{H}:T^{*}\mathrm{R}^Vb\rightarowT^{*}\mathrm{R}^n:\mathrm{H}\mathrm{a}\mthrm{ }\mathrm{i}\ athrm{l}\ athrm{t}\ athrm{o}\mathrm{n}\mathrm{i}\ athrm{a}\mthrm{n}\mathrm{f}\ athrm{l}\ athrm{o}\mathrm{w}, \end{ar y}\right.. T^{*}\mathrm{R}^{n}\rightarrow \mathrm{R}\cup\{\pm\infty\}. :. singular. not. are. purpose is to consider Hamiltonians. $\Lambda$_{h}(Ec)=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\hat{H})\cap[E-cE+c] semi‐classical difference spectrum. denotes. be defined. can. Hamiltonian function.. a. consists of. only eigenvalues by. and. (See §2),. Under suitable conditions. \#$\Lambda$_{h}(E, c)\rightarrow\infty. as. h\rightarrow 0. .. Thus the. D $\sigma$(\hat{H})\equiv\overline{\{\frac{E_{l}(h)-E_{J}(h)}{h}|$\lambda$_{i}(h)$\lambda$_{j}(h)\in$\Lambda$_{h}(E,c)\} ^{\mathrm{S}\mathrm{S} \subset \mathrm{R}. Our result is E. or. analogous. to the Helton. D $\sigma$(\hat{H})=\mathrm{R} (Theorem 2.5).. theorem,. Here the. Semiclassical operators. 2. A(x,p)\in L_{loc}^{1}(T^{*}\mathbb{R}^{n}). Let. be. a. symbol.. that is, either every complete Hamiltonian flow is near \mathrm{E} will be explained in Sect. 3.. periodic. near. terminology on. \mathbb{R}^{n}. We define the. Weyl type pseudo‐differential operators by:. Definition 2.1.. \displaystyle \hat{A}f(x)\equiv Op_{h}^{W}(A)f(x)\equiv\frac{1}{(2 $\pi$ h)^{n} \int_{T^{*}\mathrm{R}^{n} e^{\mathrm{a}\underline{x}_{h} A\perp-\text{∽}y(\frac{x+y}{2},p)f(y)dydp Followings. are. the. (Example 4). f(x)\in C_{0}^{\infty}(\mathbb{R}^{n}). .. :. .. H(x,p)=|p-A(x)|^{2}+V(x)\displaystyle \Rightarrow Op_{h}^{W}(H)=|\frac{h}{i}\nabla-A(x)|^{2}+V(x). (Example 6). assume. typical examples of Weyl type peudodifferential operators. H(xp)=xp\displaystyle \in C^{\infty}(T^{*}\mathrm{R}^{n}\rangle \Rightar ow Op_{h}^{W}(H)=\frac{h}{2i}(x\partial_{x}+\partial_{x}x). (Example 5). We. for. .. H(xp)=\displaystyle \frac{1}{2}|p|^{2}-\frac{1}{|x|}\in L_{loc}^{1}(T^{*}\mathrm{R}^{3}) \Rightar ow Op_{h}^{W}(H)=-\frac{h^{2} {2}\triangle-\frac{1}{|x|}.. that. (A1) \hat{H}_{h}=Op_{h}^{W}(H) is essentially self‐adjoint in L^{2}(\mathrm{R}^{n}) for small h>0. \langle \mathrm{A}2)$\Lambda$_{h}(E, c)\equiv \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\hat{H}_{h})\cap[E-c, E+c] consists of asymptotically infinite many eigenvalues for small h>0.. Our. guiding principle. In these cases,. (A1). Definition 2.2. is based. and. (A2). (Semiclassical. on. are. harmonic. satisfied. oscillators, hydrogen. (See. for instance. essential difference spectrum. [4, §4] near. atoms and. and. magnetic Schrödinger operators.. [13]).. E ). Under the. assumptions. (A1). and. (A2),. D $\sigma$(\displaystyle \hat{H})\equiv\frac{\wedge}{\{\frac{E_{i}(h)-E_{j}(h)}{h}|E_{i}(h),E_{J}(h)\in$\Lambda$_{h}(E\overline{c)\} \mathrm{s}\mathrm{s} where. -\ovalbox{\t \small REJECT} \mathrm{s}. h\rightarrow 0). means .. the set of. accumulating points. as. h\rightarrow 0. (i.e. a\in D $\sigma$(\hat{H}). means. that. \displaystyle \exists\frac{E_{l}(h)-E,(h)}{h}\rightar ow a. as.

