On the bottom of spectra under coverings

TL;DR

This paper proves that under certain conditions involving group actions, the lowest spectral values of a Riemannian manifold and its covering are equal, extending understanding of spectral properties in geometric analysis.

Contribution

It establishes a new link between the amenability of group actions and the equality of spectral bottoms in Riemannian coverings.

Findings

Spectral bottoms coincide under amenable group actions

Amenability of the group action is key to spectral equality

Extends spectral theory in Riemannian geometry

Abstract

For a Riemannian covering $M_1\to M_0$ of complete Riemannian manifolds with boundary (possibly empty) and respective fundamental groups $\Gamma_1\subseteq\Gamma_0$, we show that the bottoms of the spectra of $M_0$ and $M_1$ coincide if the right action of $\Gamma_0$ on $\Gamma_1\backslash\Gamma_0$ is amenable.

Equations

$$R(f)=\frac{{\int }_M(|\nabla f{|}^2+V{f}^2)}{{\int }_M{f}^2}$$

$${\lambda }_0(M,V)=\inf R(f),$$

$${\lambda }_0(M_0,V_0)\le {\lambda }_0(M_1,V_1).$$

$${\lambda }_0(M_0,V_0)={\lambda }_0(M_1,V_1).$$

$$|F\setminus Fg|\le \epsilon |F|$$

$$D_y=\{z\in M_1\mid \text{d(z,y)≤d(z,y′) for all y′∈π−1(x)}\}$$

$$d(z,y)\le d(z,y^{\prime })\le L(\gamma )=L({\gamma }_0)<r.$$

$${\phi }_1=\frac{{\psi }_1}{{\psi }_1+{\sum }_{z\in {\pi }^{-1}(x)}{\psi }_z}\text{and}{\phi }_y=\frac{{\psi }_y}{{\psi }_1+{\sum }_{z\in {\pi }^{-1}(x)}{\psi }_z}.$$

$$|{\phi }_y(z_1)-{\phi }_y(z_2)|$$

$$\le \frac{(2N+N)d(z_1,z_2)}{\chi (z_1)\chi (z_2)}\le 3Nd(z_1,z_2).\text{qed}$$

$$A^{-1}g\le h\le Ag$$

$$\begin{array}{rl}S& =\{s\in R\mid d(y,{\pi }_1(su))\le 2r+2\}\\ & =\{s\in R\mid \text{d(u,tsu))≤2r+2 for some t∈Γ1}\},\\ T& =\{t\in {\Gamma }_1\mid \text{d(u,tgu)≤2r+2 for some s∈S}\},\\ G& =TS\subseteq {\Gamma }_0.\end{array}$$

$$P=\{g\in R\mid {\Gamma }_{1}g\in F\}\subseteq R$$

$$\chi =\sum _{g\in P}{\phi }_{{\pi }_1(gu)}.$$

$$Q=\{y\in {\pi }^{-1}(x)\mid \text{(χf1)(z)=0 for some z∈Dy}\}.$$

$$P_1=\{{\pi }_1(gu)\mid g\in P\}\subseteq Q.$$

$$|P|=|P_1|\le |Q|.$$

$$Q_{-}=\{y\in Q\mid \text{0<χ(z)<1 for some z∈π−1(K)∩Dy}\}.$$

$$\chi (z)+\sum {\phi }_{{\pi }_1(g_{i}u)}(z)=1.$$

$$d(u,g^{-1}{h}_i{g}_{i}u))=d(gu,h_i{g}_{i}u)\le 2r+2.$$

$$\begin{array}{rl}|Q_{-}|& \le \epsilon |F||G|N(2r+1)\\ & =\epsilon |P||G|N(2r+1)\le \epsilon |Q||G|N(2r+1).\end{array}$$

$$\begin{array}{rl}{\int }_{D_y}\{|\nabla (\chi f_1){|}^2+V_1(\chi f_1{)}^2\}& ={\int }_{D_y}\{|\nabla f_1{|}^2+V_1{f}_1^2\}\\ & ={\int }_{M_0}\{|\nabla f{|}^2+V{f}^2\}\end{array}$$

$${\int }_{D_y}{\chi }^2{f}_1^2={\int }_{D_y}{f}_1^2={\int }_{M_0}{f}^2,$$

$${\int }_{D_y}{\chi }^2{f}_1^2\le {\int }_{M_0}{f}^2\text{and}{\int }_{D_y}|V_1|{\chi }^2{f}_1^2\le C_0{\int }_{M_0}{f}^2,$$

