Spectra of "fattened" open book structures

TL;DR

This paper proves that the spectra of the Neumann Laplacian in thin neighborhoods of 3D branching structures converge to a 2D operator spectrum, extending quantum graph concepts to physical applications.

Contribution

It introduces a new spectral convergence result for Neumann Laplacians in 3D structures, generalizing quantum graph models to more complex geometries.

Findings

Spectral convergence of Neumann Laplacian established

Extension of quantum graph analogy to 3D structures

Applicable to physics and engineering problems

Abstract

We establish convergence of spectra of Neumann Laplacian in a thin neighborhood of a branching 2D structure in 3D to the spectrum of an appropriately defined operator on the structure itself. This operator is a 2D analog of the well known by now quantum graphs. As in the latter case, such considerations are triggered by various physics and engineering applications.

Equations

$$M_{ϵ}:=\bigcup _{x\in M}B(x,ϵ)$$

$$Q_{ϵ}(u)={\int }_{M_{ϵ}}|\nabla u{|}^{2}d{M}_{ϵ}$$

$$Q(u)=\sum _k{\int }_{M_k}|{\nabla }_{M_k}u{|}^{2}dM$$

$$u{|}_{\partial M_k\cap E_m}=u{|}_{\partial M_{k^{\prime }}\cap E_m}.$$

$$Au:=-{\Delta }_{M_k}u,$$

$$||u|{|}_{L_2(M)}^2+||Au|{|}_{L_2(M)}^2<\infty ,$$

$$u{|}_{\partial M_k\cap E_m}=u{|}_{\partial M_{k^{\prime }}\cap E_m},$$

$$\sum _{k:\partial M_k\supset E_m}{D}_{{\nu }_k}u(E_m)=0,$$

$$\bigg{|}\,\big{|}\big{|}u\big{|}\big{|}_{L_{2}(M_{\epsilon})}^{2}-\big{|}\big{|}J_{\epsilon}u\big{|}\big{|}_{L_{2}(M)}^{2}\,\bigg{|}\leq o(1)\big{|}\big{|}u\big{|}\big{|}_{H^{1}(M_{\epsilon})}^{2}$$

$$Q(J_{ϵ}u)-Q_{ϵ}(u)\le o(1)Q_{ϵ}(u)$$

$$\bigg{|}\,\big{|}\big{|}u\big{|}\big{|}_{L_{2}(M)}^{2}-\big{|}\big{|}K_{\epsilon}u\big{|}\big{|}_{L_{2}(M_{\epsilon})}^{2}\,\bigg{|}\leq o(1)\big{|}\big{|}u\big{|}\big{|}_{\mathcal{G}^{1}}^{2}$$

$$Q_{ϵ}(K_{ϵ}u)-Q(u)\le o(1)Q(u)$$

$${\lambda }_n(A_{ϵ}){\to }_{ϵ\to 0}{\lambda }_n(A).$$

$${\lambda }_n(B)=\underset{W\subset D}{\min }\underset{x\in W\setminus \{0\}}{\max }\frac{q(x,x)}{(x,x)},$$

$$\dfrac{Q(J_{\epsilon}u)}{||J_{\epsilon}u||^{2}_{L_{2}(M)}}\leq\big{(}1+o(1)\big{)}\dfrac{Q_{\epsilon}(u)}{||u||^{2}_{L_{2}(M_{\epsilon})}}$$

$$\dfrac{Q_{\epsilon}(K_{\epsilon}u)}{||K_{\epsilon}u||_{L_{2}(M_{\epsilon})}^{2}}\leq\big{(}1+o(1)\big{)}\dfrac{Q(u)}{||u||_{L_{2}(M)}^{2}}$$

$${\lambda }_n=\underset{x\in W_n\setminus \{0\}}{\max }\frac{Q(x,x)}{(x,x)},$$

$${\lambda }_n^{ϵ}=\underset{x\in W_n^{ϵ}\setminus \{0\}}{\max }\frac{Q_{ϵ}(x,x)}{(x,x)},$$