(3) 31. In this note. we. further. that. assume. \langle A3) H(x, p)=\displaystyle \sum_{| $\alpha$|\leq m}a_{ $\alpha$}(x)p^{ $\alpha$} : real valued. (A4) \partial_{x}^{ $\beta$}a_{ $\alpha$}(x)\in L_{lo\mathrm{c} ^{1}(\mathbb{R}^{n}) \forall| $\beta$|\leq| $\alpha$|. (A5) \exists finite set K\in \mathbb{R}^{n} and p>0 such that. a_{ $\alpha$} is. analytic. in. G_{K}\equiv\{|{\rm Im} x|< $\rho$, {\rm Re} x\not\in K\} and for. some. C>0 and. M>0,. |a_{ $\alpha$}(x)|\leqq C(1+|x|)^{M}. in. G_{K}.. \langle A6) \exp tX_{H} : H^{-1}(E-c, E+c)\rightarrow H^{-1}(E-c, E+c) is almost complete. (i.e. \exists S\subset H^{-1}(E-c+E+c) such that the Liouville measure of S^{\mathrm{c} is 0 We. as. use. the notion of. Gevrey class. the set of all functions. f\in C^{\infty}( $\Omega$). and \exp tX_{H} is complete on S ) s\geqq 1 the Gevrey class G^{S} (of index s) is defined čompact subset K there exists a C=C_{f,K} satisfying. Given $\Omega$\subset T^{*}\mathbb{R}^{n} and. :. such that for every. .\cdot. ,. \displaystyle \max_{x\in K}|\partial^{\mathrm{a} f(x)|\leqq C^{| $\alpha$|+1}(| $\alpha$!|)^{s}, \forall $\alpha$\in \mathbb{Z}_{+}^{n}, | $\alpha$|=$\alpha$_{1}+\ldots+$\alpha$_{n}. For s>1, G_{0}^{ $\epsilon$}( $\Omega$)=G^{S}( $\Omega$)\cap C_{0}^{\infty}( $\Omega$) contains non‐zero functions. It is also known [1] that nice partitions of unity are constructed in suitable s>1 Under the assumptions (\mathrm{A}1)\sim(\mathrm{A}6) , we have Egorov type theorem for .. Gevrey class symbols (See Lemma 2.3. Let $\Omega$ be. \overline{ $\Omega$}\backslash (K\times \mathrm{R}^{n}). .. If. a. [2]).. e.g.. bounded open subset of T^{*}\mathrm{R}^{n} such that St is \exp tX_{H} invariant for every t\in \mathrm{R} and and A is Gevrey s>1 then. suppA \subset $\Omega$. e^{-\frac{eR}{h} Op_{h}^{W}(A)e^{\frac{\backslash $\iota$ H}{h} =Op_{h}^{W}( \exp tX_{H})^{*}A) \mathrm{m}\mathrm{o}\mathrm{d} h. By integrating with respect. Corollary. 2.4. Under the. to t ,. same. we. obtain. assumptions of Lemma 2.3, if. \overline{f}(t)\in C_{0}^{\infty}(\mathbb{R}). then. A_{J}^{R}\displaystyle \equiv\int_{-R}^{R}\overline{f}(t)e^{-\frac{l\mathrm{t}\mathrm{A} {h} Op_{h}^{W}(A)e^{\frac{ $\iota$ B}{h} dt=Op_{h}^{W}(\int_{-R}^{R}\tilde{f}(t)(\exp tX_{H})^{*}Adt) \mathrm{m}\mathrm{o}\mathrm{d} h. where