A lower bound on the first spectral gap of Schrödinger operators with Kato class measures

Hendrik Vogt

Fachrichtung Mathematik, Technische Universität Dresden, D-01062 Dresden, Germany.

Abstract We study Schrödinger operators on ℝ^{n} formally given by H_{μ} = −Δ − μ, where μ is a positive, compactly supported measure from the Kato class. Under the assumption that a certain condition on the μ-volume of balls is satisfied and that H_{μ} has at least two eigenvalues below the essential spectrum σ_{ess}(H_{μ}) = [0, ∞), we derive a lower bound on the first spectral gap of H_{μ}. The assumption on the μ-volume of balls is in particular satisfied if μ is of the form μ = aσ_{M}, where M is a compact (n−1)-dimensional Lipschitz submanifold of ℝ^{n}, σ_{M} the surface measure on M, and 0 ⩽ a ∈ L_{∞}(M).

MSC 2000: 35J10, 35P15, 47A55

1 Introduction and main results

There is extensive literature on estimates for the first spectral gap of Schrödinger operators. Many papers concentrate on one of the following two extreme situations: On the one hand, for Schrödinger operators with convex potentials on convex domains the gap turns out to be relatively large (see, e.g., [SWYY85], [Smi96] and the references therein); on the other hand, in tunneling situations, where the potential has two or more separated wells (minima) of equal depth, the gap becomes exponentially small with respect to a separation parameter (see, e.g., [Har78], [Har80], [Sim84]).

For Schrödinger operators on ℝ^{n} with bounded potentials, Kirsch and Simon proved in [KiSi87; Sec. 4] that the first spectral gap always admits a lower bound that is comparable to the gap size in tunnelling situations, without particular assumptions on the shape of V. In [KoVe07], Kondej and Veselic proved a similar result in dimension n = 2 for potentials given by certain measures supported on compact curves. This is the only gap estimate known to us for Schrödinger operators with a singular interaction given by a measure.

In the present paper we generalise the gap estimate of [KoVe07] to higher dimensions, for operators H_{μ} = −Δ − μ on ℝ^{n} (defined as a form sum, cf. Section 2). We assume that μ is a positive, compactly supported measure on the Borel σ-algebra of ℝ^{n} satisfying the volume bound

μ(B(x, r)) ⩽ c_{μ}r^{n−α} (x ∈ ℝ^{n}, r > 0)  (1.1)

for some c_{μ} > 0, α ∈ [0, 2). (Up to a change in the constant c_{μ}, we could equivalently assume this bound for r ⩽ 1 only since μ has compact support.) Assumption (1.1) implies that μ is in the Kato class; conversely, μ being in the Kato class implies (1.1) for α = 2 and some c_{μ} > 0 (cf. Proposition 2.2). In Example 2.3(b) we will show that the surface measure on a compact (n−1)-dimensional Lipschitz submanifold of ℝ^{n} satisfies (1.1) with α = 1.

The proof of the gap estimate in [KiSi87] uses the fact that the eigenfunctions of the Schrödinger operator are Lipschitz continuous if the potential is bounded. For singular potentials, Lipschitz continuity is no longer true in general. This difficulty is overcome in [KoVe07] by means of a method that relies on detailed knowledge of the geometry of spt μ, such as curvature bounds. Theorem 1.1 below provides a better (though not sharp) gap estimate, based only on the volume estimate (1.1).

Since μ has compact support, we have σ_{ess}(H_{μ}) = [0, ∞) by [BEKŠ94; Thm. 3.1], so σ(H_{μ}) \ [0, ∞) consists of isolated eigenvalues of finite multiplicity. In the main results of this paper, Theorems 1.1 and 1.3 below, we will assume that

(A) H_{μ} has at least two negative eigenvalues; the lowest two are denoted by λ_{0} = −κ_{0}^{2} and λ_{1} = −κ_{1}^{2} ( > λ_{0}).

The proofs of these theorems are given in Section 6.

1.1 Theorem. Let n ⩾ 2 and suppose that μ ⩾ 0 is a compactly supported measure on ℝ^{n} satisfying (1.1) for some c_{μ} > 0, 0 ⩽ α < 2. Let d denote the diameter of the smallest closed ball containing spt μ. If assumption (A) is satisfied then there exist C, p, q, β > 0 depending only on n and α such that

λ_{1} − λ_{0} ⩾ \frac{C|λ_{0}|}{(c_{μ} + 1)^{p}(d + 1)^{q}} e^{−βκ_{0}(d+1)}.  (1.2)

More precisely, for α < 1 one can choose β = n + 1, for α = 1 any β > n + 1, and β = \frac{n−1}{2−α} + 2 for α > 1. For n ⩾ 5, the factor |λ_{0}| on the right hand side of (1.2) can be omitted.

1.2 Remarks. (a) In [KoVe07; Thm. 4.3], a gap estimate similar to the above is proved for the case n = 2, μ = cσ_{im γ}, where c > 0 and γ is a C^{2}-curve without self-intersections, parameterised by arc length (cf. Example 2.3(c)). An application of Theorem 1.1 in this situation yields a slightly better estimate: Firstly, in [KoVe07] the constant C depends on the curvature of γ, and β is not given explicitly. Secondly, there is a factor |λ_{1}| in the gap estimate of [KoVe07]; replacing this factor by |λ_{0}| is in fact achieved by a simple trick (see the argument leading to equation (6.3)). Thirdly, in [KoVe07] the estimate contains an additional factor that behaves like |λ_{0}|^{8} for small |λ_{0}|.

(b) One can hardly expect that the value of β given in Theorem 1.1 is sharp. Computable examples of tunneling situations might lead to the conjecture that (1.2) always holds with β = 1, but it does not seem possible to prove this with the method presented in this paper.

(c) The attentive reader will note that the estimate (1.2) is not scaling invariant: It is easy to see that for s > 0 the operators H_{μ} and H_{μ_{s}} are unitarily equivalent, where μ_{s} is defined by μ_{s}(A) := s^{2−n}μ(sA). Moreover, under the assumptions of Theorem 1.1 one computes that μ_{s} satisfies (1.1) with c_{μ_{s}} = s^{2−α}c_{μ}, that the smallest closed ball containing spt μ_{s} has diameter d/s, and that the lowest two eigenvalues of H_{μ_{s}} are s^{2}λ_{0} and s^{2}λ_{1}. Applying Theorem 1.1 to μ_{s}, with s = εd for some ε > 0, we thus obtain the scaling invariant estimate

λ_{1} − λ_{0} ⩾ \frac{C_{ε}|λ_{0}|}{((εd)^{2−α}c_{μ} + 1)^{p}} e^{−(1+ε)βκ_{0}d},

where C_{ε} = C/(\frac{1}{ε} + 1)^{q}.

In dimension n = 1 we obtain a much simpler result.

1.3 Theorem. Let μ ⩾ 0 be a compactly supported measure on ℝ, d the diameter of spt μ. If assumption (A) is satisfied then

λ_{1} − λ_{0} ⩾ \frac{|λ_{0}|}{d||μ|| + 1}e^{−2κ_{0}d} and λ_{1} − λ_{0} ⩾ \frac{κ_{1}}{d(κ_{1}d + 1)}e^{−2κ_{0}d}.  (1.3)

The paper is organised as follows. In Section 2 we recall some basic results on form small measures and the (extended) Kato class of measures. In Section 3 we use a ground state transformation to show the representation

λ_{1} − λ_{0} = ||φ_{0}∇\frac{φ_{1}}{φ_{0}} ||_{2}^{2} ||φ_{1}||_{2}^{−2}

of the lowest spectral gap, where φ_{j} is an eigenfunction corresponding to the eigenvalue λ_{j}, for j = 1, 2. We demonstrate how this representation can be used to prove the main results, given the following two ingredients: (i) an estimate of the modulus of continuity of the eigenfunctions, (ii) a pointwise estimate from below for the ground state φ_{0} of H_{μ}. These ingredients are provided in Sections 4 and 5, respectively. Since they are of independent interest, we will prove estimates that are sharper than necessary for the proof of Theorem 1.1. The proofs of Theorems 1.1 and 1.3 are given in Section 6. In the appendix we provide an estimate on the convolution kernel of (κ^{2} − Δ)^{−1} (where κ > 0), needed in Section 5, that we did not find in the literature.