$$\lambda_{n}\leq\sup_{u\in J_{\epsilon}(W_{n}^{\epsilon})}\dfrac{Q(J_{\epsilon}u)}{||J_{\epsilon}u||^{2}_{L_{2}(M)}}\leq\big{(}1+o(1)\big{)}\lambda_{n}^{\epsilon},$$

$$\lambda_{n}^{\epsilon}\leq\sup_{u\in K_{\epsilon}(W_{n})}\dfrac{Q_{\epsilon}(K_{\epsilon}u)}{||K_{\epsilon}u||^{2}_{L_{2}(M_{\epsilon})}}\leq\big{(}1+o(1)\big{)}\lambda_{n}$$

$${\lambda }_n(A_{ϵ}){\to }_{ϵ\to 0}{\lambda }_n(A).$$

Full text

The work of both authors has been partially supported by the NSF DMS-1517938 grant.

Introduction

We consider a compact sub-variety $M$ of ${\mathbb{R}}^{3}$ that locally (in a neighborhood of any point) looks like either a smooth submanifold or an “open book” with smooth two-dimensional “pages” meeting transversely along a common smooth one-dimensional “binding,”111We do not provide here the general definition of what is called Whitney stratification, see e.g. [1, 28, 20, 41], resorting to a simple description through local models. see Fig. 1.

Clearly, any compact smooth submanifold of ${\mathbb{R}}^{3}$ (with or without a boundary) qualifies as an open book structure with a single page. Another example of such structure is shown in Fig. 2.

A “fattened” version $M_{\epsilon}$ of $M$ is an (appropriately defined) $\epsilon$–neighborhood of $M$, which we call a “fattened open book structure.”

Consider now the Laplace operator $-\Delta$ on the domain $M_{\epsilon}$ with Neumann boundary conditions (“Neumann Laplacian”), which we denote $A_{\epsilon}$. As a (non-negative) elliptic operator on a compact manifold, it has discrete finite multiplicity spectrum $\lambda^{\epsilon}_{n}:=\lambda_{n}(A^{\epsilon})$ with the only accumulation point at infinity. The result formulated in this work is that when $\epsilon\to 0$, each eigenvalue $\lambda_{n}^{\epsilon}$ converges to the corresponding eigenvalue $\lambda_{n}$ of an operator $A$ on $M$, which acts as $-\Delta_{M}$ (2D Laplace-Beltrami) on each 2D stratum (page) of $M$, with appropriate junction conditions along 1D strata (bindings).

Similar results have been obtained previously for the case of fattened graphs (see [27, 35], as well as books [2, 31] and references therein), i.e. $M$ being one-dimensional.

The case of a smooth submanifold $M\subset{\mathbb{R}}^{3}$ is not that hard and has been studied well under a variety of constraints set near $M$ (e.g., [18, 21, 2, 25]). Having singularities along strata of lower dimensions significantly complicates considerations, even in the quantum graph case [21, 5, 4, 3, 23, 27, 24, 25, 35, 39, 7].

Our considerations are driven by the similar types of applications (see, e.g. [2, 8, 10, 11, 12, 13, 14, 15, 16, 17, 22, 22, 25, 34, 35, 36, 37, 38]), as in the graph situation.

The Section 1 contains the descriptions of the main objects: open book structures and their fattened versions, the Neumann Laplacian $A$, etc. The next Section 2 contains formulation of the result. The proof is reduced to constructing two families of “averaging” and “extension” operators. This construction is even more technical than in the quantum graph case and will be provided in another, much longer text. The last Section 3 contains the final remarks and discussions.

In this article the results are obtained under the following restrictions: the width of the fattened domain shrinks “with the same speed” around all strata; no “corners” (0D strata) are present; the pages intersect transversely at the bindings. Some of them will be removed in a further work.