$${\int }_{D_y}|\nabla (\chi f_1){|}^2$$

$$\le 18N(r+1{)}^{4}A{\int }_{M_0}{f}^2+2{\int }_{M_0}|\nabla f{|}^2.$$

$${\int }_{D_y}\{|\nabla (\chi f_1){|}^2+|V_1|{\chi }^2{f}_1^2\}\le C$$

$$|Q_{-}|\le \frac{\epsilon D}{1-\epsilon D}|Q_{+}|,$$

$$R(\chi f_1)$$

$$=\frac{{\sum }_{y\in Q}{\int }_{D_y}\{|\nabla f_1{|}^2+V_1{f}_1^2\}}{{\sum }_{y\in Q}{\int }_{D_y}{f}_1^2}$$

$$\le \frac{{\sum }_{y\in Q_{+}}{\int }_{D_y}\{|\nabla f_1{|}^2+V_1{f}_1^2\}+\epsilon CD|Q_{+}|/(1-\epsilon D)}{{\sum }_{y\in Q_{+}}{\int }_{D_y}{f}_1^2}$$

$$=\frac{{\int }_{M_0}\{|\nabla f{|}^2+V{f}^2\}+\epsilon CD/(1-\epsilon D)}{{\int }_{M_0}{f}^2}$$

$$=R(f)+\frac{\epsilon CD}{(1-\epsilon D){\int }_{M_0}{f}^2}.$$

Full text

We would like to thank the Max Planck Institute for Mathematics and the Hausdorff Center for Mathematics in Bonn for their support.

1. Introduction

In this article, we study the behaviour under coverings of the bottom of the spectrum of Schrödinger operators on Riemannian manifolds.

Let $M$ be a connected Riemannian manifold, not necessarily complete, and $V\colon M\to\mathbb{R}$ be a smooth potential with associated Schrödinger operator $\Delta+V$. We consider $\Delta+V$ as an unbounded symmetric operator in the space $L^{2}(M)$ of square integrable functions on $M$ with domain $C^{\infty}_{c}(M)$, the space of smooth functions on $M$ with compact support.

For a non-vanishing Lipschitz continuous function on $M$ with compact support in $M$, we call

[TABLE]

the Rayleigh quotient of $f$. We let

[TABLE]

where $f$ runs through all non-vanishing Lipschitz continuous functions on $M$ with compact support in $M$. If $\lambda_{0}(M,V)>-\infty$, then $\Delta+V$ is bounded from below on $C^{\infty}_{c}(M)$ and $\lambda_{0}(M,V)$ is equal to the bottom of the spectrum of the Friedrichs extension of $\Delta+V$. If $\lambda_{0}(M,V)=-\infty$, then the spectrum of any self-adjoint extension of $\Delta+V$ is not bounded from below.

Recall that $\Delta+V$ is essentially self-adjoint on $C^{\infty}_{c}(M)$ if $M$ is complete and $\inf V>-\infty$. Then the unique self-adjoint extension of $\Delta+V$ is its closure. In the case where $M$ is the interior of a complete Riemannian manifold $N$ with smooth boundary and where $V$ extends smoothly to the boundary of $N$, $\lambda_{0}(M,V)$ is equal to the bottom of the Dirichlet spectrum of $\Delta+V$ on $N$.

In the case of the Laplacian, that is, $V=0$, we also write $\lambda_{0}(M)$ and call it the bottom of the spectrum of $M$. It is well known that $\lambda_{0}(M)$ is the supremum over all $\lambda\in\mathbb{R}$ such that there is a positive smooth $\lambda$-eigenfunction $f\colon M\to\mathbb{R}$ (see, e.g., [4, Theorem 7], [5, Theorem 1], or [6, Theorem 2.1].) It is crucial that these eigenfunctions are not required to be square-integrable. In fact, $\lambda_{0}(M)$ is exactly the border between the positive and the $L^{2}$ spectrum of $\Delta$ (see, e.g., [6, Theorem 2.2]).

Suppose now that $M$ is simply connected and let $\pi_{0}\colon M\to M_{0}$ and $\pi_{1}\colon M\to M_{1}$ be Riemannian subcovers of $M$. Let $\Gamma_{0}$ and $\Gamma_{1}$ be the groups of covering transformations of $\pi_{0}$ and $\pi_{1}$, respectively, and assume that $\Gamma_{1}\subseteq\Gamma_{0}$. Then the resulting Riemannian covering $\pi\colon M_{1}\to M_{0}$ satisfies $\pi\circ\pi_{1}=\pi_{0}$. Let $V_{0}\colon M_{0}\to\mathbb{R}$ be a smooth potential and set $V_{1}=V_{0}\circ\pi$.

