EXISTENCE OF THE SPECTRAL GAP FOR ELLIPTIC OPERATORS

Feng-Yu Wang

(Department of Mathematics, Beijing Normal University, Beijing 100875, China) e-mail: wangfy@bnu.edu.cn

Abstract. Let M be a connected, noncompact, complete Riemannian manifold, consider the operator L = Δ + ∇V for some V ∈ C^{2}(M) with exp[V] integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of L. As a consequence of the main result, let ρ be the distance function from a point o, then the spectral gap exists provided lim_{ρ→∞} sup Lρ < 0 while the spectral gap does not exist if o is a pole and lim_{ρ→∞} inf Lρ ≥ 0. Moreover, the elliptic operators on ℝ^{d} are also studied.

1. Introduction

Let M̃ be a d-dimensional, connected, noncompact, complete Riemannian manifold, and let M be either M̃ or an unbounded regular closed domain in M̃. Next, consider L = Δ+∇V for some V ∈ C^{∞}(M) with Z := ∫_{M} exp[V]dx < ∞. Let dμ = Z^{−1} exp[V]dx be defined on M. The spectral gap of the operator L (with Neumann boundary condition if ∂M ≠ ∅) is characterized as

λ_{1} = inf \{\frac{μ(|∇f|^{2})}{μ(f^{2}) − μ(f)^{2}} : f ∈ C^{1}(M) ∩ L^{2}(μ), f ≠ constant\}.  (1.1)

We say the spectral gap of L exists if λ_{1} > 0. From now on, we assume that L is regular in the sense that C_{0}^{∞}(M) is dense in W^{1,2}(M, dμ) with the Sobolev norm ∥·∥_{L^{2}(μ)}+∥∇·∥_{L^{2}(μ)}.

According to Wang^{[12]} and Chen-Wang^{[3]}, we have λ_{1} > 0 provided the Ricci curvature is bounded below and Hess_{V} is uniformly negatively definite out of a compact domain. Actually, the recent work by the author [14] shows that this condition implies the logarithmic Sobolev inequality which is stronger than the existence of spectral gap. Moreover, [13] proved that the logarithmic Sobolev inequality is equivalent to an exponential integrability

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 35P15, 60H30.

Key words and phrases. Spectral gap, elliptic operator, essential spectrum.

Research supported in part by NSFC(19631060), Fok Ying-Tung Educational Foundation and Scientific Research Foundation for Returned Overseas Chinese Scholars. Research at MSRI is supported in part by NSF grant DMS-9701755.

of the distance function square, which naturally refers to the negativity of Hess_{V} along the radial direction.

On the other hand, we know that the spectral gap may exist if the distance function itself is exponential integrable. For instance, let M = [0, ∞) and L = \frac{d^{2}}{dr^{2}} − c\frac{d}{dr} , c > 0, then (see [4; Example 2.8]) λ_{1} = c^{2}/4 > 0. From this we may guess that the existence of spectral gap, unlike the logarithmic Sobolev inequality, essentially depends on the first order radialdirection derivative of V rather than the second order derivative. This observation is now supported by Corollary 1.4 in the paper.

Our study is based on the fact that λ_{1} > 0 is equivalent to inf σ_{ess}(−L) > 0, where σ_{ess}(−L) denotes the essential spectrum of −L (with Neumann boundary condition if ∂M ≠ ∅). To see this one need only to show that 0 is an eigenvalue with multiplicity 1, equivalently, for any f ∈ L^{2}(μ) with Lf = 0 and νf|_{∂M} = 0 ( where ν denotes the inward unit normal vector field of ∂M when it is nonempty), one has f is constant. This is a consequence of a result in Sturm [10].

Next, for fixed o ∈ M̃, let ρ(x) be the Riemannian distance function from o. For r > 0, let B_{r} = {x ∈ M : ρ(x) < r} and B_{r}^{c} = M \ B_{r}. by Donnely-Li’s decomposition theorem (see [5]), one has inf σ_{ess}(−L) = lim_{r→∞} λ^{c}(r), where

λ^{c}(r) = inf{μ(|∇f|^{2}) : f ∈ C^{1}(M), μ(f^{2}) = 1, f = 0 on B_{r}}.