A.. \tilde{f}(t). The. [3].. denotes the Fourier transform of. analogy. H^{-1}((E-c, E+c)). singular potentials. is. Proof.. Let. Gevrey. \overline{f}\in C_{0}^{\infty}(\mathbb{R}_{t}). class. (\exp tX_{H})^{*}A(x,p)=A(\exp tX_{H}(xp)). [16].. symbol A. (\mathrm{A}1)\sim(\mathrm{A}6). ,. Either. be. proof of Theorem a. Fourier transform of. We denote the with. suppA. proved by. is the. pull‐Uack. of. M. Combescure and D. Robert. :. periodic.. Outline of the. Stones theorem. and. of Helton theorem for the smooth Hamiltonians is. Our purpose is to treat. Theorem 2.5. Under the assumptions. 3. f(t). D $\sigma$(\hat{H})=\mathrm{R} or every complete Hamiltoman flow on. 2.5. f(x) Assuming (A2), .. spectral decomposition by. \subset H^{-1}(E-cE+c). we. have. e^{\frac{l$\iota$R}{h} =\displaystyle\sum_{j}e\mathrm{f}^{E_{g}(h)t}P_{\mathcal{J}. unitary operators near. E. .. Thus for. a. e^{\frac{tR}{\hslash}. by. suitable. ,. \displaystyle\hat{A}_{f}\equiv\int_{\mathb {R}_{t} \overline{f}(t)e^{-\frac{$\iota$B}{h} Op_{h}^{W}(A)e^{\frac{\mathrm{t}H}{b} dt. =\displaystyle \sum_{i,j}\int_{\mathbb{R}_{t} .:. (1) .. (2).

(4) 32. If. $\sigma$\not\in D $\sigma$(\hat{H}). small h. .. ,. there exists. So if supp \tilde{f}\subseteq I_{ $\sigma$} ,. Âf. I_{ $\sigma$}\in$\Lambda$_{h}(E, c). sub interval. a. is. a. finite rank h^{\infty}. such that. I_{ $\sigma$}. contains finite number. smoothing operator. By Corollary 2.4,. \displaystyle \frac{E_{\mathrm{J} (h)-E.(h)}{h},\mathrm{s} for. have. we. A_{f}^{R}\displaystyle\equiv\int_{-R}^{R}\tilde{f}(t)e^{-\not\simeq}Op_{h}^{W}(A)e^{\frac{\mathrm{t}\mathrm{f}\mathrm{f} {h} dt. (3). =Op_{h}^{W}(\displaystyle \int_{-R}^{R}\tilde{f}(t)(\exp tX_{H})^{*}Adt) \mathrm{m}\mathrm{o}\mathrm{d} h. (4). .. We note that. A_{f}^{R}-A_{f}=0 (\mathrm{m}\mathrm{o}\mathrm{d} h) Considering the classical mechanics (CP), (A6). S\subset H^{-1}(E-cE+c). of. A_{f}. denotes the subset. assures. which. on. for. large. (5). R.. X=\displaystyle \frac{1}{i}X_{H}. \exp tX_{H}. is. satisfies. is essentially self‐adjoint in L^{2}(S) Here complete. From (4) and (5), the leading symbol .. \displaystyle \int_{\mathrm{R}_{t} \overline{f}(t)(\exp tX_{H})^{*}Adt=\int_{\mathb {R}_{\mathrm{t} \tilde{f}(t)(e^{i\mathrm{t}(\frac{1}{ X_{H}) Adt =f(X)A(x,p)=0. A(x,p) (with \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(A)\subset