1. The main notions

1.1. Open book structures

Simply put, an open book structure222One can find open book structures in a somewhat more general setting being discussed in algebraic topology literature, e.g. in [32, 42]. $M$ is connected and consists of finitely many connected, compact smooth submanifolds (with or without boundary) of ${\mathbb{R}}^{3}$ (strata) of dimensions two and one, such that they only intersect along their boundaries and each stratum’s boundary is the union of some lower dimensional strata [20]. We also assume that the strata intersect at their boundaries transversely. In other words, locally $M$ looks either as a smooth surface, or an “open book” with pages meeting at a non-zero angle at a “binding.” Up to a diffeomorphism, a neighborhood of the binding looks like in Fig. 3.

1.2. The fattened structure

We can now define the fattened open book structure $M_{\epsilon}$.

Let us remark first of all that there exists $\epsilon_{0}>0$ so small that for any two points $x_{1},x_{2}$ on the same page of $M$, the closed intervals of radius $\epsilon_{0}$ normal to $M$ at these points do not intersect. This ensures that the $\epsilon<\epsilon_{0}$-fattened neighborhoods do not form a “connecting bridge” between two points that are otherwise far away from each other along $M$. We will assume that in all our considerations $\epsilon<\epsilon_{0}$, which is not a restriction, since we will be interested in the limit $\epsilon\to 0$.

We denote the ball of radius $r$ about $x$ as $B(x,r)$.

Definition 1.1**.**

Let $M$ denote an open book structure in ${\mathbb{R}}^{3}$ and $\epsilon_{0}>0$, as defined above. We define for any $\epsilon<\epsilon_{0}$ the corresponding fattened domain $M_{\epsilon}$ as follows:

[TABLE]

1.3. Quadratic forms and operators

We adopt the standard notation for Sobolev spaces (see, e.g. [29]). Thus, $H^{1}(\Omega)$ denotes the space of square integrable with respect to the Lebesgue measure functions on a domain $\Omega\subset\mathbb{R}^{n}$ with square integrable first order weak derivatives.

Definition 1.2**.**

Let $Q_{\epsilon}$ be the closed non-negative quadratic form with domain $H^{1}(M_{\epsilon})$, given by

[TABLE]

We also refer to $Q_{\epsilon}(u)$ as the energy of $u$.

This form is associated with a unique self-adjoint operator $A_{\epsilon}$ in $L_{2}(M_{\epsilon})$. The following statement is standard (see, e.g. [6, 29]):

Proposition 1.3**.**

The form $Q_{\epsilon}$ corresponds to the Neumann Laplacian $A_{\epsilon}=-\Delta$ on $M_{\epsilon}$ with its domain consisting of functions in $H^{2}(M_{\epsilon})$ whose normal derivatives at the boundary $\partial M_{\epsilon}$ vanish.

Its spectrum $\sigma(A_{\epsilon})$ is discrete and non-negative.

Moving now to the limit structure $M$, we equip it with the surface measure $dM$ induced from ${\mathbb{R}}^{3}$.

Definition 1.4**.**

Let $Q$ be the closed, non-negative quadratic form (energy) on $L_{2}(M)$ given by

[TABLE]

with domain $\mathcal{G}^{1}$ consisting of functions $u$ for whose $Q(u)$ is finite and that are continuous across the bindings between pages $M_{k}$ and $M_{k^{\prime}}$:

[TABLE]

Here $\nabla_{M_{k}}$ is the gradient along $M_{k}$ and restrictions in (4) to the binding $E_{m}$ coincide as elements of $H^{1/2}(E_{m})$.

Unlike the fattened graph case, by the Sobolev embedding theorem [6] the restriction to the binding is not continuous as an operator from $\mathcal{G}^{1}$ to $C(E_{m})$, it only maps to $H^{1/2}(E_{m})$. This distinction significantly complicates the analysis of fattened stratified surfaces in comparison with fattened graphs.

Proposition 1.5**.**

The operator $A$ associated with the quadratic form $Q$ acts on each $M_{k}$ as

[TABLE]

with the domain $\mathcal{G}^{2}$ consisting of functions on $M$ such that the following conditions are satisfied:

• •

[TABLE]

continuity across common bindings $E_{m}$ of pairs of pages $M_{k},M_{k^{\prime}}$:

[TABLE]

• •

Kirchhoff condition* at the bindings:*

[TABLE]

where $-\Delta_{M_{k}}$ is the Laplace-Beltrami operator on $M_{k}$ and $D_{\nu_{k}}$ denotes the normal derivative to $\partial M_{k}$ along $M_{k}$.