Thence, λ_{1} > 0 is equivalent to λ^{c}(r) > 0 for large r. More precisely, we have the following.

Theorem 1.1. (1) If μ(B_{r}) > 0, then

λ_{1} ≤ λ^{c}(r)/μ(B_{r}).  (1.2)

(2) Let λ(R) be the smallest positive Neumann eigenvalue of −L on B_{R}. If λ^{c}(r) > 0, then

λ_{1} ≥ sup_{R>r} \frac{λ^{c}(r)λ(R)μ(B_{R})(R − r)^{2} − 2λ(R)(1 − μ(B_{R}))}{2λ(R)(R − r)^{2} + λ^{c}(r)(R − r)^{2}μ(B_{R}) + 2μ(B_{R})} > 0.  (1.3)

We now go to estimate the quantity λ^{c}(r). For D ≥ 0, define

γ(r) = sup_{ρ(x)=r,x∉cut(o)} Lρ(x), C(r) = ∫_{D+1}^{r} γ(s)ds, r > D.

Here and in what follows, the point x runs over M.

Theorem 1.2. Suppose that there exists D > 0 such that either ∂M ⊂ B_{D} or νρ ≤ 0 on ∂M ∩ (B_{D}^{c} \ cut(o)). For any r_{0} > D and positive function f ∈ C[r_{0}, ∞), we have

λ^{c}(r_{0}) ≥ inf_{t≥r_{0}} f(t){ ∫_{r_{0}}^{t} exp[−C(r)]dr ∫_{r}^{∞} exp[C(s)]f(s)ds}^{−1}.  (1.4)

Consequently, we have λ_{1} > 0 provided there exists positive f ∈ C[D + 1, ∞) such that

sup_{t≥D+1} \frac{1}{f(t)} ∫_{D+1}^{t} exp[−C(r)]dr ∫_{r}^{∞} exp[C(s)]f(s)ds < ∞.  (1.5)

We remark that the assumption in Theorem 1.2 holds if either ∂M is bounded or o ∈ M and M is convex. Especially, for the case M = [0, ∞) and o = 0, we have C(r) = V (r). By [4; Theorem 2.1], if f′ > 0 then

λ_{1} ≥ inf_{t≥0} f′(t) exp[C(t)]{ ∫_{t}^{∞} exp[C(s)]f(s)ds}^{−1}.  (1.6)

Hence, the second assertion in Theorem 1.2 can be regarded as an extension of [4; Theorem 2.1] to Riemannian manifolds.

Corollary 1.3. Under the assumption of Theorem 1.2. We have λ_{1} > 0 if

sup_{t≥D+1} exp[−C(t)] ∫_{t}^{∞} exp[C(s)]ds < ∞.

Consequently, λ_{1} > 0 provided ∫_{D+1}^{∞}(γ + ε)^{+}(r)dr < ∞ for some ε > 0.

Now, it is the time to state the result mentioned in the abstract.

Corollary 1.4. (1) Under the assumption of Theorem 1.2. We have λ_{1} > 0 provided lim_{r→∞} sup_{ρ(x)=r,x∉cut(o)} Lρ(x) < 0.

(2) Suppose that ∂M is bounded and o is a pole. If lim_{r→∞} inf_{ρ(x)=r} Lρ(x) ≥ 0, then λ_{1} = 0.

Remark. 1) The first part of Corollary 1.4 follows from Corollary 1.3 directly. It was pointed out to the author by the referee that this can also be proved by using Cheeger’s inequality (c.f. [1]), the argument goes as follows. Define the isoperimetric constant by

h_{r} = inf \frac{A(∂Ω ∩ interior(M))}{μ(Ω)},

where Ω runs over all bounded open subset of B_{r}^{c} and A denotes the measure on ∂Ω induced by μ. Then, one has λ^{c}(r) ≥ h_{r}^{2}/4.