2 Form small measures and Kato class measures

Throughout this section let μ ⩾ 0 be a measure on the Borel σ-algebra of ℝ^{n}. We recall the definition of the operator H_{μ} for form small μ and some results on the (extended) Kato class of measures. In the following we write for brevity L_{p} for L_{p}(ℝ^{n}), and similarly W_{2}^{1}, C_{c}^{∞}, etc.

The measure μ is called form small with respect to the Laplacian on ℝ^{n} if μ does not charge sets of zero capacity and there exist γ ∈ [0, 1), c ∈ ℝ such that

∫ |u|^{2} dμ ⩽ γ ∫ |∇u|^{2} dx + c ∫ |u|^{2} dx (u ∈ W_{2}^{1}).  (2.1)

Here and in the following, we tacitly assume that a quasi-continuous representative of u is chosen if we write u ∈ W_{2}^{1}; then the integral ∫ |u|^{2} dμ is unambiguously defined. It is well-known that, under condition (2.1),

D(τ_{μ}) := W_{2}^{1},  τ_{μ}(u) := ∫ |∇u|^{2} dx − ∫ |u|^{2} dμ  (2.2)

defines a closed quadratic form τ_{μ} in L_{2}. The domain D(τ_{μ}) is dense in L_{2}, so we can define the Schrödinger operator H_{μ} as the selfadjoint operator in L_{2} associated with τ_{μ}.

In accordance with [StVo96; p. 114] we say that μ is in the extended Kato class if there exists κ > 0 such that G_{κ} ∗ μ ∈ L_{∞}, where G_{κ} is the convolution kernel of the free resolvent (κ^{2} − Δ)^{−1}. (It is automatic that then μ does not charge sets of zero capacity.) We say that μ is Kato small if lim_{κ→∞} ||G_{κ} ∗μ||_{∞} < 1. By [StVo96; Thm. 3.1], a Kato small measure is also form small. The measure μ is in the (proper) Kato class if ||G_{κ} ∗ μ||_{∞} → 0 as κ → ∞.

In the following let k_{t} denote the convolution kernel of e^{tΔ}, i.e., k_{t}(x) = (4πt)^{− \frac{n}{2}} e^{− \frac{|x|^{2}}{4t}} for t > 0, x ∈ ℝ^{n}. Moreover, for an operator B on L_{2} and p, q ∈ [1, ∞] we denote by ||B||_{p→q} the norm of B|_{L_{p}∩L_{2}} regarded as an operator from L_{p} to L_{q}.

2.1 Proposition. Let α > 0, γ ∈ [0, 1) and assume that

||∫_{0}^{α} k_{t} ∗ μ dt||_{∞} ⩽ γ.

Then μ is Kato small, and ||e^{−tH_{μ}}||_{1→1} ⩽ \frac{1}{1−γ} for all t ∈ [0, α].

Proof. Arguing as in [Voi86; Prop. 4.7(b)] one finds κ > 0 such that

||G_{κ} ∗ μ||_{∞} = ||∫_{0}^{∞} e^{−κ^{2}t}k_{t} ∗ μ dt||_{∞} < 1,

so μ is Kato small. We recall the approximation of H_{μ} given in [StVo96]: For j, m ∈ ℕ, m ⩾ κ let μ_{j} := 11_{B(0,j)}μ, V_{j,m} := (m^{2} − κ^{2})G_{m} ∗ μ_{j}. Then

||∫_{0}^{α} e^{tΔ}V_{j,m} dt||_{∞} = ||(m^{2} − κ^{2})G_{m} ∗ ∫_{0}^{α} k_{t} ∗ μ_{j} dt||_{∞}
⩽ ||m^{2}(m^{2} − Δ)^{−1}||_{∞→∞}||∫_{0}^{α} k_{t} ∗ μ dt||_{∞} ⩽ γ,

and by [Voi77; Thm. 1(c) and the last formula line of part (i) of its proof] we obtain that ||e^{−tH_{V_{j,m}}}||_{1→1} ⩽ \frac{1}{1−γ} for all t ∈ [0, α], j, m ∈ ℕ, m ⩾ κ. Moreover, H_{V_{j,m}} → H_{μ_{j}} in the strong resolvent sense as m → ∞, by [StVo96; Cor. 2.4(a)], and H_{μ_{j}} → H_{μ} in the strong resolvent sense as j → ∞, by [StVo96; Thm. 3.3(a)]. We thus conclude that ||e^{−tH_{μ}}||_{1→1} ⩽ \frac{1}{1−γ} for all t ∈ [0, α]; cf. [StVo96; proof of Cor. 2.4(b)].

In the next result we show the relation between the Kato class condition and the volume bound (1.1). We will need the following fact: Assume that m: [0, ∞) → [0, ∞) is increasing, m(0) = 0, μ(B(x, r)) ⩽ m(r) for all x ∈ ℝ^{n}, r > 0. Then for any decreasing function f: [0, ∞) → [0, ∞) and all x ∈ ℝ^{n}, one can estimate

∫_{ℝ^{n}} f(|x − y|) dμ(y) ⩽ ∫_{0}^{∞} f(r) dm(r);  (2.3)

cf. [Dav95; p. 179].

2.2 Proposition. Let 0 ⩽ α ⩽ 2. Assume that there exists c ⩾ 0 such that

||∫_{0}^{t} k_{s} ∗ μ ds||_{∞} ⩽ ct^{1− \frac{α}{2}}  (t > 0).  (2.4)

Then the volume estimate (1.1) holds with c_{μ} = c/g(e_{1}), where g(x) := ∫_{0}^{1} k_{s}(x) ds. Conversely, if α < 2 (α ⩽ 1 in the case n = 1) then (1.1) implies (2.4) with c = \frac{2}{2−α}2^{−α}π^{− \frac{n}{2}} Γ(\frac{n−α}{2} + 1)c_{μ}; in particular, μ is in the Kato class.

Proof. Let r > 0. For y ∈ ℝ^{n} \ {0} we compute, substituting s = r^{2}t, that

∫_{0}^{r^{2}} k_{s}(y) ds = ∫_{0}^{1}(4πr^{2}t)^{− \frac{n}{2}} e^{− \frac{|y|^{2}}{4r^{2}t}} r^{2}dt = r^{2−n} ∫_{0}^{1} k_{t}(\frac{y}{r}) dt = r^{2−n}g(\frac{y}{r}) =: g_{r}(y).

Since g_{r} ∗ μ is lower semicontinuous, we thus obtain from assumption (2.4) that sup(g_{r} ∗ μ) ⩽ cr^{2−α}. Moreover, g_{r}(y) ⩾ g_{r}(re_{1}) = r^{2−n}g(e_{1}) for 0 < |y| ⩽ r, so we infer for all x ∈ ℝ^{n} that

r^{2−n}g(e_{1})μ(B(x, r)) ⩽ ∫_{B(x,r)} g_{r}(x − y) dμ(y) ⩽ g_{r} ∗ μ(x) ⩽ cr^{2−α},

and the first assertion follows.