The spectrum of $A$ is discrete and non-negative.

The proof is simple, standard, and similar to the graph case. We thus omit it.

2. The main result

Definition 2.1**.**

We denote the ordered in non-decreasing order eigenvalues of $A$ as $\{\lambda_{n}\}_{n\in\mathbb{N}}$, and those of $A_{\epsilon}$ as $\{\lambda_{n}^{\epsilon}\}_{n\in\mathbb{N}}$.

For a real number $\Lambda$ not in the spectrum of $A_{\epsilon}$, we denote by $\mathcal{P}_{\Lambda}^{\epsilon}$ the spectral projector of $A_{\epsilon}$ in $L_{2}(M_{\epsilon})$ onto the spectral subspace corresponding to the half-line $\{\lambda\in{\mathbb{R}}\,|\,\lambda<\Lambda\}$.

Similarly, $\mathcal{P}_{\Lambda}$ denotes the analogous spectral projector for $A$. We then denote the corresponding (finite dimensional) spectral subspaces as $\mathcal{P}_{\Lambda}^{\epsilon}L_{2}(M_{\epsilon})$ and $\mathcal{P}_{\Lambda}L_{2}(M)$ for $M_{\epsilon}$ and $M$ respectively.

We now introduce two families of operators needed for the proof of the main result.

Definition 2.2**.**

A family of linear operators $J_{\epsilon}$ from $H^{1}(M_{\epsilon})$ to $\mathcal{G}^{1}$ is called averaging operators if for any $\Lambda\notin\sigma(A_{\epsilon})$ there is an $\epsilon_{0}$ such that for all $\epsilon\in(0,\epsilon_{0}]$ the following conditions are satisfied:

• •

For $u\in\mathcal{P}_{\Lambda}^{\epsilon}L_{2}(M_{\epsilon})$, $J_{\epsilon}$ is “nearly an isometry” from $L_{2}(M_{\epsilon})$ to $L_{2}(M)$ with an $o(1)$ error, i.e.

[TABLE]

where $o(1)$ is uniform with respect to $u$.

• •

For $u\in\mathcal{P}_{\Lambda}^{\epsilon}L_{2}(M_{\epsilon})$, $J_{\epsilon}$ asymptotically “does not increase the energy,” i.e.

[TABLE]

where $o(1)$ is uniform with respect to $u$.

Definition 2.3**.**

A family of linear operators $K_{\epsilon}$ from $\mathcal{G}^{1}$ to $H^{1}(M_{\epsilon})$ is called extension operators if for any $\Lambda\notin\sigma(A)$ there is an $\epsilon_{0}$ such that for all $\epsilon\in(0,\epsilon_{0}]$ the following conditions are satisfied:

• •

For $u\in\mathcal{P}_{\Lambda}L_{2}(M)$, $K_{\epsilon}$ is “nearly an isometry” from $L_{2}(M)$ to $L_{2}(M_{\epsilon})$ with $o(1)$ error, i.e.

[TABLE]

where $o(1)$ is uniform with respect to $u$.

• •

For $u\in\mathcal{P}_{\Lambda}L_{2}(M)$, $K_{\epsilon}$ asymptotically “does not increase” the energy, i.e.

[TABLE]

where $o(1)$ is uniform with respect to $u$.

Existence of such averaging and extension operators is known to be sufficient for spectral convergence of $A_{\epsilon}$ to $A$ (see [31]). For the sake of completeness, we formulate and prove this in our situation.

Theorem 2.4**.**

Let $M$ be an open book structure and its fattened partner $\{M_{\epsilon}\}_{\epsilon\in(0,\epsilon_{0}]}$ as defined before. Let $A$ and $A_{\epsilon}$ be the operators on $M$ and $M_{\epsilon}$ as in Definitions 1.3 and 1.5.