Next, let ν be the inward normal vector field of ∂Ω. Noting that νρ ≤ 0 on ∂M ∩ ∂Ω, we obtain

A(∂Ω ∩ interior(M)) ≥ ∫_{∂Ω} νρdA − ∫_{∂Ω∩∂M} νρdA ≥ ∫_{∂Ω} νρdA = − ∫_{Ω} Lρdμ,

where Lρ is understood in distribution sense in the case that cut(o) ≠ ∅. Then, under the condition we have lim_{r→∞} λ^{c}(r) > 0.

2) The proof of Corollary 1.4 (2) is based on the following upper bound estimate (c.f. [9; Proposition 2.13]):

λ_{1} ≤ \frac{1}{4} sup\{ε^{2} : μ(exp[ερ]) < ∞\}.  (1.7)

This estimate can be proved by taking the test function f_{n} = exp[ε(ρ ∧ n)/2] and then letting n → ∞, refer to the proof of Theorem 3.2 below.

The proofs of the above results are given in the next section, and along the same line, the spectral gap of elliptic operators on ℝ^{d} is studied in section 3.

2. Proofs

Proof of Theorem 2.1. We prove (1.2) and (1.3) respectively.

a) The proof of (1.2) is modified from Thomas [10] which studies the upper bound of the spectral gap for discrete systems.

For any ε > 0, choose f_{ε} ∈ C^{1}(M) with f_{ε}|_{B_{r}} = 0 such that μ(f_{ε}^{2}) = 1 and μ(|∇f_{ε}|^{2}) ≤ ε + λ^{c}(r). Noting that μ(f_{ε}) = μ(f_{ε}1_{B_{r}^{c}}) ≤ \sqrt{μ(B_{r}^{c})}, we have μ(f_{ε}^{2}) − μ(f_{ε})^{2} ≥ μ(B_{r}), then

λ_{1} ≤ \frac{μ(|∇f_{ε}|^{2})}{μ(f_{ε}^{2}) − μ(f_{ε})^{2}} ≤ \frac{ε + λ^{c}(r)}{μ(B_{r})}.

This proves (1.2) by letting ε ↓ 0.

b) Next, we go to prove (1.3). It suffices to show that for any f ∈ C^{1}(M) with μ(f^{2}) = 1, μ(f) = 0 and any R > r,

μ(|∇f|^{2}) ≥ \frac{λ^{c}(r)λ(R)μ(B_{R})(R − r)^{2} − 2λ(R)(1 − μ(B_{R}))}{2λ(R)(R − r)^{2} + λ^{c}(r)(R − r)^{2}μ(B_{R}) + 2μ(B_{R})}.  (2.1)

Let a = μ(f^{2}1_{B_{R}}). Noting that μ(f) = 0, we obtain

\frac{1}{λ(R)}μ_{B_{R}}(|∇f|^{2}) ≥ μ_{B_{R}}(f^{2}1_{B_{R}}) − μ_{B_{R}}(f1_{B_{R}})^{2} = \frac{a}{μ(B_{R})} − \frac{μ(f1_{B_{R}^{c}})^{2}}{μ(B_{R})^{2}}

≥ \frac{a}{μ(B_{R})} − \frac{(1 − a)(1 − μ(B_{R}))}{μ(B_{R})^{2}} = \frac{a + μ(B_{R}) − 1}{μ(B_{R})^{2}},

where μ_{B_{R}} = μ/μ(B_{R}). This implies

μ(|∇f|^{2}) ≥ \frac{λ(R)}{μ(B_{R})}(a + μ(B_{R}) − 1) =: g_{1}(a).  (2.2)

Next, define

h(x) = \{ \begin{matrix}0, & if ρ(x) ≤ r, \\ 1, & if ρ(x) ≥ R, \\ \frac{ρ(x)−r}{R−r} , & otherwise. \end{matrix}

Then fh = 0 on {x ∈ M : ρ(x) = r}. By the definition of λ^{c}(r),

1 − a ≤ μ(f^{2}h^{2}) ≤ \frac{μ(|∇(fh)|^{2})}{λ^{c}(r)} ≤ \frac{2}{λ^{c}(r)}(μ(|∇f|^{2}) + \frac{a}{(R − r)^{2}} ).