H^{-1}(E-c, E+c)\backslash (K\times \mathbb{R}^{n}. for all Thus. f(X)=0. flow. 4. on. I_{ $\sigma$}\cap \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)=\emptyset. (Spectrum. Lemma 3.1. For the. that is,. of this. proofs periodic.. of X ) Either. lemma,. see. and. so. \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)\subset D $\sigma$(\hat{H}). \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(X)=\mathbb{R}. e.g.. S is. [7].. Thus. We need the. every Hamiltonian flow. D $\sigma$(\hat{H})\neq \mathbb{R}. means. that. following. on. S is. \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(X)\neq \mathbb{R}. lemma. :. periodic. and every Hamiltonian \square. Examples. In this section. we. introduce concrete. examples. These examples. (See e.g. [14]). (Example 7) ( 2-\dim Harmonic oscillator). mechanics Let. or. .. H(x,p)=\displaystyle \frac{1}{2}|p|^{2}+Ax_{1}^{2}+Bx_{2}^{2}(AB>0). For fixed c>0. We compare flow \exp tX_{H}. ( even. D $\sigma$(\hat{H}) :. in the. c=h^{1- $\delta$}). case. .. Then. are. fundamental. physical objects. of quantum. \displaystyle \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\hat{H}_{h})=\{\sqrt{2A}(i+\frac{1}{2})h+\sqrt{2B}(j+\frac{1}{2})h|i,j\in \mathrm{N}\geqq 0\}.. \left{bginary}{l D$\sigma$(\ht{H})\neqmathr{R}&\mathr{f}\mathr{o}\mathr{}\sqrt{facB}{A\inmathr{Q},\ D$\sigma$(\ht{H})=\mathr{R}&\mathr{f}\mathr{o}\mathr{}\sqrt{facB}{A\noti\mathr{Q}. \end{ary}\ight.. with the classical mechanics: For. $\Sigma$_{E}=\{(x,p)\in T^{*}(\mathrm{R}^{n})|H(xp)=E\}. $\Sigma$_{E}\rightarrow$\Sigma$_{E} satisfies. ,. the Hamiltonian. \left{bginary}{l \exptX_{H}\mathr{i} ms}\athrm{}\athrm{l w}\mathr{}\mathr{y}\mathr{s}\mathr{p}\mathr{e}\mathr{}\mathr{i mo}\athrm{d}\athrm{i c}\mathr{f} mo}\athrm{}\sqtfrac{B}A\inmathr{Q}\ exptX_{H}\mathr{i} ms}\athrm{n}\athrm{o}\athrm{n}-\athrm{p}\athrm{e}\athrm{}\athrm{i o}\mathr{d}\mathr{i mc}\athrm{f} o}\mathr{}\sqtfrac{B}A\notimahr{Q}. \endary}\ight. (Example 8)(Hydrogen atom) Let H(x,p)=\displaystyle \frac{1}{2}|p|^{2}-\frac{1}{r} Then \displaystyle \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\hat{H}_{h})\cap \mathrm{R}_{-}=\{E_{\mathrm{J} (h)=-\frac{1}{2h_{J^{j3} ^{A} |n\in \mathrm{N}_{>0}\}. instance, taking $\Lambda$(-1.5,0.5) (i.e. -2<E_{j}(h)<-1 ), we have .. For.