Conversely, assume that (1.1) holds and that α < 2. Then for s > 0 we obtain by (2.3), substituting r = (4sρ)^{\frac{1}{2}} , that

||k_{s} ∗ μ||_{∞} ⩽ ∫_{0}^{∞}(4πs)^{− \frac{n}{2}} e^{− \frac{r^{2}}{4s}} c_{μ} dr^{n−α} = c_{μ}(4πs)^{− \frac{n}{2}} ∫_{0}^{∞} e^{−ρ}(4s)^{\frac{n−α}{2}} dρ^{\frac{n−α}{2}}
= c_{μ}(4s)^{− \frac{α}{2}} π^{− \frac{n}{2}} Γ(\frac{n−α}{2} + 1).

(This computation is also valid in the case n = α = 1 if one replaces dr^{n−α} and dρ^{\frac{n−α}{2}} with the Dirac measure at 0.) Integration from 0 to t now yields the second assertion.

If n = 1 and μ is finite, then assumption (1.1) is trivially satisfied with α = 1 and c_{μ} = ||μ||. We now give examples in dimension n ⩾ 2 where assumption (1.1) is satisfied with α = 1.

2.3 Example. (a) (cf. [BEKŠ94; Thm. 4.1(iv)]) Let U ⊆ ℝ^{n−1} be open and bounded, f: \overline{U} → ℝ^{n} Lipschitz continuous, and

|f(x) − f(y)| ⩾ |x − y| (x, y ∈ \overline{U}).

Let μ = σ_{f(U)}, i.e.,

∫ φ dμ = ∫_{U} φ(f(y))g(y) dy (φ ∈ C_{c}(ℝ^{n})),

with g := (det(f^{′⊤}f′))^{\frac{1}{2}} the square root of the Gram determinant. We show that then (1.1) holds with

α = 1, c_{μ} = 2^{n−1}ω_{n−1}||g||_{∞},

where ω_{n−1} denotes the volume of the (n−1)-dimensional unit ball.

Indeed, let x ∈ ℝ^{n}, r > 0. There exists y_{0} ∈ \overline{U} such that r_{0} := dist(x, f(\overline{U})) = |x − f(y_{0})|. If r < r_{0} then μ(B(x, r)) = 0 ⩽ c_{μ}r^{n−1}. If r ⩾ r_{0} then for y ∈ U with f(y) ∈ B(x, r) we have

|y − y_{0}| ⩽ |f(y) − f(y_{0})| ⩽ |f(y) − x| + |x − f(y_{0})| ⩽ r + r_{0} ⩽ 2r,

and it follows that

μ(B(x, r)) = ∫_{U} 11_{B(x,r)}(f(y))g(y) dy ⩽ ∫_{B(y_{0},2r)} ||g||_{∞} dy = ω_{n−1}(2r)^{n−1}||g||_{∞}.

(b) Let M be a compact (n−1)-dimensional Lipschitz submanifold of ℝ^{n}, σ_{M} the surface measure on M and 0 ⩽ a ∈ L_{∞}(M). Then M can be covered by finitely many relatively open subsets that can be parameterised as in (a), so μ = aσ_{M} satisfies (1.1) with α = 1.

(c) Let n = 2, N ∈ ℕ, γ_{1}, …, γ_{N} Lipschitz curves in ℝ^{2}, γ_{j}: I_{j} → ℝ^{2} parameterised by arc length on a compact interval I_{j} ⊆ ℝ (j = 1, …, N), and assume that |γ_{j}(s) − γ_{j}(t)| ⩾ \frac{1}{2}|s − t| for all j ∈ \{1, …, N\}, s, t ∈ I_{j}. In particular, each curve γ_{j} is intersection free, but the different curves may intersect. If, e.g., γ: I → ℝ^{2} is a curve (with a compact interval I ⊆ ℝ) that is piecewise C^{1} and parameterised by arc length, then γ can be split into finitely many parts γ_{1}, …, γ_{N} satisfying the above.

Let now

Γ := ∑_{j=1}^{N} γ_{j}, a ∈ L_{∞}(im Γ), μ := aσ_{im Γ}.

By part (a), applied with f_{j}(t) := γ_{j}(2t), we obtain that then (1.1) holds with α = 1, c_{μ} = N||a||_{∞}2^{1}ω_{1} · 2 = 8N||a||_{∞}. It follows that the spectral gap estimate of [KoVe07] can be obtained as a special case of Theorem 1.1.

The following representation of the eigenfunctions of H_{μ} is extracted from [BEKŠ94; proof of Cor. 2.3]. We include the proof for the reader’s convenience.

2.4 Lemma. Assume that μ is form small, φ an eigenfunction of H_{μ}, H_{μ}φ = −κ^{2}φ for some κ > 0. Then φ = G_{κ} ∗ (φμ).

Proof. The form smallness of μ implies that D(H_{μ}) ⊆ W_{2}^{1} ⊆ L_{2}(μ), so φ ∈ L_{2}(μ). By [BEKŠ94; Lemma 2.2] we obtain that u := G_{κ} ∗ (φμ) ∈ W_{2}^{1} and

⟨u, v⟩_{κ} := ⟨∇u, ∇v⟩ + κ^{2}⟨u, v⟩ = ∫ φ\overline{v} dμ (v ∈ W_{2}^{1}).  (2.5)

Moreover, for v ∈ W_{2}^{1} we have

0 = ⟨(H_{μ} + κ^{2})φ, v⟩ = (τ_{μ} + κ^{2})(φ, v) = ⟨φ, v⟩_{κ} − ∫ φ\overline{v} dμ.  (2.6)

Combining (2.5) and (2.6) we conclude that ⟨φ, v⟩_{κ} = ⟨u, v⟩_{κ} for all v ∈ W_{2}^{1} and therefore φ = u.

We use the above representation for proving the following estimates of the L_{2}-norm against the L_{∞}-norm of the eigenfunctions of H_{μ} (cf. [KoVe07; Lemma 5.5]); these estimates will be needed in the proofs of the main results.

2.5 Proposition. Assume that μ is form small, φ a bounded eigenfunction of H_{μ}, H_{μ}φ = −κ^{2}φ for some κ > 0.

(a) If μ is finite then ||φ||_{2}^{2} ⩽ κ^{−2}||μ||||φ||_{∞}^{2}.

(b) If n = 1 and spt μ ⊆ B[0, R] for some R > 0 then ||φ||_{2}^{2} ⩽ (2R + \frac{1}{κ})||φ||_{∞}^{2}.

(c) If n ⩾ 5 and spt μ ⊆ B[0, R] for some R > 0 then ||φ||_{2}^{2} ⩽ \frac{2n−4}{n−4}ω_{n}R^{n}||φ||_{∞}^{2}.

Proof. Without loss of generality assume that ||φ||_{∞} = 1. By Lemma 2.4 we have φ = G_{κ} ∗ (φμ).

(a) Since ||G_{κ}||_{1} = ||(κ^{2} − Δ)^{−1}||_{1→1} = κ^{−2}, we obtain that

||φ||_{2}^{2} ⩽ ||φ||_{1}||φ||_{∞} = ||G_{κ} ∗ (φμ)||_{1} ⩽ κ^{−2}||μ||.