Therefore,

μ(|∇f|^{2}) ≥ \frac{λ^{c}(r)}{2} − (\frac{λ^{c}(r)}{2} + \frac{1}{(R − r)^{2}} )a =: g_{2}(a).  (2.3)

By combining (2.2) with (2.3) we obtain

μ(|∇f|^{2}) ≥ inf_{ε∈[0,1]} max{g_{1}(ε), g_{2}(ε)}.  (2.4)

Since g_{1}(ε) is increasing in ε while g_{2}(ε) is decreasing in ε, the above infimum is attained at

ε_{0} = \frac{λ^{c}(r)/2 + λ(R)(1 − μ(B_{R}))/μ(B_{R})}{λ(R)/μ(B_{R}) + λ^{c}(r)/2 + 1/(R − r)^{2}}

which solves g_{1}(ε) = g_{2}(ε). Then μ(|∇f|) ≥ g_{1}(ε_{0}) = g_{2}(ε_{0}) which is equal to the right-hand side of (2.1). □

Proof of Theorem 1.2. For any m > r_{0}, let Ω_{m} = B_{m} \ B̄_{r_{0}}. Since L is regular,

λ^{c}(r_{0}) = lim_{m→∞} λ_{0}(Ω_{m}),  (2.5)

where λ_{0}(Ω_{m}) denotes the smallest eigenvalue of −L on Ω_{m} with Neumann condition on interior(∂M ∩ ∂Ω_{m}) and Dirichlet condition on the remainder of ∂Ω_{m}. Let u(> 0) be the corresponding eigenfunction. Define

F (t) = ∫_{r_{0}}^{t} exp[−C(r)]dr ∫_{r}^{m} exp[C(s)]f(s)ds, t ∈ [r_{0}, m].

We claim that there exists c(m) > 0 such that u(x) ≤ c(m)F (ρ(x)) on Ω_{m}. Actually, since |∇u| is bounded on Ω_{m}, it suffices to show that u = 0 on S := {x ∈ ∂Ω_{m} : ρ(x) = r_{0}}. If there exists x_{0} ∈ S such that u(x_{0}) > 0, then x_{0} ∈ interior(∂M ∩ S) by the boundary conditions. This means that ν(x_{0}) = ∇ρ(x_{0}) which contradicts the assumption that νρ(x_{0}) ≤ 0.

Next, let c := inf_{t≥r_{0}} f(t)/F (t), and let x_{t} be the L-diffusion process with reflecting boundary on ∂B_{m}. By the assumption and the Itô’s formula for ρ(x_{t}) (see [7]), we have, before the time τ := {t ≥ 0 : ρ(x_{t}) = r_{0}},

dρ(x_{t}) = \sqrt{2}db_{t} + Lρ(x_{t})dt − dL_{t}, x_{0} ∈ Ω_{m},  (2.6)

where b_{t} is an one-dimensional Brownian motion, Lρ is taken to be zero on cut(o) and L_{t} is an increasing process with support contained in {t ≥ 0 : x_{t} ∈ cut(o) ∪ ∂B_{m}}. Noting that Lρ(x_{t}) ≤ γ(ρ(x_{t})) for x_{t} ∉ cut(o), by (2.6) and Itô’s formula we obtain

dF ∘ ρ(x_{t}) ≤ \sqrt{2}F′ ∘ ρ(x_{t})db_{t} − f(x_{t})dt
≤ \sqrt{2}F′ ∘ ρ(x_{t})db_{t} − cF ∘ ρ(x_{t})dt.

This then implies

E^{x}F ∘ ρ(x_{t∧τ}) ≤ F ∘ ρ(x) exp[−ct].