Since the lift of a positive $\lambda$-eigenfunction of $\Delta$ on $M_{0}$ to $M_{1}$ is a positive $\lambda$-eigenfunction of $\Delta$, we always have $\lambda_{0}(M_{0})\leq\lambda_{0}(M_{1})$ by the above characterization of the bottom of the spectrum of $\Delta$ by positive eigenfunctions. In Section 4, we present a short and elementary proof of the inequality which does not rely on the characterization of $\lambda_{0}$ by positive eigenfunctions:

Theorem 1.3**.**

For any Riemannian covering $\pi\colon M_{1}\to M_{0}$ as above,

[TABLE]

Brooks showed in [3, Theorem 1] that $\lambda_{0}(M_{0})=\lambda_{0}(M_{1})$ in the case where $M_{0}$ is complete, has finite topological type, and $\pi$ is normal with amenable group $\Gamma_{1}\backslash\Gamma_{0}$ of covering transformations. Bérard and Castillon extended this in [1, Theorem 1.1] to $\lambda_{0}(M_{0},V_{0})=\lambda_{0}(M_{1},V_{1})$ in the case where $M_{0}$ is complete, $\pi_{1}(M_{0})$ is finitely generated (this assumption occurs in point (1) of their Section 3.1), and the right action of $\Gamma_{0}$ on $\Gamma_{1}\backslash\Gamma_{0}$ is amenable. We generalize these results as follows:

Theorem 1.4**.**

If the right action of $\Gamma_{0}$ on $\Gamma_{1}\backslash\Gamma_{0}$ is amenable, then

[TABLE]

Here a right action of a countable group $\Gamma$ on a countable set $X$ is said to be amenable if there exists a $\Gamma$-invariant mean on $L^{\infty}(X)$. This holds if and only if the action satisfies the Følner condition: For any finite subset $G\subseteq\Gamma$ and $\varepsilon>0$, there exists a non-empty, finite subset $F\subseteq X$, a Følner set, such that

[TABLE]

for all $g\in G$. By definition, $\Gamma$ is amenable if the right action of $\Gamma$ on itself is amenable, and then any action of $\Gamma$ is amenable.

In comparison with the results of Brooks, Bérard, and Castillon, the main point of 1.4 is that we do not need any assumptions on metric and topology of $M_{0}$. A main new point of our arguments is that we adopt our constructions more carefully to the different competitors for $\lambda_{0}$ separately.

2. Fundamental domains and partitions of unity

Choose a complete Riemannian metric $h$ on $M_{0}$. In what follows, geodesics, distances, and metric balls in $M_{0}$, $M_{1}$, and $M$ are taken with respect to $h$ and its lifts to $M_{1}$ and $M$, respectively.

Fix a point $x$ in $M_{0}$. For any $y\in\pi^{-1}(x)$, let

[TABLE]

be the fundamental domain of $\pi$ centered at $y$. Then $D_{y}$ is closed in $M_{1}$, the boundary $\partial D_{y}$ of $D_{y}$ has measure zero in $M_{1}$, and $\pi\colon D_{y}\setminus\partial D_{y}\to M_{0}\setminus C$ is an isometry, where $C$ is a subset of the cut locus ${\rm Cut}(x)$ of $x$ in $M_{0}$. Recall that ${\rm Cut}(x)$ is of measure zero. Moreover, $M_{1}=\cup_{y\in\pi^{-1}(x)}D_{y}$, $y\in\pi^{-1}(x)$.

Lemma 2.2**.**

For any $\rho>0$, there is an integer $N(\rho)$ such that any $z$ in $M_{1}$ is contained in at most $N(\rho)$ metric balls $B(y,\rho)$, $y\in\pi^{-1}(x)$.

Proof.