(b) Outside spt μ we have (κ^{2}−Δ)φ = 0 and hence (κ^{2}−Δ)|φ| ⩽ 0 in the distributional sense, i.e., |φ| is a subsolution of the equation (κ^{2} − Δ)u = 0. Moreover, u(x) := e^{−κ(|x|−R)} defines a solution of this equation on ℝ\{0}, and |φ| ⩽ u on [−R, R]. Since φ = G_{κ} ∗(φμ) vanishes at ∞, we conclude that |φ| ⩽ u on ℝ and therefore

||φ||_{2}^{2} ⩽ 2 ∫_{0}^{R} 1 dx + 2 ∫_{R}^{∞} e^{−2κ(x−R)} dx = 2R + \frac{1}{κ}.

(c) Arguing as in (b) we obtain that |φ| is subharmonic on ℝ^{n} \ B[0, R] and vanishes at ∞. Therefore, |φ(x)| ⩽ (\frac{R}{|x|})^{n−2} for |x| ⩾ R and hence

||φ||_{2}^{2} ⩽ ∫ (\frac{R}{|x|} ∧ 1)^{2(n−2)} dx = ω_{n}R^{n} + nω_{n} ∫_{R}^{∞} R^{2(n−2)}r^{−2(n−2)} r^{n−1}dr
= ω_{n}R^{n} + nω_{n}R^{2(n−2)} \frac{1}{n−4}R^{4−n} = (1 + \frac{n}{n−4})ω_{n}R^{n}.

3 The ground state transformation

Throughout this section, μ ⩾ 0 is a Kato class measure on ℝ^{n}. It follows from [BlMa90; Thm. 3.2.(ii)] that then the eigenfunctions of H_{μ} are in C_{0}(ℝ^{n}), the space of continuous functions vanishing at ∞.

Assume that λ_{0} := inf σ(H_{μ}) < 0 is an eigenvalue of H_{μ}. It is well-known that λ_{0} is non-degenerate and that a corresponding eigenfunction φ_{0} can be chosen such that φ_{0} ⩾ 0. Lemma 2.4 implies that then inf_{B[0,R]} φ_{0} > 0 for all R > 0. We define the unitarily transformed (and shifted) form τ̃_{μ} in L_{2}(φ_{0}^{2}) := L_{2}(ℝ^{n}, φ_{0}^{2}dx) by

D(τ̃_{μ}) := {u ∈ L_{2}(φ_{0}^{2}); φ_{0}u ∈ D(τ_{μ})}, τ̃_{μ}(u) := (τ_{μ} − λ_{0})(φ_{0}u).

In the case μ = V dx, with V from the Kato class, the following result is already proved in [DaSi84; Prop. 4.4S]; see [KoVe07; Thm. 3.1] for the case μ = cσ_{M}, with M ⊆ ℝ^{n} a compact C^{2}-manifold of codimension 1 (cf. Example 2.3(b)).

3.1 Proposition. For all u ∈ D(τ̃_{μ}) one has ∇u ∈ L_{2}(φ_{0}^{2})^{n} and

τ̃_{μ}(u) = ∫ |∇u|^{2} φ_{0}^{2}dx.  (3.1)

Proof. We first assume that u ∈ W_{2}^{1} ∩ L_{∞}. Then φ_{0}u ∈ W_{2}^{1} and φ_{0}u\overline{u} ∈ W_{2}^{1} since φ_{0} ∈ W_{2}^{1} ∩ L_{∞}. Therefore, by the product rule,

∫ ∇(φ_{0}u) · ∇(\overline{φ_{0}u}) dx = ∫ |∇u|^{2} φ_{0}^{2}dx + ∫ ∇φ_{0} · ∇(φ_{0}u\overline{u}) dx,

and hence

τ̃_{μ}(u) − ∫ |∇u|^{2} φ_{0}^{2}dx = τ_{μ}(φ_{0}u) − λ_{0} ∫ |φ_{0}u|^{2} dx − ∫ |∇u|^{2} φ_{0}^{2}dx
= ∫ ∇φ_{0} · ∇(φ_{0}u\overline{u}) dx − ∫ φ_{0} · φ_{0}u\overline{u} dμ − λ_{0} ∫ φ_{0} · φ_{0}u\overline{u} dx
= (τ_{μ} − λ_{0})(φ_{0}, φ_{0}\overline{u}u) = ∫ (H_{μ} − λ_{0})φ_{0} · φ_{0}u\overline{u} dx = 0.

Thus we have shown the assertion for u ∈ D := W_{2}^{1} ∩ L_{∞}.

Now observe that D ⊇ φ_{0}^{−1}C_{c}^{∞} (ψ ∈ C_{c}^{∞} implies φ_{0}^{−1}ψ ∈ D since φ_{0}^{−1} ∈ W_{2,loc}^{1} ∩L_{∞,loc}). Since C_{c}^{∞} is a core for τ_{μ}, it follows that D is a core for τ̃_{μ}. Let u ∈ D(τ̃_{μ}), (u_{k}) ⊆ D, u_{k} → u in D(τ̃_{μ}). Then u_{k} → u in L_{2}(φ_{0}^{2}), and by (3.1) applied to u_{k} − u_{k′} ∈ D we obtain that (∇u_{k})_{k} is a Cauchy sequence in L_{2}(φ_{0}^{2})^{n}. This implies ∇u ∈ L_{2}(φ_{0}^{2})^{n} in the distributional sense, ∇u_{k} → ∇u in L_{2}(φ_{0}^{2})^{n}, and we conclude that (3.1) holds for all u ∈ D(τ̃_{μ}).

In the following assume in addition that μ has compact support and that λ_{1} := inf(σ(H_{μ})\{λ_{0}}) < 0; then λ_{1} is an eigenvalue of H_{μ} since σ_{ess}(H_{μ}) = [0, ∞) by [BEKŠ94; Thm. 3.1]. Let φ_{1} be an associated eigenfunction, φ_{1} real-valued. By Proposition 3.1 we then obtain

(λ_{1} − λ_{0})||φ_{1}||_{2}^{2} = (τ_{μ} − λ_{0})(φ_{1}) = τ̃_{μ}(\frac{φ_{1}}{φ_{0}} ) = ||φ_{0}∇\frac{φ_{1}}{φ_{0}} ||_{2}^{2}.  (3.2)

We now describe how this formula can be used for the estimation of λ_{1} − λ_{0}. The ansatz is largely the same as in [KiSi87] and [KoVe07]; we will indicate the differences below.

As in [KiSi87; p. 405] we normalise φ_{0}, φ_{1} such that ||φ_{0}||_{∞} = ||φ_{1}||_{∞} = 1, sup φ_{1} = 1. Note that then inf φ_{1} < 0 since φ_{0}, φ_{1} are orthogonal. Since φ_{1} ∈ C_{0}(ℝ^{n}), there exist x_{0}, x_{1} ∈ ℝ^{n} such that φ_{1}(x_{0}) = min φ_{1}, φ_{1}(x_{1}) = max φ_{1}. By Lemma 2.4 we have φ_{1} = G\sqrt{|λ_{1}|} ∗ (φ_{1}μ), and hence (|λ_{1}| − Δ)φ_{1} = 0 on ℝ^{n} \ spt μ. This implies that φ_{1} has no positive maxima and no negative minima outside spt μ, and thus x_{0}, x_{1} ∈ spt μ.