Let τ′ = inf\{t ≥ 0 : x_{t} ∈ \overline{∂Ω_{m} \ ∂M}\}, we have τ′ ≤ τ and u(x_{t∧τ′}) ≤ u(x_{t} ∧ τ). Noting that E^{x}u(x_{t∧τ′}) = u(x) exp[−λ_{0}(Ω_{m})t], we obtain

u(x) exp[−λ_{0}(Ω_{m})t] ≤ c(m)E^{x}F ∘ ρ(x_{t} ∧ τ) ≤ c(m)F ∘ ρ(x) exp[−ct].

This implies λ_{0}(m) ≥ c for any m > r_{0}. Therefore, λ^{c}(r_{0}) ≥ c. □

It was pointed out by the referee that there is an equivalent analysis proof of Theorem 1.2 (refer to [6; Lemma 1.1]). Let F and u be as in above with ∫_{Ω_{m}} u^{2} = 1, then LF ∘ρ ≤ −cF ∘ρ on Ω_{m} in the distribution sense. Let f = u/F, then f is bounded as was shown in the proof of Theorem 1.2. We have ufνF ≤ 0 on ∂Ω_{m} since νρ ≤ 0 on ∂M and u = 0 on ∂Ω_{m} \ ∂M. Therefore, by Green’s formula, we obtain

λ_{0}(Ω_{m}) = − ∫_{Ω_{m}} uLudμ = − ∫_{Ω_{m}} uL(fF)dμ
= − ∫_{Ω_{m}}[ufLF + uF Lf + 2u⟨∇f, ∇F ⟩]dμ
≥ c ∫_{Ω_{m}}[ufF + ⟨∇(fF^{2}), ∇f⟩ − 2u⟨∇f, ∇F ⟩]dμ + ∫_{∂Ω_{m}}[uF νf]dA
= c + ∫_{Ω_{m}} |∇f|^{2}F dμ + ∫_{∂Ω_{m}}[uνu − ufνF]dA ≥ c.

Here, we have used the fact that uνu = 0 by the mixed boundary condition.

Proof of Corollary 1.3. The prove of the first assertion is essentially due to [4]. Under the condition we have

∫_{t}^{∞} exp[C(s)]ds ≤ c exp[C(t)], t ≥ D + 1

for some constant c > 0. This implies (see [4; Lemma 6.1])

∫_{t}^{∞} exp[εs + C(s)]ds ≤ \frac{c}{1 − cε} exp[C(t) + εt], ε ∈ (0, c^{−1}).

By taking f(r) = exp[r/(2c)] in (1.4), we prove the first assertion.

Next, if there exists ε > 0 such that c_{1} := ∫_{D+1}^{∞}(γ + ε)^{+}ds < ∞. Let C_{ε}(r) = C(r) − ∫_{D+1}^{r}(γ + ε)^{+}(s)ds. Then C′_{ε}(r) = γ(r) − (γ + ε)^{+}(r) ≤ −ε. Therefore,

exp[−C(t)] ∫_{t}^{∞} exp[C(s)]ds ≤ exp[−C_{ε}(t)] ∫_{t}^{∞} exp[C_{ε}(r) + c_{1}]dr
≤ exp[−C_{ε}(t)] ∫_{t}^{∞} exp[C_{ε}(t) − ε(r − t) + c_{1}]dr
= exp[c_{1}]/ε < ∞.

Hence λ_{1} > 0 by the first assertion. □

Remark. ¿From (1.3) we may derive explicit lower bounds of λ_{1}. For instance, assume that B_{R} is convex for any R, let K ≥ 0 be such that Ric−Hess_{V} ≥ −K. We have^{[3]}

λ(R) ≥ \frac{π^{2}}{8} K\{ exp [KR^{2}/2] − 1\}^{−1}.  (2.7)

Next, if lim_{r→∞} sup γ(r) < 0, let β(r) = inf_{s≥r}(−γ(s))^{+}, by taking f(t) = exp[β(r)t/2] in (1.4), we obtain

