The main objectives of this course

• is to look at strong convergence and spectrum of different geometric examples,
• speak about the recent polynomial method of Chen, Garza-Vargas, Trop, Van Handel.

Lecture 1: Spectral convergence and fundamental notes

We will always speak of spectrum of self-adjoint operators. Most of the time it will be the Laplacian \(\Delta\).

Example 1

Consider a box of size \(\varepsilon \times 1\). The Dirichlet spectrum is \[\begin{equation} (\frac{\Pi m}{ \varepsilon})^{2} + (\pi k)^{2}, \text{ for each } k,m \in \mathbb{Z}_{>0}. \end{equation}\] The is the spectrum of \(\sin (\pi k x) \sin (\pi m y / \varepsilon)\). The Neumann spectrum has functions \(\cos( \pi k x) \cos(\pi m y /\varepsilon)\) for \(k,m \geq 0\).

1. When \(m \neq 0\), the spectral part is at infinity.
2. When \(m=0\), the eigenvalues are \((\pi k)^{2}\) which are the eigenvalues of the interval \([0,1]\) with respect to \(-\partial^{2}/\partial x^{2}\) with Neumann conditions. \[\begin{equation} \text{ for each } M, \mathop{\mathrm{sp}}(\Delta_{\varepsilon}) \cap [-M,M] = \mathop{\mathrm{sp}}( - \partial^{2}/\partial x^{2}) \cap [-M,M]. \end{equation}\] Here, with periodic conditions, we have convergence of spectrum to the convergence of the circle.

Example 2

Consider a cone of angel \(\alpha\) as \(\alpha \rightarrow 0\). Call it \(C_{\alpha}\). Then \(\Delta\) on the functions \(C_{0}^{\infty}( C_{\alpha} \setminus \{0\})\) is not necessarily self-adjoint. We have to choose the conditions for limits. This gives us the Neumann Laplacian. The quadratic form \(\mathcal{E}(f,g) = \int_{} \nabla f \cdot \nabla g r \mathop{\mathrm{\,d}}r \mathop{\mathrm{\,d}}\theta\) on \[\begin{equation} \mathop{\mathrm{Dom}}( \mathcal{E}) = \{ f , \nabla f \in L^{2}\}. \end{equation}\] The associated operator \(\Delta\) has \[\begin{equation} \mathop{\mathrm{Dom}}(\Delta) = \{f, g \mapsto \mathcal{E}(f,g) \text{ bounded on }L^{2}\}. \end{equation}\]

The eigenfunctions of the form are \(\varphi(r) e^{-i m \theta/\alpha}\) where \(\varphi(r)\) is a Bessel function \(J_{m/\alpha}( \sqrt{\lambda} r)\). Then again

1. \(m \neq 0\) is the spectral part at \(\infty\)
2. \(m=0\) are the proper radial functions.

The radial spectrum converges to the spectrum associated to the quadratic form. The limit geometric object is finite. \[\begin{equation} \mathcal{E}_{1}(f,g) = \int_{0}^{1} \frac{\partial f}{ \partial r} \frac{\partial g}{ \partial r} r\mathop{\mathrm{\,d}}r. \end{equation}\] \[\begin{equation} \Delta^{(1)}= \frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{ \partial r}. \end{equation}\] The spectre of \(\Delta_{\alpha}\) converges towards the spectrum of \(\Delta^{(1)}\).

Notions of convergence of spectra

(Beckus-Bellisard 2016)
This is the convergence of spectrum as sets. \((\mathcal{H}_{t})_{t \in T}\) are Hilbert spaces indexed by topological spaces \(T\). \(A_{t}\) are self-adjoint operators and \(F_{t} = \mathop{\mathrm{sp}}(A_{t})\) is a closed subset of \(\mathbb{R}\). As \(t \rightarrow t_{0}\), we define

1. Hausdorff distance:

   \(d_{H}(F_{t},F_{t_{0}}) \rightarrow 0\) if \[\begin{align} \text{ for each } \varepsilon>0 & \quad F_{t} \subseteq U_{\varepsilon}(F_{t_{0}}) \\ & \quad F_{t_{0}} \subseteq U_{\varepsilon}(F_{t}), \end{align}\] such that \(t\) is close enough to \(t_{0}\).

2. Convergence in the sense of Victoris

   \[\begin{equation} F_{t} \rightarrow F_{t_{0}} \end{equation}\] if \[\begin{align} & \text{ for each } K \subseteq R \text{ closed such that } K \cap F_{t_{0}} \neq \phi \\ & \text{ then } K \cap F_{t} = \phi \text{ for a disk } t \text{ close to } t_{0} \\ & \text{ for }\mathcal{O} \subseteq \mathbb{R} \text{ open such that } \mathcal{O} \cap F_{t_{0}} \neq \phi \\ & \text{ then } \mathcal{O} \cap F_{t} \neq \phi \text{ for each } t \text{ close to } t_{0} \end{align}\]

3. Convergence in Fell topology: Identical to Vietoris but with \(K\) compact. In the given examples, we have a convergnce in the Fell topology but not in the sense of Vietoris.
4. “Convergence with respect to spectral edges”

   A spectral gap of \({F}_{t}\) is the connected component of \(F_{t}^{c}\). The spectral edge of \(F_{t}\) converges to that of \(F_{t_{0}}\) if