Let $z\in B(z_{1},\rho)\cap B(y_{2},\rho)$ with $y_{1}\neq y_{2}$ in $\pi^{-1}(x)$ and $\gamma_{1},\gamma_{2}\colon[0,1]\to M_{1}$ be minimal geodesics from $y_{1}$ to $z$ and $y_{2}$ to $z$, respectively. Then $\sigma_{1}=\pi\circ\gamma_{1}$ and $\sigma_{2}=\pi\circ\gamma_{2}$ are geodesic segments form $x$ to $\pi(z)$. Since $y_{1}\neq y_{2}$, $\sigma_{1}$ and $\sigma_{2}$ are not homotopic relative to $\{0,1\}$. Hence, if $z$ lies in in the intersection of $n$ pairwise different balls $B(y_{i},\rho)$ with $y_{1},\dots,y_{n}\in\pi^{-1}(x)$, then the concatenations $\sigma_{1}^{-1}*\sigma_{i}$ represent $n$ pairwise different homotopy classes of loops at $x$ of length at most $2\rho$. Hence $n$ is at most equal to the number $N(\rho)$ of homotopy classes of loops at $x$ with representatives of length at most $2\rho$. ∎

Lemma 2.3**.**

If $K\subseteq M_{0}$ is compact, then $\pi^{-1}(K)\cap D_{y}$ is compact. More precisely, if $K\subseteq B(x,r)$, then $\pi^{-1}(K)\cap D_{y}\subseteq B(y,r)$.

Proof.

Choose $r>0$ such that $K\subseteq B(x,r)$. Let $z\in\pi^{-1}(K)\cap D_{y}$ and $\gamma_{0}$ be a minimal geodesic from $\pi(z)\in K$ to $x$. Let $\gamma$ be the lift of $\gamma_{0}$ to $M_{1}$ starting in $z$. Then $\gamma$ is a minimal geodesic from $z$ to some point $y^{\prime}\in\pi^{-1}(x)$. Since $z\in D_{y}$, this implies

[TABLE]

Hence $\pi^{-1}(K)\cap D_{y}\subseteq B(y,r)$. ∎

Let $K\subseteq M_{0}$ be a compact subset and choose $r>0$ such that $K\subseteq B(x,r)$. Let $\psi\colon\mathbb{R}\to\mathbb{R}$ be the function which is equal to $1$ on $(-\infty,r]$, to $t+1-r$ for $r\leq t\leq r+1$, and to [math] on $[r+1,\infty]$. For $y\in\pi^{-1}(x)$, let $\psi_{y}=\psi_{y}(z)=\psi(d(z,y))$. Note that $\psi_{y}=1$ on $\pi^{-1}(K)\cap D_{y}$ and that $\operatorname{supp}\psi_{y}=\bar{B}(y,r+1)$.

Lemma 2.4**.**

Any $z$ in $M_{1}$ is contained in the support of at most $N(r+1)$ of the functions $\psi_{y}$, $y\in\pi^{-1}(x)$.

Proof.

This is clear from 2.2 since $\operatorname{supp}\psi_{y}$ is contained in the ball $B(y,r+1)$. ∎

In particular, each point of $M_{1}$ lies in the support of only finitely many of the functions $\psi_{y}$. Therefore the function $\psi_{1}=\max\{1-\sum\psi_{y},0\}$ is well defined. By 2.3, we have $\operatorname{supp}\psi_{1}\cap\pi^{-1}(K)=\emptyset$. Together with $\psi_{1}$, the functions $\psi_{y}$ lead to a partition of unity on $M_{1}$ with functions $\varphi_{1}$ and $\varphi_{y}$, $y\in\pi^{-1}(x)$, given by

[TABLE]

Note that $\operatorname{supp}\varphi_{1}=\operatorname{supp}\psi_{1}$ and $\operatorname{supp}\varphi_{y}=\operatorname{supp}\psi_{y}$ for all $y\in\pi^{-1}(x)$.

Lemma 2.6**.**

The functions $\varphi_{y}$, $y\in\pi^{-1}(x)$, are Lipschitz continuous with Lipschitz constant $3N(r+1)$.

Proof.

The functions $\psi_{y}$, $y\in\pi^{-1}(x)$, are Lipschitz continuous with Lipschitz constant $1$ and take values in $[0,1]$. Hence $\psi_{1}$ is Lipschitz continuous with Lipschitz constant $N=N(r+1)$, by 2.4, and takes values in $[0,1]$. Therefore the denominator $\chi=\psi_{1}+\sum_{z\in\pi^{-1}(x)}\psi_{z}$ in the fraction defining the $\varphi_{y}$ is Lipschitz continuous and takes values in $[1,N]$. Hence

[TABLE]

As a consequence of 2.6, we get that $\varphi_{1}=1-\sum\varphi_{y}$ is also Lipschitz continuous with Lipschitz constant $6N(r+1)^{2}$.

3. Pulling up

Let $f$ be a non-vanishing Lipschitz continuous function on $M_{0}$ with compact support and let $f_{1}=f\circ\pi$. We will construct a cutoff function $\chi$ on $M_{1}$ such that $R(\chi f_{1})$ is close to $R(f)$.