λ^{c}(r) ≥ inf_{t≥r} exp [\frac{β(r)t}{2}]\{ ∫_{r}^{t} ds ∫_{s}^{∞} exp [\frac{β(r)u}{2} + ∫_{s}^{u} γ(v)dv]du\}^{−1}

≥ inf_{t≥r} exp [\frac{β(r)t}{2}]\{ ∫_{r}^{t} ds ∫_{s}^{∞} exp [\frac{β(r)u}{2} − β(r)(u − s)]du\}^{−1}

= \frac{β(r)^{2}}{4} .  (2.8)

Here, in the second step, we have assumed that β(r) > 0 so that γ(v) ≤ −β(r) for v ≥ r. Then the estimate λ^{c}(r) ≥ β(r)^{2}/4 is true for any r since β(r) is nonnegative. The explicit lower bound of λ_{1} then follows from (1.3), (2.7) and (2.8).

Proof of Corollary 1.4 (2). Suppose that ∂M ⊂ B_{D}. Under the polar coordinate at o, we have x = (r, ξ) for r = ρ(x) and ξ ∈ 𝕊^{d−1}, the (d − 1)-dimensional unit sphere which is considered as the bundle of unit tangent vectors at o. Under this coordinate, the Riemannian volume element can be written as dx = g(r, ξ)drdξ and Δρ = \frac{∂}{∂r} (log g(r, ξ))|_{r=ρ}. Suppose that lim_{r→∞} inf_{ρ(x)=r} Lρ ≥ 0. Then, for any ε > 0 there exists r_{1} > D such that

\frac{∂}{∂r} log g(r, ξ) ≥ −\frac{ε}{2} − \frac{∂}{∂r} V (r, ξ), r ≥ r_{1}.

This implies

g(r, ξ) ≥ g(r_{1}, ξ) exp [ − \frac{ε}{2}(r − r_{1}) − V (r, ξ) + V (r_{1}, ξ)]

≥ c exp [ − \frac{ε}{2}ρ − V ], r ≥ r_{1}

for some constant c > 0. Therefore

μ(exp[ερ]) ≥ ∫_{[D,∞)×𝕊^{d−1}} exp[εr + V (r, ξ)]g(r, ξ)drdξ

≥ c ∫_{[r_{1},∞)×𝕊^{d−1}} exp [\frac{ε}{2}r]drdξ = ∞.

By (1.7), we have λ_{1} = 0. □

Remark. 1) According to the above proof, the function ρ in Corollary 1.4 (2) can be replaced by the distance from any bounded regular domain such that the outward-pointing normal exponential map on the boundary induces a diffeomorphism. See e.g. Kumura [8] for some discussions on such manifolds.

2) In general, for any r > D > 0, let

Ξ_{r}^{D} = {ξ ∈ 𝕊^{d−1} : exp[sξ]|_{[0,r]} is minimal and exp[sξ] ∈ M, s ∈ [D, r]}.

Then Ξ_{r}^{D} is nonincreasing in r. Let ν = dξ be the standard measure on 𝕊^{d−1}, the assumption of Corollary 1.4 (2) can be replaced by: there exists D > 0 such that

lim_{r→∞} ν(Ξ_{r}^{D}) exp[εr] = ∞ for any ε > 0.  (2.9)

3. Spectral gap for elliptic operators on ℝ^{d}

This section is a continuation of [2] and [4] in which the lower bound estimates are studied for the spectral gap of elliptic operators on ℝ^{d}.