In the following we first assume that n ⩾ 2. Let R be the radius of the smallest closed ball containing spt μ; for simplicity suppose that spt μ ⊆ B[0, R]. Let ε:= \frac{1}{4} inf_{B(0,R+1)} φ_{0}. Then 0 < ε ⩽ \frac{1}{4}. Since φ_{1} ∈ C_{0}(ℝ^{n}), there exists δ ∈ (0, 1] such that |φ_{1}(x) − φ_{1}(y)| ⩽ ε for |x − y| ⩽ δ. (Explicit estimates from below for ε and δ will be given in the next two sections.) It follows that

\frac{φ_{1}}{φ_{0}} ⩽ \frac{ε}{φ_{0}} ⩽ \frac{1}{4} on B(x_{0}, δ),  \frac{φ_{1}}{φ_{0}} ⩾ φ_{1} ⩾ 1 − ε ⩾ \frac{3}{4} on B(x_{1}, δ).  (3.3)

In [KiSi87] and [KoVe07], similar estimates were proved by means of gradient estimates on φ_{0} and φ_{1}.

Let now T̃ be the convex hull of B(x_{0}, δ) ∪ B(x_{1}, δ), and T := {x ∈ T̃; 0 ⩽ ⟨x − x_{0}, x_{1} − x_{0}⟩ ⩽ |x_{1} − x_{0}|^{2}}. Then T is a tube connecting the two points x_{0} and x_{1}. By the Cauchy-Schwarz inequality we obtain from (3.2) that

(λ_{1} − λ_{0})||φ_{1}||_{2}^{2} ⩾ ∫_{T} φ_{0}^{2}|∇\frac{φ_{1}}{φ_{0}} |^{2} ⩾ \frac{1}{|T |} inf_{T} φ_{0}^{2} (∫_{T} |∇\frac{φ_{1}}{φ_{0}} |)^{2},

where |T | denotes the volume of T. (This estimate leads to better results than the corresponding estimate in [KiSi87; eq. (4.4)] and [KoVe07; eq. (16)].) Using the fundamental theorem of calculus, we infer from (3.3) that ∫_{T} |∇\frac{φ_{1}}{φ_{0}} | ⩾ \frac{1}{2}ω_{n−1}δ^{n−1}. Moreover, |T | = |x_{1} − x_{0}|ω_{n−1}δ^{n−1}, so we conclude that

(λ_{1} − λ_{0})||φ_{1}||_{2}^{2} ⩾ \frac{1}{2Rω_{n−1}δ^{n−1}} (4ε)^{2}(\frac{1}{2}ω_{n−1}δ^{n−1})^{2} = \frac{2ω_{n−1}}{R} ε^{2}δ^{n−1}.  (3.4)

For n = 1, the above estimation simplifies considerably: If spt μ ⊆ [−R, R] then from (3.2) we obtain, again using the Cauchy-Schwarz inequality and the fundamental theorem of calculus, that

(λ_{1} − λ_{0})||φ_{1}||_{2}^{2} ⩾ ∫_{−R}^{R} φ_{0}^{2}|∇\frac{φ_{1}}{φ_{0}} |^{2} ⩾ \frac{1}{2R} inf_{[−R,R]} φ_{0}^{2} (∫_{−R}^{R} |∇\frac{φ_{1}}{φ_{0}} |)^{2} ⩾ \frac{1}{2R} inf_{[−R,R]} φ_{0}^{2}. (3.5)

4 Hölder continuity of the eigenfunctions

Throughout this section we assume that n ⩾ 2 and that μ ⩾ 0 is a Kato class measure on ℝ^{n}. Then the eigenfunctions of H_{μ} are uniformly continuous since they are in C_{0}(ℝ^{n}) by [BlMa90]. In the main result of this section, Theorem 4.3 below, we will show that the eigenfunctions are Hölder continuous if the volume bound (1.1) is satisfied for some c_{μ} > 0, α ∈ [0, 2), with a Hölder exponent depending on α.

We start with an estimate of the modulus of continuity of the eigenfunctions of H_{μ} that is rather simple but holds in great generality. Let G_{0} be the fundamental solution of −Δ given by

G_{0}(x) = \begin{cases}\frac{1}{2π} ln \frac{1}{|x|} & if n = 2, \\ \frac{1}{(n−2)σ_{n−1}} |x|^{2−n} & if n ⩾ 3. \end{cases}

For r_{0} > 0 let

ρ_{1,r_{0}} := (G_{0} − G_{0}(r_{0}e_{1}))^{+}, ρ_{2,r_{0}} := |∇(G_{0} − ρ_{1,r_{0}})|.

Then ρ_{2,r_{0}}(x) = \frac{1}{σ_{n−1}} |x|^{1−n} for |x| > r_{0}, ρ_{2,r_{0}}(x) = 0 otherwise.

4.1 Proposition. Let φ be an eigenfunction of H_{μ}, H_{μ}φ = −κ^{2}φ for some κ > 0. Let ε > 0. If r_{0} > 0 satisfies ||ρ_{1,r_{0}} ∗ μ||_{∞} ⩽ \frac{ε}{4} then

|φ(x) − φ(y)| ⩽ ε||φ||_{∞} for all x, y ∈ ℝ^{n} such that |x − y| ⩽ \frac{ε}{2}||ρ_{2,r_{0}} ∗ μ||_{∞}^{−1}.

Proof. Let g_{1} := (G_{κ} − G_{κ}(r_{0}e_{1}))^{+}, g_{2} := G_{κ} − g_{1}. By Lemma 2.4 we have

φ = G_{κ} ∗ (φμ) = (g_{1} + g_{2}) ∗ (φμ).

Since g_{1} ∗(φμ) = φ−g_{2} ∗(φμ) is continuous, we can estimate |g_{1} ∗(φμ)(x)−g_{1} ∗(φμ)(y)| ⩽ 2||g_{1} ∗ (φμ)||_{∞} and hence

|φ(x) − φ(y)| ⩽ |g_{2} ∗ (φμ)(x) − g_{2} ∗ (φμ)(y)| + 2||g_{1} ∗ (φμ)||_{∞}
⩽ |x − y| · |||∇g_{2}| ∗ (φμ)||_{∞} + 2||g_{1} ∗ (φμ)||_{∞}
⩽ (|x − y| · |||∇g_{2}| ∗ μ||_{∞} + 2||g_{1} ∗ μ||_{∞})||φ||_{∞}  (4.1)

for all x, y ∈ ℝ^{n}. Observe that

∇G_{κ}(z) = ∫_{0}^{∞}(4πt)^{− \frac{n}{2}} (−\frac{2z}{4t})e^{− \frac{|z|^{2}}{4t} −κ^{2}t}dt

for all z ∈ ℝ^{n} \ {0}, and that this formula also holds for κ = 0. (For n ⩾ 3 the latter is clear; for n = 2 it can easily be seen by computing the integral.) We infer that |∇g_{2}| ⩽ ρ_{2,r_{0}} and g_{1} ⩽ ρ_{1,r_{0}}. By the assumption on r_{0} we thus obtain from (4.1) that

|φ(x) − φ(y)| ⩽ (|x − y| · ||ρ_{2,r_{0}} ∗ μ||_{∞} + 2 · \frac{ε}{4})||φ||_{∞},

and the assertion follows.

4.2 Remarks. (a) The above estimate of the modulus of continuity does not depend on κ. It is clear that for large κ one obtains a better estimate using (4.1) without further estimating g_{1} and |∇g_{2}|.

(b) Let r_{0} > 0 be as in Proposition 4.1 and assume that μ is finite. Obviously, ||ρ_{2,r_{0}} ∗ μ||_{∞} ⩽ ||ρ_{2,r_{0}}||_{∞}||μ|| = \frac{1}{σ_{n−1}} r_{0}^{1−n}||μ||, so it follows that

|φ(x) − φ(y)| ⩽ ε||φ||_{∞} if |x − y| ⩽ \frac{σ_{n−1}}{2||μ||} εr_{0}^{n−1}.

