On the bottom of spectra under coverings

Abstract

For a Riemannian covering $M_1\rightarrow M_0$ of connected Riemannian manifolds with 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.

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 $V:M\rightarrow \mathbb {R}$ 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

$$\begin{aligned} R(f) = \frac{\int _M ( |\nabla f|^2 + Vf^2 )}{\int _M f^2} \end{aligned}$$

(1.1)

the Rayleigh quotient of f. We let

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

(1.2)

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 $f:M\rightarrow \mathbb {R}$ (see, e.g., [3, Theorem 7], [4, Theorem 1], or [5, 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., [5, Theorem 2.2]).

Suppose now that M is simply connected and let $\pi _0:M\rightarrow M_0$ and $\pi _1:M\rightarrow M_1$ 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 $\pi :M_1\rightarrow M_0$ satisfies $\pi \circ \pi _1=\pi _0$. Let $V_0:M_0\rightarrow \mathbb {R}$ 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 $\lambda _0(M_0)\le \lambda _0(M_1)$ by the above characterization of the bottom of the spectrum of $\Delta$ by positive eigenfunctions. In Sect. 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 1.1

For any Riemannian covering $\pi :M_1\rightarrow M_0$ as above,

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

Brooks showed in [2, 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 1.2

If the right action of $\Gamma _0$ on $\Gamma _1\backslash \Gamma _0$ is amenable, then

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

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

$$\begin{aligned} |F{\setminus } Fg| \le \varepsilon |F| \end{aligned}$$

(1.3)

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 Theorem 1.2 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

$$\begin{aligned} D_y = \{ z\in M_1 \mid d(z,y)\le d(z,y') \text { for all } y'\in \pi ^{-1}(x) \} \end{aligned}$$

(2.1)

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 $\pi :D_y{\setminus }\partial D_y\rightarrow M_0{\setminus } C$ is an isometry, where C is a subset of the cut locus $\mathrm{Cut}(x)$ of x in $M_0$. Recall that $\mathrm{Cut}(x)$ is of measure zero. Moreover, $M_1=\cup _{y\in \pi ^{-1}(x)}D_y$, $y\in \pi ^{-1}(x)$.

Lemma 2.1

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 $z\in B(y_1,\rho )\cap B(y_2,\rho )$ with $y_1\ne y_2$ in $\pi ^{-1}(x)$ and $\gamma _1,\gamma _2:[0,1]\rightarrow M_1$ 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 from x to $\pi (z)$. Since $y_1\ne y_2$, $\sigma _1$ 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 $2\rho$. $\square$

Lemma 2.2

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 $y'\in \pi ^{-1}(x)$. Since $z\in D_y$, this implies

$$\begin{aligned} d(z,y) \le d(z,y') \le L(\gamma ) = L(\gamma _0) < r. \end{aligned}$$

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

Let $K\subseteq M_0$ be a compact subset and choose $r>0$ such that $K\subseteq B(x,r)$. Let $\psi :\mathbb {R}\rightarrow \mathbb {R}$ be the function which is equal to 1 on $(-\infty ,r]$, to $t+1-r$ for $r\le t\le r+1$, and to 0 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 $\mathrm{supp}\,\psi _y=\bar{B}(y,r+1)$.

Lemma 2.3

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 Lemma 2.1 since $\mathrm{supp}\,\psi _y$ is contained in the ball $B(y,r+1)$. $\square$

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 Lemma 2.2, we have $\mathrm{supp}\,\psi _1\cap \pi ^{-1}(K)=\emptyset$. 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

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

(2.2)

Note that $\mathrm{supp}\,\varphi _1=\mathrm{supp}\,\psi _1$ and $\mathrm{supp}\,\varphi _y=\mathrm{supp}\,\psi _y$ for all $y\in \pi ^{-1}(x)$.

