数理解析研究所講究録
第 1628 巻 2009 年 58-62

58$\pi_{1}$ -and$\pi_{2}$ -theories of operators

Shunsaku Nii*
In this talk, a topological index theory which can be seen as -theory ofoperators is introduced. This terminology is inspired by the one -theory ofoperators by Sanson [6] referring to infinite dimensional Maslov index theory. This viewpoint begins by seeing the classical theory of Strum-Liouville
operators as -theory of .
$\pi_{2}$$\pi_{1}$$S^{1}$$\pi_{1}$1The theory of Strum-Liouville operators:a -theory of$S^{1}$$\pi_{1}$Consider the eigenvalue problem of a Strum-Liouville operator:

on $I=[-1,1]$ or$-p”+f(x)p=\lambda p$ ,$\mathbb{R}$This equation is written as a system of first order equations:$\{\begin{array}{l}p’=qq’=(f(x)-\lambda)p.\end{array}$In the polar coordinate $p=r\cos\theta,$ $q=r\sin\theta$ , this system becomes:$\{\begin{array}{l}r’=(1-\lambda+f(x))r\sin\theta\cos\theta\theta’=1+(\lambda-f(x)-1)\sin^{2}\theta\end{array}$

Because the right hand side of the equation is monotone in , the numberwinds
increase as does. Therefore for each eigenvalue
ofidentifies the eigenfunction. That is, the eigenfunctions are orderedby the number of humps.
$\theta$$\theta(I)$$S^{1}$$\lambda$$\pi_{1}(S^{1})$*Faculty of Mathematics, Kyushu University$\lambda$$\lambda,$$\theta(I)\in$592Maslov index: a$\pi_{1}$-theory of matrices

The first natural generalization of the above theory is what is called (KellerMaslov( Arnol d) index for the eigenvalue problem of a Schr\"o dinger operator:$)$$-$$-p”+M(x)p=\lambda p$ ,,on $I=[-1,1]$ orThis system is equivalent to the following Hamiltonian system:$p\in \mathbb{R}^{n}$$\mathbb{R}$$\{\begin{array}{l}p’= \frac{\partial H}{\partial q}q’=-\frac{\partial H}{\partial p},\end{array}$where the Hainiltonian is given by $H(p, q)= \frac{1}{2}\{|q|^{2}+t_{p}(\lambda I-M(x))p\}$ .
Because a Hamiltonian system preserves the symplectic structure, thissystem induces a flow on the Lagrangian GraJ3mannian manifold $\Lambda(n)=$$Sp(n)(\mathbb{R}^{n}\cross\{0\})$ ,where $Sp(n)$ is the symplectic group.

Fact$\pi_{1}(\Lambda(n))\cong \mathbb{Z}$Therefore $(p, q)(I)\in\pi_{1}(\Lambda(n))$ characterizes the eigenfunctions. $((p, q)(x)$ is
not necessarily monotone.) This is what is called Maslov index.

3Infinite dimensional Maslov index:
a -theory of operators$\pi_{1}$There are several infinite dimensional generalization of Maslov index. Oneby Swanson [6] is among the earliests.
Let $E=H\cross H^{*}$ for a Hilbert space $H$ and its dual
. Define a$E$
symplectic structure on by
Then the IFhredholm
Lagrangian GraJ3mannian manifold
is defined by $\mathcal{F}\Lambda_{H}=Sp_{C}(E)H$ ,where $Sp_{C}(E):=$ { $id+$ compact preserves } $\subset GL(E)$$H^{*}$$\omega((e, \alpha), (f, \beta))=\alpha\cdot f-\beta\cdot e$$\mathcal{F}\Lambda_{H}$

$|$

Fact

$\omega$$\pi_{1}(\mathcal{F}\Lambda_{H})\cong \mathbb{Z}$Swanson applied this fact for deformations of elliptic operators, and call
his theory as -theory of operators contrasting it to Ftiredholm index$\pi_{1}$$(\pi_{0^{-}}$

theory of operators) which distinguishes connected components of operators.

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Recently, Deng [2] reformulated this theory on
and applied to the boundary value problem offor a star-shaped domain
an elliptic operator.
$E=H^{\frac{1}{2}}(\partial\Omega)\cross H^{-\frac{1}{2}}(\partial\Omega)$$\Omega$

The Stability index: a4$\pi_{2}$-theory of matrices

In spite of early development of the -theory for selfadjoint operators, any
analogous theory for non-selfadjoint operators has not appeared until recently. The obstacles were that the eigenvalues are not real and the systems
are no longer Hamiltonian. The first step for this direction seems to be theStability index theory by Alexander-Gardner-Jones [1, 5] explained below.
Consider the eigenvalue problem for a not-selfadjoint operator:$\pi_{1}$$-p”+M(x)p’+N(x)p=\lambda p$ ,$p\in \mathbb{C}^{n}$on,This system is equivalent to the following system on$\mathbb{C}^{2n}$

$I$

.