(5) 33. -2<E_{j}(h)<-1\displaystyle \Leftrightarrow-2<-\frac{1}{2h^{2}j^{2}}<-1\Leftrightarrow\frac{1}{2h}<j<\frac{1}{\sqrt{2}h}.. |\displaystyle \frac{E_{j_{1} (h)-E_{j_{2} (h)}{h}|=\frac{1}{2h^{3} |\frac{1}{j_{1}^{2} -\frac{1}{j_{2}^{2} | \displaystyle \geq \frac{1}{2h^{3} |\frac{1}{j_{1}^{2} -\frac{1}{(j_{1}+1)^{2} |. >\displaystyle \frac{1}{2h^{3} |\frac{1}{(_{\sqrt{2h} 1)^{2} -\frac{1}{(_{\sqrt{2h} 1+1)^{2} | >\displaystyle \frac{2h+2\sqrt{2} {2h^{2}+2\sqrt{2}h+1}\rightar ow 2\sqrt{2}. It follows. D $\sigma$(\hat{H})\neq \mathrm{R}.. Regarding. the classical mechanics whose orbits. Theorem 4.1. (J.. If all bound orbits Theorem 4.2. ([12]).. $\gamma$($\alpha$^{2}-$\beta$^{2}\rangle^{1/2}]\}^{2} One such. as. can. The. .. Let. three. closed, followings. orbits, then. V(r)=\displaystyle \frac{A}{r}. or. $\alpha,\ \beta$,. typical. where. with magnetic field. [10],. by. theorems.. H(x,p)=\displaystyle \frac{1}{2}|p|^{2}+V(r). .. U(x)=($\alpha$^{2}-$\beta$^{2})^{-2}\{\mathrm{a}x- $\beta$[x^{2}+ periodic.. $\gamma$ , every Hamiltonian flow for low energy is all. 2.5 for quantum mechanics with above. charged particles. are. V(r)=Br^{2}.. H(x_{1}, x_{2}, p_{1}, p_{2})=\displaystyle \frac{1}{2}|p|^{2}+U(x_{1})+U(x_{2}). For suitable. apply Theorem. \displaystyle\frac{1}{2}\sum_{J^{=1}^{n}a_{$\theta$}x) 5. also closed. all. h\rightarrow 0.. Define 3-\dim Hamiltonian with central force. (1873)).. Bertrand are. are. as. potentials. Many. Lotka‐Volterra system. other systems. are. known,. H(xp)=\displaystyle \sum_{i=1}^{ $\tau \iota$}(r_{l}x_{l}-\exp(p_{ $\iota$}+. [6]. and etc.. Remark assumption (A5). (\mathrm{A}\mathrm{S})\exists. is too strong. We can replace (A5) by a bounded subset L\supset K and $\rho$>0 s.t.. finite set K\in \mathbb{R}^{n}. ,. a_{ $\alpha$}\in C^{\infty}(T^{*}\mathbb{R}^{n}). is. analytic. in. G_{L\backslash K}\equiv\{|{\rm Im} x|<\cdot $\rho$, {\rm Re} x\in L\backslash K\} and for. some. C>0 and M>0. |a_{ $\alpha$}(x)|\leqq C(1+|x|)^{M} Let. \mathrm{N} ). in. G_{L\backslash K}.. introduce two C^{\infty}‐cutoff functions $\chi$ and $\zeta$ where $\zeta$ is supported in and $\chi$=1 on supp $\zeta$ The distance between the supports of $\zeta$ and. us. .. pseudodifferential operator Â. with. symbol suppA \subset \mathrm{T}^{*}\mathb {R}^{\mathrm{n} \backslash (\mathrm{K}\times \mathb {R}^{\mathrm{n} ). a. small. 1- $\chi$. neighborhood of T^{*}\mathbb{R}^{n}\backslash (K\times positive. Consider the. is then. and write the commutator. [\hat{H}. ,. Â]. as. follows. :. [\hat{H}. ,. Â]. \hat{}. =. [H. \hat{}. ,. \hat{} $\zeta$ Â(l— $\chi$ ) + [H (1- $\zeta$) Â] (l— $\chi$ ). Â] $\chi$+ H ,. right‐hand side, each operator has a smooth symbol so that we can use the same in the C^{\infty} pseudodiferential operators. The first terms give negligible contributions. Since. In the last term of the. computations. as. \Vert [ \hat{H}. ,. Â]x || L^{2}+\Vert\hat{H}\hat{A}(1-x)\Vert_{L^{2}}\leqq Ch.. proof techniques of Egorov theorem are applicable (See [2] for partitions of unitiy of A we obtain the Egorov theorem as in Lemma 2.3. The standard. more. precise).. Thus. taking suitable. ,. 6. Conclusion. Simple Helton like theorems are discussed including when singular potentials. It is emphasized that the periods explicitly characterized by D $\sigma$(\hat{H}) for the smooth potentials (See e.g. [5]). By using Kustaanheimo‐Stiefel transforms, the Hamiltonians with coulombic potentials are presumed to have the same properties. We would like to mention about it in the future.. of closed orbits will be.