For the proof of Theorem 1.1 we will need the following application of Proposition 4.1. In the case of Example 2.3(c) (where n = 2 and α = 1), a related result is given in [KoVe07; Prop. 6.8].

4.3 Theorem. Assume that the measure μ is finite and that (1.1) holds for some c_{μ} > 0, 0 ⩽ α < 2. Let φ be an eigenfunction of H_{μ} and ε > 0. Then

|φ(x) − φ(y)| ⩽ ε||φ||_{∞} for all |x − y| ⩽ δ,

where

δ = \begin{cases}\frac{σ_{n−1}}{2} \frac{1−α}{n−α}||μ||^{− \frac{1−α}{n−α}} c_{μ}^{− \frac{n−1}{n−α}} ε & if α < 1, \\ \frac{σ_{n−1}}{2n} \frac{ε}{c_{μ}} [ln(1 + \frac{4}{σ_{n−1}} ||μ||^{\frac{1}{n−1}} cμ^{\frac{n−2}{n−1}} \frac{1}{ε})]^{−1} & if α = 1, \\ \frac{1}{2}(σ_{n−1})^{\frac{1}{2−α}} \frac{α−1}{n−1}(\frac{2−α}{4})^{\frac{α−1}{2−α}} (\frac{ε}{c_{μ}} )^{\frac{1}{2−α}} & if α > 1. \end{cases}

Proof. Let r_{0} > 0. In order to apply Proposition 4.1, we have to estimate ||ρ_{1,r_{0}} ∗ μ||_{∞} and ||ρ_{2,r_{0}} ∗ μ||_{∞}. Let m(r) := c_{μ}r^{n−α} (r ⩾ 0). Using (2.3) we estimate

||ρ_{1,r_{0}} ∗ μ||_{∞} ⩽ ∫_{0}^{r_{0}} ρ_{1,r_{0}}(re_{1}) dm(r) = ρ_{1,r_{0}}(re_{1})m(r)|_{0+}^{r_{0}} − ∫_{0}^{r_{0}} m(r) dρ_{1,r_{0}}(r)
= ∫_{0}^{r_{0}} m(r)\frac{1}{σ_{n−1}} r^{1−n} dr = \frac{c_{μ}}{σ_{n−1}} ∫_{0}^{r_{0}} r^{1−α} dr = \frac{c_{μ}}{σ_{n−1}(2−α)}r_{0}^{2−α}.

Setting

r_{0} := (\frac{2−α}{4}σ_{n−1} \frac{ε}{c_{μ}} )^{\frac{1}{2−α}}  (4.2)

we thus obtain ||ρ_{1,r_{0}} ∗ μ||_{∞} ⩽ \frac{ε}{4}, so

|φ(x) − φ(y)| ⩽ ε||φ||_{∞} for all |x − y| ⩽ \frac{ε}{2}||ρ_{2,r_{0}} ∗ μ||_{∞}^{−1}  (4.3)

by Proposition 4.1.

Recall that σ_{n−1}ρ_{2,r_{0}}(x) ⩽ (|x| ∨ r_{0})^{1−n} for all x ∈ ℝ^{n}. In the case α > 1 we obtain by (2.3) that

σ_{n−1}||ρ_{2,r_{0}} ∗ μ||_{∞} ⩽ ∫_{0}^{∞}(r ∨ r_{0})^{1−n} dm(r) = r_{0}^{1−n}m(r_{0}) + ∫_{r_{0}}^{∞} r^{−α}(n − α)c_{μ} dr
= c_{μ}r_{0}^{1−α} + c_{μ} \frac{n−α}{α−1}r_{0}^{1−α} = c_{μ} \frac{n−1}{α−1}r_{0}^{1−α},

hence

\frac{ε}{2}||ρ_{2,r_{0}} ∗ μ||_{∞}^{−1} ⩾ \frac{1}{2} \frac{α−1}{n−1}σ_{n−1} \frac{ε}{c_{μ}} r_{0}^{α−1} = \frac{1}{2} \frac{α−1}{n−1}(\frac{2−α}{4})^{\frac{α−1}{2−α}} (σ_{n−1} \frac{ε}{c_{μ}} )^{1+ \frac{α−1}{2−α}}

by (4.2), and the assertion follows from (4.3).

Let now r_{1} := (\frac{||μ||}{c_{μ}})^{\frac{1}{n−α}}. Then m(r_{1}) = c_{μ}r_{1}^{n−α} = ||μ|| and hence, again by (2.3),

σ_{n−1}||ρ_{2,r_{0}} ∗ μ||_{∞} ⩽ ∫_{0}^{r_{1}}(r ∨ r_{0})^{1−n} dm(r).

In the case α < 1 we estimate

σ_{n−1}||ρ_{2,r_{0}} ∗ μ||_{∞} ⩽ ∫_{0}^{r_{1}}r^{1−n} dm(r) = ∫_{0}^{r_{1}}r^{−α}(n − α)c_{μ} dr = c_{μ} \frac{n−α}{1−α}r_{1}^{1−α} = \frac{n−α}{1−α}||μ||^{\frac{1−α}{n−α}} cμ^{\frac{n−1}{n−α}} ,

so as above the assertion follows from (4.3).

Finally, let α = 1. Then

σ_{n−1}||ρ_{2,r_{0}} ∗ μ||_{∞} ⩽ r_{0}^{1−n}m(r_{0} ∧ r_{1}) + ∫_{r_{0}∧r_{1}}^{r_{1}} r^{−1}(n − 1)c_{μ} dr
= r_{0}^{1−n}c_{μ}(r_{0} ∧ r_{1})^{n−1} + (n − 1)c_{μ} ln \frac{r_{1}}{r_{0}∧r_{1}}
= c_{μ}[(\frac{r_{1}}{r_{0}} ∧ 1)^{n−1} + (n − 1) ln(\frac{r_{1}}{r_{0}} ∨ 1)] = c_{μ}f(\frac{r_{1}}{r_{0}} ),

with f(x) := (x ∧ 1)^{n−1} + (n − 1) ln(x ∨ 1). We show that f(x) ⩽ n ln(1 + x) by distinguishing the cases x < 1 and x ⩾ 1: If x < 1 then

f(x) = x^{n−1} + 0 ⩽ x ⩽ \frac{1}{ln 2} ln(1 + x) ⩽ n ln(1 + x)

since n ⩾ 2. If x ⩾ 1 then f(x) = 1 + (n − 1) ln x, so we have to show that h_{n}(x) := n ln(1 + x) − (n − 1) ln x ⩾ 1. Observe that h_{n}(x) ⩾ h_{2}(x) and that h_{2} is increasing on [1, ∞), hence h_{n}(x) ⩾ h_{2}(1) = 2 ln 2 − ln 1 ⩾ 1. We conclude that

||ρ_{2,r_{0}} ∗ μ||_{∞} ⩽ \frac{c_{μ}}{σ_{n−1}} f(\frac{r_{1}}{r_{0}} ) ⩽ \frac{nc_{μ}}{σ_{n−1}} ln(1 + \frac{r_{1}}{r_{0}} ).

By (4.3) this implies the assertion since r_{1} = (\frac{||μ||}{c_{μ}})^{\frac{1}{n−1}} and r_{0} = \frac{σ_{n−1}}{4} \frac{ε}{c_{μ}}.

4.4 Remark. Theorem 4.3 implies that the eigenfunctions of H_{μ} are Lipschitz continuous if α < 1. In general, the eigenfunctions are not Lipschitz continuous if α = 1: One can show that for n ⩾ 2 and μ = cσ_{M}, where M = B_{ℝ^{n−1}}(0, 1) × {0} and c > 0 is large enough for the ground state φ_{0} to exist, the logarithmic factor given in Theorem 4.3 reflects the correct behaviour of the modulus of continuity of φ_{0}.