Lemma 2.4

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 Lemma 2.3, 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

$$\begin{aligned} |\varphi _y(z_1) - \varphi _y(z_2)|&\le \frac{|(\chi (z_2)-\chi (z_1))\psi _y(z_1) + \chi (z_1)(\psi _y(z_1)-\psi _y(z_2))|}{\chi (z_1)\chi (z_2)} \\&\le \frac{(2N+N) d(z_1,z_2)}{\chi (z_1)\chi (z_2)} \le 3N d(z_1,z_2). \end{aligned}$$

$\square$

As a consequence of Lemma 2.4, 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 $M_0$ and h be a complete background Riemannian metric on $M_0$ as in Sect. 2. Then there is a constant $A\ge 1$ such that

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

(3.1)

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 $K=\mathrm{supp}\,f$ 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 $\pi _0^{-1}(x)=\Gamma _0u$ as the union of $\Gamma _1$-orbits $\Gamma _1gu$, 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

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

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 (1.3). Let

$$\begin{aligned} P = \{ g \in R \mid \Gamma _1g \in F \} \subseteq R \end{aligned}$$

and set

$$\begin{aligned} \chi = \sum _{g\in P}\varphi _{\pi _1(gu)}. \end{aligned}$$

Since $|P|=|F|<\infty$, $\mathrm{supp}\,\chi$ is compact. Hence, by Lemma 2.4, $\chi f_1$ is compactly supported and Lipschitz continuous on $M_1$. Let

$$\begin{aligned} Q = \{ y \in \pi ^{-1}(x) \mid (\chi f_1)(z)\ne 0 \text { for some } z\in D_y \}. \end{aligned}$$

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

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

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

$$\begin{aligned} |P| = |P_1| \le |Q|. \end{aligned}$$

The problematic subset of Q is

$$\begin{aligned} Q_- = \{ y \in Q \mid 0<\chi (z)<1 \text { for some } z\in \pi ^{-1}(K)\cap D_y \}. \end{aligned}$$

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 $\varphi _{\pi _1(g_iu)}(z)\ne 0$ and

$$\begin{aligned} \chi (z) + \sum \varphi _{\pi _1(g_iu)}(z) =1. \end{aligned}$$

Furthermore, there has to be a $g\in P$ with $\varphi _{\pi _1(gu)}(z)\ne 0$. Then the supports of the functions $\varphi _{\pi _1(gu)}$ and $\varphi _{\pi _1(g_iu)}$ intersect and we get $d(\pi _1(gu),\pi _1(g_iu))\le 2r+2$. That is, we have $d(gu,h_ig_iu)\le 2r+2$ for some $h_i\in \Gamma _1$. We conclude that

$$\begin{aligned} d(u,g^{-1}h_ig_iu) = d(gu,h_ig_iu) \le 2r+2. \end{aligned}$$

Since $\pi _1$ is distance non-increasing, we get that there are $s_i\in S$ and $t_i\in T$ such that $g^{-1}h_ig_i=t_is_i$, and then $h_ig_i=gt_is_i$. Since $g_i\notin P$, we conclude that $\Gamma _1gt_is_i\notin F$, i.e., $\Gamma _1 g \in F {\setminus } F(t_is_i)^{-1}$. Since $(t_is_i)^{-1}\in G$, there are at most $\varepsilon |F||G|$ such elements $g\in P$. Since $d(y,z)\le r$ and $d(z,\pi _1(gu))\le r+1$, we conclude with Lemma 2.1 that for fixed $g \in P$ there are at most $N(2r+1)$ such $y\in Q$. We conclude that

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

(3.2)

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

$$\begin{aligned} \begin{aligned} \int _{D_y} \{|\nabla (\chi f_1)|^2 + V_1(\chi f_1)^2 \}&= \int _{D_y} \{|\nabla f_1|^2 + V_1f_1^2 \} \\&= \int _{M_0} \{ |\nabla f|^2 + V_{0} f^2 \} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \int _{D_y} \chi ^2f_1^2 = \int _{D_y} f_1^2 = \int _{M_0} f^2, \end{aligned}$$

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

$$\begin{aligned} \int _{D_y} \chi ^2f_1^2 \le \int _{M_0} f^2 \quad \text {and}\quad \int _{D_y} |V_1|\chi ^2f_1^2 \le C_0 \int _{M_0} f^2, \end{aligned}$$

where $C_0$ is the maximum of $|V_0|$ on $\mathrm{supp}\,f=K$. By Lemma 2.3, Lemma 2.4, and (3.1), we have $|\nabla \chi |^2\le 9N(r+1)^4A$ on the support of f. Therefore

$$\begin{aligned} \int _{D_y} |\nabla (\chi f_1)|^2&\le 2\int _{D_y} \{|\nabla \chi |^2 f^2 + \chi ^2|\nabla f\circ \pi |^2| \} \\&\le 18N(r+1)^4A \int _{M_0} f^2 + 2\int _{M_0} |\nabla f|^2. \end{aligned}$$