   If \(A_{t}\) are bounded operators then as \(t \rightarrow t_{0}\) we have \[\begin{align} \text{ CV of spectral edges } '' & \Leftrightarrow \text{ CV of Vietoris } \\ & \Leftrightarrow \|P(A_{t})\| \rightarrow \|P(A_{t_{0}})\| \text{ as }t \rightarrow t_{0} \forall \text{ polynomial }P, \deg(P) \leq 2. \end{align}\]

   If the \(A_{t}\) are not necessarily bounded, \[\begin{equation} \text{ CV of spectral edges } \Leftrightarrow \text{ CV of Fell } \Leftrightarrow \text{ for all } $z \in \mathbb{C} \setminus \mathbb{R} \|(z-A_{t})^{-1}\| \rightarrow \|(z-A_{t_{0}})^{-1}\|. \end{equation}\]

Example 3

“Graphes épaissis” Exner-Post 2000 - ongoing

Metric graph \(G =(V,E)\). \(L_{e} =\) length of the arête \(e\). We take \(\frac{\partial^{2}}{ \partial x^{2}}\) on each edge. Choice of conditions are taken on the limits of the vertices. We have Kirchoff conditions. \(f\) is continuous on vertices. We define \[\begin{equation} \mathcal{E}(f,g) = \sum_{e}^{} \int_{}^{} f' g'\mathop{\mathrm{\,d}}x, \end{equation}\] where \(f,g \in H^{1}\) on each edge and continues to the vertices.

We suppose that \(G\) is embedded isometrically in \(\mathbb{R}^{n}\). Then \(G_{\varepsilon} =\) neighbourhood \(\varepsilon \subseteq \mathbb{R}^{n}\), \(\Delta_{\varepsilon}\) = Neumann’s Laplacian.

One has \[\begin{equation} P_{0}^{\hat{}} = \frac{1}{ 2 i\pi} \int_{\Gamma}^{} (z- A_{0})^{-1}\mathop{\mathrm{\,d}}z = \lim_{ \varepsilon \rightarrow 0} P_{\varepsilon}^{\hat{}} \end{equation}\]

We generalize notions of convergence of norm of spectrum by Post-Weidman. Take \(A_{\varepsilon}\) on \(\mathcal{H}_{\varepsilon}\), take \(A_{0}\) on \(H_{0}\). We have \[\begin{equation} A_{\varepsilon} \xrightarrow{\text{ generalized resolvent }} A_{0} \end{equation}\] if there exists a Hilbert space \(\mathcal{H}\) and injections \(i_{\varepsilon} : \mathcal{H}_{\varepsilon} \rightarrow \mathcal{H}\) and \(i_{0} : \mathcal{H}_{0} \rightarrow \mathcal{H}\) with \(i_{\varepsilon}^{*} i_{\varepsilon} = I\) and \(i_{0}^{*} i_{0} = I\) with \[\begin{equation} \| i_{\varepsilon} ( z- A_{\varepsilon})^{-1} i_{\varepsilon}^{*} - i_{0} ( z- A_{0})^{-1} i_{0}^{*} \|_{\mathcal{B}(\mathcal{H})} \xrightarrow[]{{\varepsilon \rightarrow 0}} 0. \end{equation}\]

Ex 4) Gromov-Hausdorff convergence. One defines \((X,d,x,m)\) a pointed metric space. Take \((X,d)\) complete, separable and a point \(x \in X\), \(m\) is a Radon measure, non-zero and \(x \in \sup(m)\).

That is, \[\begin{equation} i_{n \sharp} m_{n} \rightarrow i_{\infty \sharp} m_{\infty} \end{equation}\] \[\begin{equation} \int_{}^{} \varphi \circ i_{n}\mathop{\mathrm{\,d}}m_{n} \rightarrow \int_{}^{} \varphi \circ i_{\infty}\mathop{\mathrm{\,d}}m_{\infty} \end{equation}\] for all \(\varphi\) that are continuous and have bounded support.

Here are some references

• Fukaya 85
• Colding Naber
• Lott-Villani
• Sturm
• Ambrosio-Gigli-Savaré

Case where the \(X_{n}\) are varieties of dimension \(N\), Ricci \(X_{n} \geq K\) and \(d_{n}\) is Ramanujan distance. We have \(m_{n} = \mathop{\mathrm{vol}}{x_{n}}/\mathop{\mathrm{vol}}(B_{X_{n}}(x_{n},1))\). The limit object \((X_{\infty},d_{\infty},x_{\infty},m_{\infty})\).

• \(\exists R \subseteq X_{\infty}\) with “ensemble régulier” and \(m_{\infty}(X_{\infty} \setminus R) = 0\).
• \(\exists k \leq N\) such that \(R\) is equipped with bi-Lipshitz maps of values in \(\mathbb{R}^{k}\)
• \(m_{\infty}\) is a Hausdorff measure that is \(k\)-dimensional
• \(X_{\infty}\) satisfies a condition \(R \subseteq D(K,N)\) which is the synthetic curvature. This is given by Sturm/Lott-Villani. This is “Riemannian”.

(Cheeger-Colding 2000)

(Gigli-Mondino-Savaré)