5 Estimate of φ_{0} from below

In this section we prove a pointwise estimate of the ground state φ_{0} of H_{μ} from below. Throughout the section we assume that

||∫_{0}^{t} k_{s} ∗ μ ds||_{∞} ⩽ ct^{θ}  (t > 0)  (5.1)

for some c > 0, θ ∈ (0, 1] (cf. Proposition 2.2); in particular, μ is in the Kato class.

As mentioned in Section 3, μ being in the Kato class implies that the eigenfunctions of H_{μ} are in C_{0}(ℝ^{n}). The following result provides an estimate of the L_{∞}-norm of the eigenfunctions.

5.1 Proposition. Assume that (5.1) holds for some c > 0, θ ∈ (0, 1]. Let φ be an eigenfunction of H_{μ}, H_{μ}φ = −κ^{2}φ for some κ > 0. If ∫ |φ| dμ < ∞ then

||φ||_{∞} ⩽ c_{n}(c, θ, κ) ∫ |φ| dμ,

where

c_{1}(c, θ, κ) = \frac{1}{2κ},
c_{2}(c, θ, κ) = \frac{e}{2π} ln(1 + (4c)^{\frac{1}{θ}} κ^{−2}),
c_{n}(c, θ, κ) = \frac{1}{n−2}ω_{n}^{−2/n}(4π)^{− \frac{n−2}{2}} (4c)^{\frac{n−2}{2θ}}  (n ⩾ 3).

Before proving this proposition, we state and prove the main result of this section, the announced pointwise estimate of φ_{0}.

5.2 Theorem. Assume that spt μ ⊆ B[0, R] for some R > 0 and that (5.1) holds for some c > 0, θ ∈ (0, 1]. Let 0 ⩽ φ_{0} ∈ D(H_{μ}) be the ground state of H_{μ}, ||φ_{0}||_{∞} = 1, κ > 0 such that H_{μ}φ_{0} = −κ^{2}φ_{0}.

(a) If n = 1 then φ_{0}(x) ⩾ e^{−κ(|x|+R)} for all x ∈ ℝ.

(b) If n = 2 then

φ_{0}(x) ⩾ \frac{1}{2e}([(4c)^{− \frac{1}{2θ}} (|x| + R)^{−1}] ∧ 1)e^{−κ(|x|+R)} (x ∈ ℝ^{2}).

(c) If n ⩾ 3 then

φ_{0}(x) ⩾ c_{n}(4c)^{− \frac{n−2}{2θ}} (|x| + R)^{2−n}e^{−κ(|x|+R)} (x ∈ ℝ^{n}),

where c_{n} = \frac{1}{σ_{n−1}} ω_{n}^{2/n}(4π)^{\frac{n−2}{2}} = 2^{n−3}(\frac{2}{n})^{\frac{2}{n}} Γ(\frac{n}{2})^{\frac{n−2}{n}}.

Proof. Lemma 2.4 implies that

φ_{0}(x) = ∫_{B(0,R)} G_{κ}(x − y)φ_{0}(y) dμ(y) ⩾ G_{κ}((|x| + R)e_{1}) ∫ φ_{0} dμ  (5.2)

for all x ∈ ℝ^{n}.

(a) For n = 1 we have G_{κ}(x) = \frac{1}{2κ}e^{−κ|x|} (x ∈ ℝ), and ∫ φ_{0} dμ ⩾ 2κ by Proposition 5.1, so the assertion follows from (5.2).

(b) Let x ∈ ℝ^{2} \ {0}. Proposition A.1(b) from the appendix and Proposition 5.1 yield

G_{κ}(x) ∫ φ_{0} dμ ⩾ \frac{\frac{1}{2π} ln(1 + \frac{1}{κ|x|})e^{−κ|x|}}{\frac{e}{2π} ln(1 + (4c)^{\frac{1}{2θ}} \frac{1}{κ})^{2}} = \frac{1}{2e} \frac{ln(1 + \frac{a}{κ})}{ln(1 + \frac{b}{κ})}e^{−κ|x|},

with a = \frac{1}{|x|} and b = (4c)^{\frac{1}{2θ}}. By (5.2) the assertion follows if ln(1 + x)/ ln(1 + y) ⩾ \frac{x}{y} ∧ 1 for all x, y > 0. In the case x ⩾ y the latter inequality is clear; for x < y it is equivalent to \frac{1}{x} ln(1 + x) ⩾ \frac{1}{y} ln(1 + y), which is a consequence of the concavity of ln, so the proof of (b) is complete.

(c) Let x ∈ ℝ^{n} \ {0}. Proposition A.1(a) from the appendix and Proposition 5.1 yield

G_{κ}(x) ∫ φ_{0} dμ ⩾ \frac{1}{(n−2)σ_{n−1}} |x|^{2−n}e^{−κ|x|} · (n − 2)ω_{n}^{2/n}(4π)^{\frac{n−2}{2}} (4c)^{− \frac{n−2}{2θ}} ,

so the assertion follows from (5.2).

For the proof of Proposition 5.1 we need the following estimate of ||T_{μ}(t)||_{p→∞}.

5.3 Lemma. Assume that (5.1) holds for some c > 0, θ ∈ (0, 1]. Let t_{0} := 2(4c)^{− \frac{1}{θ}}. Then

||T_{μ}(t_{0})||_{p→∞} ⩽ 2(2πt_{0})^{− \frac{n}{2p}} = 2(4π)^{− \frac{n}{2p}} (4c)^{\frac{n}{2θp}} (1 ⩽ p ⩽ ∞).

Proof. Let t_{1} := \frac{1}{2}t_{0} = (4c)^{− \frac{1}{θ}}. By assumption (5.1) we have

||∫_{0}^{t_{1}} k_{t} ∗ μ dt||_{∞} ⩽ ct_{1}^{θ} = \frac{1}{4}.

From Proposition 2.1 it follows that ||T_{aμ}(t_{1})||_{1→1} ⩽ 1/(1 − \frac{a}{4}) for all 0 ⩽ a < 4. Moreover, ||T_{0μ}(t_{1})||_{1→∞} = ||e^{t_{1}Δ}||_{1→∞} = (4πt_{1})^{− \frac{n}{2}} , so by the Stein interpolation theorem we infer as in [StVo96; Thm. 5.1] that

||T_{μ}(t_{1})||_{1→2} ⩽ ||T_{2μ}(t_{1})||1→1^{\frac{1}{2}}||T_{0μ}(t_{1})||1→∞^{\frac{1}{2}} ⩽ 2^{\frac{1}{2}} (4πt_{1})^{− \frac{n}{4}} .

Hence, by duality,

||T_{μ}(t_{0})||_{1→∞} ⩽ ||T_{μ}(t_{1})||_{1→2}^{2} ⩽ 2(4πt_{1})^{− \frac{n}{2}} = 2(2πt_{0})^{− \frac{n}{2}} .

The assertion now follows from Riesz-Thorin interpolation with the estimate

||T_{μ}(t_{0})||_{∞→∞} ⩽ ||T_{μ}(t_{1})||_{1→1}^{2} ⩽ (\frac{1}{1−1/4})^{2} < 2.