Consider the operator L = ∑_{i,j=1}^{d} a_{ij}(x)∂_{i}∂_{j} + ∑_{i=1}^{d} b_{i}(x)∂_{i}, where ∂_{i} = \frac{∂}{∂x_{i}} , a(x) := (a_{ij}(x)) is positively definite, a_{ij} ∈ C^{2}(ℝ^{d}) and b_{i} = ∑_{j=1}^{d}(a_{ij}∂_{j}V + ∂_{j}a_{ij}) for some V ∈ C^{2}(ℝ^{d}) with Z := ∫ exp[V]dx < ∞. The specific form of b implies that L is symmetric with respect to dμ = Z^{−1} exp[V]dx. In the present setting, the spectral gap of L is described as

λ_{1}(a, V) = inf{μ(⟨a∇f, ∇f⟩) : f ∈ C^{1}(ℝ^{d}), μ(f) = 0, μ(f^{2}) = 1}.  (4.1)

Moreover, we assume that L is regular in the sense that C_{0}^{∞}(ℝ^{d}) is dense in W^{1,2}(ℝ^{d}, dμ) with the Sobolev norm ∥ · ∥_{L^{2}(μ)} + ∥\sqrt{⟨a∇·, ∇·⟩}∥_{L^{2}(μ)}.

Obviously, if a ≥ αI for some constant α > 0, then λ_{1}(a, V) ≥ αλ_{1}(I, V). From this one may transform the present setting to the manifold case. But this comparison only works for the case a is uniformly positively definite, and it will lead to some loss if a is very different from I, see e.g. Examples 3.1 and 3.2 below. Hence, it should be worthy to study L directly as in previous sections.

Define

γ(r) = sup_{|x|=r} \frac{r(tr(a(x)) + ⟨b(x), x⟩)}{⟨a(x)x, x⟩} − \frac{1}{r},

C(r) = ∫_{1}^{r} γ(s)ds, α(r) = inf_{|x|=r} \frac{1}{r^{2}} ⟨a(x)x, x⟩, r > 0.

The main result in this section is the following.

Theorem 3.1. If there exists positive f ∈ C[1, ∞) such that

sup_{t≥1} \frac{1}{f(t)} ∫_{1}^{t} exp[−C(r)]dr ∫_{r}^{∞} exp[C(s)]\frac{f(s)}{α(s)}ds < ∞,  (3.2)

then λ_{1} > 0.

Proof. For g ∈ C^{2}(ℝ), we have

Lg(|x|) =\frac{1}{|x|^{3}} (|x|^{2}tr(a(x)) + |x|^{2}⟨b(x), x⟩ − ⟨a(x)x, x⟩)g′(|x|)

+ \frac{1}{|x|^{2}} ⟨a(x)x, x⟩g′′(|x|), |x| > 0.

For positive f ∈ C[1, ∞) with ∫_{1}^{∞} exp[C(r)]\frac{f(r)}{α(r)}dr < ∞, let

g(t) = ∫_{1}^{t} exp[−C(r)]dr ∫_{r}^{∞} exp[C(s)]\frac{f(s)}{α(s)}ds.

Then

Lg(|x|) ≤ −f(|x|), |x| ≥ 1.

Therefore, the proof of Theorem 1.2 implies that λ^{c}(1) > 0 provided (3.2) holds, then λ_{1} > 0. □

Remark. Theorem 3.1 remains true for unbounded regular domains with bounded boundary. As for the unbounded boundary case, for the estimation of λ^{c}(r), one has to consider the normal vector field induced by the metric ⟨∂_{i}, ∂_{j}⟩ = (a^{−1})_{ij}, this will cause difficulty for general a.

For the case M = [0, ∞), one has γ = \frac{b}{a}, α = a. Then, by Theorem 3.1, we have λ_{1} > 0 if there exists positive f ∈ C[1, ∞) with f′ < 0 such that

sup_{t>1} \frac{1}{f′(t)} exp[−C(t)] ∫_{t}^{∞} exp[C(s)]\frac{f(s)}{α(s)}ds < ∞.  (3.3)

This is just the condition in [4; Theorem 2.1]. Therefore, Theorem 3.1 is the exact extension of [4; Theorem 2.1] to high dimensions.

Next, the following examples shows that Theorem 3.1 can be better than comparing a with a constant matrix.