:

$\{\begin{array}{l}p’=qq’=(N(x)-\lambda I)p+M(x)q.\end{array}$This time, the system induces a flow on the complex Graflmannian manifold$G_{n}(\mathbb{C}^{2n})=GL(2n)(\mathbb{C}^{n}\cross\{0\})$ . For a disc, this flow induces a map$D\subset \mathbb{C}$$\Phi:S^{2}\cong(D\cross\partial I)\cup(\partial D\cross I)arrow G_{n}(\mathbb{C}^{2n})$Fact$\pi_{2}(G_{n}(\mathbb{C}^{2n}))\cong \mathbb{Z}$Then Alexander-Gardner-Jones proved sort of$\pi_{2}$-theory of matrices.

Theorem (Alexander-Gardner-Joned [1],Gardner-Jones [5])represents the number of eigenvalues in $D$ including the multiplicity.
$\Phi(S^{2})\in\pi_{2}(G_{n}(\mathbb{C}^{2n}))$This theory is sometimes referred to as Alexander-Gardner-Jones bundle theory, as it is formulated by the terminology of line bundles and the Chern class.

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The infinite dimensional Stability index:
a -theory of operators$\pi_{2}$It is natural to think about infinite dimensional generalization of the Stability
index from the viewpoints both in pure mathematics and in application. One

such example is the following eigenvalue problem:$\{\begin{array}{l}u_{xx}+\Delta_{y}u+\beta(y)u_{x}+f(x, y)u=\lambda u, (x, y)\in \mathbb{R}\cross\Omega\frac{\partial u}{\partial\nu}=0, on \mathbb{R}\cross\partial\Omega,\end{array}$where
is a bounded domain. This equation can be written as an
ordinary differential equation in x-variable on an appropriate Hilbert space$\Omega\subset \mathbb{R}^{m}$$H_{\Omega}$.

Here is a difficulty: $GL(H)$ is contractible for an infinite dimensional
Hilbert space $H$ . This means that a naive generalization of the Stability
index becomes trivial and does not detect any information.
Fortunately, we can exploit compactness of the problem: Let $GL_{C}(H)$ $:=$
{ $id+$ compact invertible} $\subset GL(H)$ and fix a polarization $H=H_{-}\oplus H_{+}$ ,then the $\mathbb{R}edholm$ GraBmannian manifold $F(H_{+})$ is the orbit ofunder$F(H_{+})=GL_{C}(H)H_{+}$
the action of $GL_{C}(H)i.e$ .
.
Under this setting, the problem induces a system on $F(H_{+})$ .
$|$$H_{+}$

Remark In this case, the system does not generate a flow, as the problem
is ill-posed.
Then, for a disc$D\subset \mathbb{C}$, this system induces a map$\Phi:S^{2}\cong(D\cross\partial I)\cup(\partial D\cross I)arrow F(H_{+})$,

and we have the following theorem.
Theorem (Deng-N. [3])$\Phi(S^{2})\in\pi_{2}(F(H_{+}))$ represents the numberof eigenvalues in $D$ including
the multiplicity.

We also have a similar result for an elliptic operator posed on a boundeddomain [4]These results can be called -theory of operators in the Swanson’s ex$\pi_{2}$pression.

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References[1]

J. Alexander, R. Gardner and C. Jones, A topologicalarising in the stability analysis of travelling waves J. Reine Angew. Math.
410(1990), pp. 167-212.
$invar\dot{\tau}ant$[2] J. Deng, Instability of waves and pattems in Hamiltonian systems andthe Maslov Index, Ph.D thesis, Brown University (2002).

[3]J. Deng and S. Nii, Infinite dimensional Evans function
forelliptic eigenvalue problems in a channel, J. Diff. Eqn. 226(2006) 57-89.
$theo\dot{r}y$[4] –, An Infinite-dimensional Evans function theory for Elliptic
Boundary Value Problems, J. Diff. Eqn. 244(2008) 753-765.
[5] R. Gardner, C. Jones, A Stability Index for Steady State Solutions ofBoundary Value Problems for Parabolic Systems, J. Diff. Eqn. 91(1991)

181-203.

[6] R. Swanson, Fredholm intersection theory and elliptic boundary deformation problems (I), (II), J. D. E., 28, 189-201, 202-219. (1978).