Suppose there exist averaging operators $\{J_{\epsilon}\}_{\epsilon\in(0,\epsilon_{0}]}$ and extension operators $\{K_{\epsilon}\}_{\epsilon\in(0,\epsilon_{0}]}$ as stated in Definitions 2.2 and 2.3.

Then, for any $n$

[TABLE]

We start with the following standard (see, e.g. [33]) min-max characterization of the spectrum.

Proposition 2.5**.**

Let $B$ be a self-adjoint non-negative operator with discrete spectrum of finite multiplicity and $\lambda_{n}(B)$ be its eigenvalues listed in non-decreasing order. Let also $q$ be its quadratic form with the domain $D$. Then

[TABLE]

where the minimum is taken over all $n$-dimensional subspaces $W$ in the quadratic form domain $D$

Proof of Theorem 2.4 now employs Proposition 2.5 and the averaging and extension operators $J,\,K$ to “replant” the test spaces $W$ in (17) between the domains of the quadratic forms $Q$ and $Q_{\epsilon}$.

Let us first notice that due to the definition of these operators (the near-isometry property), for any fixed finite-dimensional space $W$ in the corresponding quadratic form domain, for sufficiently small $\epsilon$ the operators are injective on $W$. Since we are only interested in the limit $\epsilon\to 0$, we will assume below that $\epsilon$ is sufficiently small for these operators to preserve the dimension of $W$. Thus, taking also into account the inequalities (9)-(12), one concludes that on any fixed finite dimensional subspace $W$ one has the following estimates of Rayleigh ratios:

[TABLE]

[TABLE]

Let now $W_{n}\subset\mathcal{G}^{1}$ and $W_{n}^{\epsilon}\subset H^{1}(M_{\epsilon})$ be $n$, such that

[TABLE]

and

[TABLE]

Due to the min-max description and inequalities (14) and (15), one gets

[TABLE]

and

[TABLE]

Thus, $\lambda_{n}-\lambda_{n}^{\epsilon}=o(1)$, which proves the theorem. $\Box$

The long technical task, to be addressed elsewhere, consists in proving the following statement:

Theorem 2.6**.**

Let $M$ be an open book structure and its fattened partner $\{M_{\epsilon}\}_{\epsilon\in(0,\epsilon_{0}]}$ as defined before. Let $A$ and $A_{\epsilon}$ be operators on $M$ and $M_{\epsilon}$ as in Definitions 1.3 and 1.5. There exist averaging operators $\{J_{\epsilon}\}_{\epsilon\in(0,\epsilon_{0}]}$ and extension operators $\{K_{\epsilon}\}_{\epsilon\in(0,\epsilon_{0}]}$ as stated in Definitions 2.2 and 2.3.

This leads to the main result of this text:

Theorem 2.7**.**

Let $M$ be an open book structure and its fattened partner $\{M_{\epsilon}\}_{\epsilon\in(0,\epsilon_{0}]}$. Let $A$ and $A_{\epsilon}$ be operators on $M$ and $M_{\epsilon}$ as in Definitions 1.3 and 1.5.

Then, for any $n$

[TABLE]

3. Conclusions and final remarks

• •

As the quantum graph case teaches [24, 31], allowing the volumes of the fattened bindings to shrink when $\epsilon\to 0$ slower than those of fattened pages, is expected to lead to interesting phase transitions in the limiting behavior. This is indeed the case, as it will be shown in yet another publication.

• •

It is more practical to allow presence of zero-dimensional strata (corners). The analysis and results get more complex, as we hope to show in yet another work, with more types of phase transitions.

• •

Resolvent convergence, rather than weaker local convergence of the spectra, as done in [31] in the graph case, would be desirable and probably achievable.

• •

One can allow some less restrictive geometries of the fattened domains.

• •

The case of Dirichlet Laplacian is expected to be significantly different in terms of results and much harder to study, as one can conclude from the graph case considerations [21].

4. Acknowledgments

The work of both authors was partially supported by the NSF DMS-1517938 Grant.