In conclusion,

$$\begin{aligned} \int _{D_y} \{ |\nabla (\chi f_1)|^2 + |V_1|\chi ^2f_1^2 \} \le C \end{aligned}$$

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

$$\begin{aligned} |Q_-| \le \frac{\varepsilon D}{1-\varepsilon D}|Q_+|, \end{aligned}$$

and conclude that

$$\begin{aligned} R(\chi f_1)&= \frac{\int \{ |\nabla (\chi f_1)|^2 + V_1\chi ^2f_1^2 \}}{\int (\chi f_1)^2} \\&= \frac{\sum _{y\in Q} \int _{D_y} \{ |\nabla f_1|^2 + V_1f_1^2 \}}{\sum _{y\in Q} \int _{D_y} f_1^2} \\&\le \frac{\sum _{y\in Q_+} \int _{D_y} \{ |\nabla f_1|^2 + V_1f_1^2 \} + \varepsilon CD |Q_+|/(1-\varepsilon D) }{\sum _{y\in Q_+} \int _{D_y} f_1^2} \\&= \frac{\int _{M_0} \{ |\nabla f|^2 + V_{0} f^2 \} + \varepsilon CD/(1-\varepsilon D) }{\int _{M_0} f^2} \\&= R(f) + \frac{\varepsilon CD}{(1-\varepsilon D)\int _{M_0} f^2}. \end{aligned}$$

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

Proof of Theorem 1.2

By Theorem 1.1, we have $\lambda _0(M_0,V_0)\le \lambda _0(M_1,V_1)$. By (1.2), 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 $\lambda _0(M_0,V_0)\ge \lambda _0(M_1,V_1)$.$\square$

4 Pushing down

Let f be a Lipschitz continuous function on $M_1$ with compact support. Define the push down $f_0:M_0\rightarrow \mathbb {R}$ of f by

$$\begin{aligned} f_0(x) = \left( \sum _{y\in \pi ^{-1}(x)}f(y)^2 \right) ^{1/2}. \end{aligned}$$

Since $\mathrm{supp}\,f$ 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 $\mathrm{supp}\,f_0=\pi (\mathrm{supp}\,f)$, and hence $\mathrm{supp}\,f_0$ is compact. Furthermore, $f_0$ is differentiable at each point x, where f is differentiable at all $y\in \pi ^{-1}(x)$ and $f(y)\ne 0$ for some $y\in \pi ^{-1}(x)$, and then

$$\begin{aligned} \nabla f_0(x) = \frac{1}{f_0(x)}\sum _{y\in \pi ^{-1}(x)}f(y)\pi _*(\nabla f(y)). \end{aligned}$$

For the norm of the differential of $f_0$ at x, we get

$$\begin{aligned} |\nabla f_0(x)|^2&\le \frac{1}{f_0(x)^2}\left| \sum _{y\in \pi ^{-1}(x)}f(y)\pi _*(\nabla f(y)) \right| ^2\\&\le \frac{1}{f_0(x)^2} \sum _{y\in \pi ^{-1}(x)}f(y)^2\sum _{y\in \pi ^{-1}(x)}|\nabla f(y)|^2 \\&= \sum _{y\in \pi ^{-1}(x)}|\nabla f(y)|^2. \end{aligned}$$

Furthermore, $f_0$ is differentiable with vanishing differential at almost any point of $\{f_0=0\}$. Therefore $f_0$ is Lipschitz continuous and

$$\begin{aligned} \int _{M_0} f_0^2 = \int _{M_1} f^2, \quad \int _{M_0} V_0f_0^2 = \int _{M_1} V_1f^2, \quad \int _{M_0} |\nabla f_0|^2 \le \int _{M_1} |\nabla f|^2. \end{aligned}$$

In particular, we have $R(f_0)\le R(f)$.

Proof of Theorem 1.1

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 $R(f_0)\le R(f)$. The asserted inequality follows now from the characterization of the bottom of the spectrum by Rayleigh quotients as in (1.2). $\square$

5 Final remarks

It is well-known that any countable group is the fundamental group of a smooth four-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 five-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, 2], 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 $x\rightarrow \infty$.