Let $g$ be the given Riemannian metric on $M$ and $h$ be a complete background Riemannian metric on $M$ as in Section 2. Then there is a constant $A\geq 1$ such that

[TABLE]

on the support of $f$. We continue to take distances and metric balls in $M_{0}$, $M_{1}$, and $M$ with respect to $h$ and its respective lifts to $M_{1}$ and $M$.

Fix a point $x$ in $M_{0}$. With $K=\operatorname{supp}f$ and $r>0$ such that $K\subseteq B(x,r)$, we get a partition of unity with functions $\varphi_{1}$ and $\varphi_{y}$, $y\in\pi^{-1}(x)$, as above.

Fix preimages $u\in M$ and $y=\pi_{1}(u)\in M_{1}$ of $x$ under $\pi_{0}$ and $\pi$, respectively. Write $\pi_{0}^{-1}(x)=\Gamma_{0}u$ as the union of $\Gamma_{1}$-orbits $\Gamma_{1}gu$, where $g$ runs through a set $R$ of representatives of the right cosets of $\Gamma_{1}$ in $\Gamma_{0}$, that is, of the elements of $\Gamma_{1}\backslash\Gamma_{0}$. Then $\pi^{-1}(x)=\{\pi_{1}(gu)\mid g\in R\}$. Let

[TABLE]

Since the fibres of $\pi$ and $\pi_{0}$ are discrete, $S$ and $T$ are finite subsets of $\Gamma_{0}$, hence also $G$.

Let $\varepsilon>0$ and $F\subseteq\Gamma_{1}\backslash\Gamma_{0}$ be a Følner set for $G$ and $\varepsilon$ satisfying (1.5). Let

[TABLE]

and set

[TABLE]

Since $|P|=|F|<\infty$, $\operatorname{supp}\chi$ is compact. Hence, by 2.6, $\chi f_{1}$ is compactly supported and Lipschitz continuous on $M_{1}$. Let

[TABLE]

To estimate the Rayleigh quotient of $\chi f_{1}$, it suffices to consider $\chi f_{1}$ on the union of the $D_{y}$, $y\in Q$. We first observe that

[TABLE]

To show this, let $y=\pi_{1}(gu)$ and observe that $f_{1}$ does not vanish identically on $\pi^{-1}(K)\cap D_{y}$ and that $\varphi_{y}$ is positive on $\pi^{-1}(K)\cap D_{y}$. Since $R$ is a set of representatives of the right cosets of $\Gamma_{1}$ in $\Gamma_{0}$, there exists a one-to-one correspondence between $P$ and $P_{1}$, and hence

[TABLE]

The problematic subset of $Q$ is

[TABLE]

Let now $y\in Q_{-}$ and $z\in\pi^{-1}(K)\cap D_{y}$ with $0<\chi(z)<1$. Since $\pi_{1}(gu)$, $g\in R$, runs through all points of $\pi^{-1}(x)$, we have $\sum_{g\in R}\varphi_{\pi_{1}(gu)}(z)=1$. Hence there are $g_{1},\dots,g_{k}\in R\setminus P$ such that $\varphi_{\pi_{1}(g_{i}u)}(z)\neq 0$ and

[TABLE]

Furthermore, there has to be a $g\in P$ with $\varphi_{\pi_{1}(gu)}(z)\neq 0$. Then the supports of the functions $\varphi_{\pi_{1}(gu)}$ and $\varphi_{\pi_{1}(g_{i}u)}$ intersect and we get $d(\pi_{1}(gu),\pi_{1}(g_{i}u))\leq 2r+2$. That is, we have $d(gu,h_{i}g_{i}u)\leq 2r+2$ for some $h_{i}\in\Gamma_{1}$. We conclude that

[TABLE]

Since $\pi_{1}$ is distance non-increasing, we get that there are $s_{i}\in S$ and $t_{i}\in T$ such that $g^{-1}h_{i}g_{i}=t_{i}s_{i}$, and then $h_{i}g_{i}=gt_{i}s_{i}$. Since $g_{i}\notin P$, we conclude that $\Gamma_{1}gt_{i}s_{i}\notin F$, i.e., $\Gamma_{1}g\in F\setminus F(t_{i}s_{i})^{-1}$. Since $(t_{i}s_{i})^{-1}\in G$, there are at most $\varepsilon|F||G|$ such elements $g\in P$. Since $d(y,z)\leq r$ and $d(z,\pi_{1}(gu))\leq r+1$, we conclude with 2.2 that for fixed $g\in P$ there are at most $N(2r+1)$ such $y\in Q$. We conclude that

[TABLE]

We now estimate the Rayleigh quotient of $\chi f_{1}$. For any $y\in Q_{+}=Q\setminus Q_{-}$, we have $\chi=1$ on $\pi^{-1}(K)\cap D_{y}$ and therefore

[TABLE]

and

[TABLE]

where, here and below, integrals, gradients, and norms are taken with respect to the original Riemannian metric $g$ on $M$.