(6) 34. Acknowledgements The author would like to thank the. organizers of this encouragement.. conference for the kind invitation. The author also wishes. to thank Professor T. Suzuki for his. References [1]. N.. [2]. A. BOUZOUINA. [3]. M. COMBESCURE and D.. ANANTHARAMAN, Entropy. and the localization. Anal.. of eigenfunctions,. AND D. ROUERT, Uniform semiclassical ables, Duke Math. J., 8(2) (2002), pp. 223‐252.. esitmates. Math.,. 168. (2008),. pp. 435‐475.. for the propagetion of quantum observ‐. ROBERT, Distribution of matrix elements and level spacing for classically chaotic systems, Annales de 1\mathrm{I}.\mathrm{H}.\mathrm{P} Physique théorique, 61(4) (1994) pp. 443‐483. .. [4]. M. DIMASSI and. [5]. S. DOZIAS, Clustering for the spectrum of h ‐pseudodifferential operators with surface, J. Funct. Anal., 145 (1997) pp. 296‐311.. J.SCHÖSTRAND, Spectral asymptotics. in the semi‐classical. limit, Cambridge. Univ.. Press,. (1999).. [6]. R. L. FERNANDES and W. M.. of the International Conference. [7]. V.. [8]. J. W.. Guillemin, Lectures. and. [9]. L.. [10]. HÖRMANDER,. T. IWAI and N.. R.. [13]. spectral theory of elliptic operators,. Duke Math J. 44. The. analysis of. linear. (1977),. 487‐517.. partial differential operators rv, Springer‐Verlag, (1984).. KATAYAMA, Multifold Kepler systems‐Dynamical systems all of whose bounded trajectories Phys., 36 (1995), pp. 1790−1811.. J. Math.. KUWABARA, Difference spectrum of the Schrodinger operator in. E. ONOFRI and M. \mathrm{P}\mathrm{A}\mathrm{U}\mathrm{R}\mathrm{J}. Math.. Phys.. 19. (1978),. a. magnetic field, Math.. \mathrm{Z}. M. REED and B.. L. I.. [15]. T. \mathrm{T}\mathrm{A}\mathrm{T}\mathrm{E}. Search. for periodic. Hamiltonian. SIMON,. Methods. A. generalized. of Modern. Mathematical. Physics,. II. Fourier. (1972).. Bertrands. (2000). theorem, J.. Analysis, Self‐Adjointness,. SCHIFF, Quantum mechanics, McGraw‐Hill, (1968).. Asymptotic behavior of eigenfunctions Pub., No.12(1999).. and. eigenvalues for ergodic. Tohoku Math.. K. \mathrm{Y} Functional. Y.Miyanishi: ‐mail:. flows:. 23$. pp. 1850‐1858.. Academic Press, New York,. [14]. [16]. energy. OLIVA, Hamiltonian dynamics of the Lotka‐ Volterra Equations, Proceedings on Differential Equations, (1995).. pp. 579‐599.. [12]. an. HELTON, An operator algebra approach to partial differential equations; Propagation of singularities spectral theorey, Indiana Univ. Math. J, (1963), pp. 997‐1018.. are.closed,. [11]. on. periodic flow on. and. periodic systems, Thesis,. Analysis, Springer Verlag, (1968). Center for Mathematical Modeling and Data Science, Osaka University, Machikaneyamacho 1‐3, Toyonakashi 560‐8531,. Japan;. miyanishi@sigmath.es.osaka u.ac.jp.

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