Example 3.1. Take a(x) = (1 + |x|^{2})^{α}I, b(x) = 0, α ≥ (1 + d)/2. Then L is regular. It is easy to see that V = −α log(1 + |x|^{2}). Noting that a ≥ I, by the comparison procedure, we may consider the operator L̄ = Δ − ∇V. But by (1.7) the spectral gap of L̄ does not exists since μ(exp[ε|x|]) = ∞ for any ε > 0. Hence the comparison procedure does not work for this example.

On the other hand, one has α(r) = (1 + r^{2})^{α}, γ(r) = \frac{d−1}{r}, C(r) = r^{d−1}. Take f = \sqrt{t}, we obtain

∫_{1}^{t} exp[−C(r)]dr ∫_{r}^{∞} exp[C(s)]\frac{f(s)}{α(s)}ds ≤ 2 ∫_{1}^{t} r^{3/2−2α}dr ≤ 4f(t)

since 2α ≥ 1 + d. By Theorem 3.1 we have λ_{1} > 0. Example 3.2. Take a(x) = \frac{1}{|x|+1}I, V (x) = −|x|^{2} for |x| ≥ 1. Then the comparison procedure does not apply. Now we go to check the condition of Theorem 3.1. Obviously, α(r) = \frac{1}{1+r} , ⟨b(x), x⟩ = −\frac{r}{(1+r)^{2}} − \frac{2r^{2}}{1+r} . Then

γ(r) = \frac{d − 1}{r} − \frac{1}{1 + r} − 2r, exp[C(r)] = \frac{c_{1}r^{d−1}}{r + 1} exp[−4r^{2}], r ≥ 1

for some c_{1} > 0. Take f(r) = r^{1−d} exp[−r + 4r^{2}], there exists c_{2} > 0 such that

\frac{1}{f(t)} ∫_{1}^{t} exp[−C(r)]dr ∫_{r}^{∞} exp[C(s)]f(s)ds ≤ c_{2} \frac{1}{f(t)} ∫_{1}^{t} exp[−r + 4r^{2}]\frac{1 + r}{r^{d−1}} dr

which goes to c_{2}/8 as t → ∞. Therefore, Theorem 3.1 implies that λ_{1} > 0.

Finally, we present an upper bound estimate as (1.7). Theorem 3.2. Let β(r) = sup_{|x|=r} \frac{1}{r^{2}} ⟨a(x)x, x⟩, we have

λ_{1} ≤ \frac{1}{4} sup \{ε^{2} : μ( exp [ε ∫_{0}^{|x|} \frac{1}{\sqrt{β(r)}}dr]) < ∞\}.

Proof. Let h(r) = ∫_{0}^{r} \frac{1}{\sqrt{β(s)}}ds. If μ(exp[εh(|x|)]) = ∞, we go to prove λ_{1} ≤ ε^{2}/4. Let f(x) = exp[\frac{ε}{2}(h(|x|) ∧ n)], n ≥ 1. By (3.1), we have

λ_{1} ≤ \frac{ε^{2}μ(f^{2})}{4(μ(f^{2}) − μ(f)^{2})}.  (3.4)

Next, for any m > 1, choose r_{m} > 0 such that μ({h(|x|) ≥ r_{m}}) = 1/m, we have

μ(1_{\{|x|≥r_{m}\}}f^{2})^{1/2} ≥ \sqrt{m}μ(1_{\{|x|≥r_{m}\}}f) ≥ \sqrt{m}μ(f) − \sqrt{m} exp[εh(r_{m})/2].

Then

μ(f)^{2} ≤ (\sqrt{μ(f^{2})}/\sqrt{m} + exp [εh(r_{m})/2])^{2}.  (3.5)

Noting that μ(f^{2}) → ∞ as n → ∞, by combining (3.4) with (3.5), we obtain

λ_{1} ≤ \frac{ε^{2}}{4(1 − 1/m)}, m > 1.

Therefore, λ_{1} ≤ ε^{2}/4 since m is arbitrary. □

Acknowledgement. The author is grateful to the referee whose suggestions improved the quality of the paper. The revised version was finished during the author’s visit to MSRI in Berkeley.

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