For any $y\in Q_{-}$, we have

[TABLE]

where $C_{0}$ is the maximum of $|V_{0}|$ on $\operatorname{supp}f=K$. By 2.4, 2.6, and (3.1), we have $|\nabla\chi|^{2}\leq 9N(r+1)^{4}A$. Therefore

[TABLE]

In conclusion,

[TABLE]

for any $y\in Q_{-}$, where $C>0$ is an appropriate constant, which depends on $f$, but not on $y$ or the choice of $\varepsilon$ and $F$. With $D=|G|N(2r+1)$, we obtain from (3.2) that

[TABLE]

and conclude that

[TABLE]

For $\varepsilon\to 0$, the right hand side converges to $R(f)$.

Proof of 1.4.

By 1.3, we have $\lambda_{0}(M_{0},V_{0})\leq\lambda_{0}(M_{1},V_{1})$. By (1.1), the bottom of the spectrum of Schrödinger operators is given by the infimum of corresponding Rayleigh quotients $R(f)$ of Lipschitz continuous functions with compact support. The arguments above show that, for any such function $f$ on $M_{0}$ and any $\delta>0$, there is a Lipschitz continuous function $\chi f_{1}$ on $M_{1}$ with compact support and Rayleigh quotient at most $R(f)+\delta$. Therefore we also have $\lambda_{0}(M_{0},V_{0})\geq\lambda_{0}(M_{1},V_{1})$. ∎

4. Pushing down

Let $f$ be a Lipschitz continuous function on $M_{1}$ with compact support. Define the push down $f_{0}\colon M_{0}\to\mathbb{R}$ of $f$ by

[TABLE]

Since $\operatorname{supp}f$ is compact, the sum on the right hand side is finite for all $x\in M_{0}$, and hence $f_{0}$ is well defined. We have $\operatorname{supp}f_{0}=\pi(\operatorname{supp}f)$, and hence $\operatorname{supp}f_{0}$ is compact. Furthermore, $f_{0}$ is differentiable at each point $x$, where $f$ is differentiable at all $y\in\pi^{-1}(x)$ and $f(y)\neq 0$ for some $y\in\pi^{-1}(x)$, and then

[TABLE]

For the norm of the differential of $f_{0}$ at $x$, we get

[TABLE]

Furthermore, $f_{0}$ is differentiable with vanishing differential at almost any point of $\{f_{0}=0\}$. Therefore $f_{0}$ is Lipschitz continuous and

[TABLE]

In particular, we have $R(f_{0})\leq R(f)$.

Proof of 1.3.

For any non-vanishing Lipschitz continuous function $f$ on $M_{1}$ with compact support, the push down $f_{0}$ as above is a Lipschitz continuous function on $M_{0}$ with compact support and Rayleigh quotient $R(f_{0})\leq R(f)$. The asserted inequality follows now from the characterization of the bottom of the spectrum by Rayleigh quotients as in (1.1). ∎

5. Final remarks

It is well-known that any countable group is the fundamental group of a smooth $4$-manifold. (A variant of the usual argument for finitely presented groups, taking connected sums of $S^{1}\times S^{3}$ and performing surgeries, can be used to produce $5$-manifolds with fundamental group any countable group.) In particular, for a non-finitely generated, amenable group $G$, e.g., $G=\bigoplus_{n\in\mathbb{N}}\mathbb{Z}$ or $G=\mathbb{Q}$, there is a smooth manifold $M$ with $\pi_{1}(M)\cong G.$ In contrast to the results in [1, 3], our main result also applies to such examples.

Moreover, we do not assume $\lambda_{0}(M_{0},V_{0})>-\infty.$ Given any non-compact manifold $M_{0}$, it is indeed easy to construct a smooth potential $V_{0}$ such that $\lambda_{0}(M_{0},V_{0})=-\infty.$ In fact, it suffices that $V_{0}(x)$ tends to $-\infty$ sufficiently fast as $x\to\infty$.