Proof of Proposition 5.1. By Lemma 2.4 we have φ = G_{κ} ∗ (φμ) = (κ^{2} − Δ)^{−1}(φμ). In the case n = 1, the assertion follows since ||G_{κ}||_{∞} = \frac{1}{2κ}. For n ⩾ 2 we estimate, with t_{0} = 2(4c)^{− \frac{1}{θ}} as in Lemma 5.3,

||φ||_{∞} = ||e^{−κ^{2}t_{0}}T_{μ}(t_{0})φ||_{∞} ⩽ ||T_{μ}(t_{0})(κ^{2} − Δ)^{−1} : M(ℝ^{n}) → L_{∞}(ℝ^{n})|| ∫ |φ| dμ,

where M(ℝ^{n}) is the Banach space of finite Borel measures on ℝ^{n}. We now have to show that the norm of T_{μ}(t_{0})(κ^{2} − Δ)^{−1} is less or equal c_{n}(c, θ, κ).

First suppose that n = 2. Then for all p ∈ [1, ∞) we have

||(κ^{2} − Δ)^{−1} : M(ℝ^{n}) → L_{p}(ℝ^{n})|| ⩽ ∫_{0}^{∞} e^{−κ^{2}t}||e^{tΔ} : M(ℝ^{n}) → L_{p}(ℝ^{n})|| dt
⩽ ∫_{0}^{∞}(4πt)^{− \frac{2}{2} (1− \frac{1}{p})}e^{−κ^{2}t} dt = κ^{− \frac{2}{p}} (4π)^{\frac{1}{p} −1} ∫_{0}^{∞} s^{\frac{1}{p} −1}e^{−s} ds.

Since ∫_{0}^{∞} s^{\frac{1}{p} −1}e^{−s} ds = Γ(\frac{1}{p}) = pΓ(1 + \frac{1}{p}) ⩽ p, we conclude by Lemma 5.3 that

||T_{μ}(t_{0})(κ^{2} − Δ)^{−1} : M(ℝ^{n}) → L_{∞}(ℝ^{n})|| ⩽ 2(4π)^{− \frac{2}{2p}} (4c)^{\frac{2}{2θp}} · κ^{− \frac{2}{p}} (4π)^{\frac{1}{p} −1}p
= \frac{1}{2π}p((4c)^{\frac{1}{θ}} κ^{−2})^{\frac{1}{p}} =: f(p).

For the proof of the case n = 2 it remains to show that there exists p ∈ [1, ∞) such that f(p) ⩽ c_{2}(c, θ, κ). With a := (4c)^{\frac{1}{θ}} κ^{−2} this inequality reads pa^{\frac{1}{p}} ⩽ e ln(1 + a). If a ⩽ e then a ⩽ e ln(1 + a), so the inequality holds with p = 1. For a > e we take p = ln a; then

pa^{\frac{1}{p}} = pe^{\frac{1}{p} ln a} = ln a · e^{1} ⩽ e ln(1 + a).

Let now n ⩾ 3. For m ∈ M(ℝ^{n}) we have (κ^{2} − Δ)^{−1}m = ∫ G_{κ}(· − y) dm(y) (Bochner integral of y ↦ G_{κ}(· − y) ∈ L_{1}(ℝ^{n})) and hence

||T_{μ}(t_{0})(κ^{2} − Δ)^{−1}m||_{∞} ⩽ ∫ ||T_{μ}(t_{0})G_{κ}(· − y)||_{∞} dm(y) ⩽ sup_{y∈ℝ^{n}} ||T_{μ}(t_{0})G_{0}(· − y)||_{∞} · ||m||.

Using Lemma 5.4 below and Lemma 5.3, we deduce that

||T_{μ}(t_{0})(κ^{2} − Δ)^{−1} : M(ℝ^{n}) → L_{∞}(ℝ^{n})|| ⩽ \frac{1}{2(n−2)}ω_{n}^{− \frac{2}{n}} ||T_{μ}(t_{0})||1→∞^{\frac{n−2}{n}}||T_{μ}(t_{0})||∞→∞^{\frac{2}{n}}
⩽ \frac{1}{2(n−2)}ω_{n}^{− \frac{2}{n}} · 2(4π)^{− \frac{n−2}{2}} (4c)^{\frac{n−2}{2θ}}

and conclude the proof.

In the following interpolation lemma we implicitly use the fact that G_{0} is in the weak Lebesgue space L_{\frac{n}{n−2} ,w}.

5.4 Lemma. Let n ⩾ 3. Let T: L_{1} + L_{∞}(ℝ^{n}) → L_{∞}(ℝ^{n}) be a bounded linear operator, y ∈ ℝ^{n}, f(x) := (n − 2)G_{0}(x − y) = \frac{1}{σ_{n−1}} |x − y|^{2−n} (x ∈ ℝ^{n} \ \{y\}). Then

||T f||_{∞} ⩽ \frac{1}{2}ω_{n}^{− \frac{2}{n}} ||T ||1→∞^{\frac{n−2}{n}}||T ||∞→∞^{\frac{2}{n}} .

Proof. Without loss of generality assume that y = 0. Let r_{0} > 0, f_{1} := (f − f(r_{0}e_{1}))^{+}, f_{2} := f − f_{1}. Then ||f_{2}||_{∞} = f(r_{0}e_{1}) = \frac{1}{σ_{n−1}} r_{0}^{2−n},

||f_{1}||_{1} = ∫_{0}^{r_{0}}(r^{2−n} − r_{0}^{2−n}) r^{n−1}dr = ∫_{0}^{r_{0}}(r − r_{0}^{2−n}r^{n−1}) dr = (\frac{1}{2} − \frac{1}{n})r_{0}^{2},

and hence

||T f||_{∞} ⩽ ||T f_{1}||_{∞} + ||T f_{2}||_{∞} ⩽ ||T ||_{1→∞} \frac{n−2}{2n}r_{0}^{2} + ||T ||_{∞→∞} \frac{1}{σ_{n−1}} r_{0}^{2−n}.

Let now a := \frac{1}{n}||T ||_{1→∞}, b := \frac{1}{σ_{n−1}} ||T ||_{∞→∞} and r_{0} := (\frac{b}{a})^{1/n} (without loss of generality T ≠ 0, so a ≠ 0). Then we obtain that

||T f||_{∞} ⩽ r_{0}^{2}(\frac{n−2}{2}a + br_{0}^{−n}) = r_{0}^{2}(\frac{n−2}{2}a + a) = (\frac{n}{σ_{n−1}} \frac{||T ||_{∞→∞}}{||T ||_{1→∞}} )^{\frac{2}{n}} \frac{n}{2} \frac{1}{n}||T ||_{1→∞},

and the assertion follows.

6 Proofs of the main results

We are now in a position to prove our main theorems stated in Section 1.

Proof of Theorem 1.1. Let R := \frac{d}{2} and assume without loss of generality that spt μ ⊆ B[0, R]. As in Section 3 we normalise φ_{0}, φ_{1} such that ||φ_{0}||_{∞} = ||φ_{1}||_{∞} = 1 and sup φ_{0} = sup φ_{1} = 1. Let ε := \frac{1}{4} inf_{B[0,R+1]} φ_{0}. Then ε ∈ (0, \frac{1}{4}]. Let δ̃ > 0 be as in Theorem 4.3, so that |φ_{1}(x) − φ_{1}(y)| ⩽ ε for |x − y| ⩽ δ̃, and let δ := δ̃ ∧ 1. According to (3.4) we can estimate

λ_{1} − λ_{0} ⩾ \frac{2ω_{n−1}}{R} ε^{2}δ^{n−1}||φ_{1}||_{2}^{−2}.  (6.1)

By Proposition 2.5(a) we have ||φ_{1}||_{2}^{−2} ⩾ |λ_{1}|||μ||^{−1}. Moreover,

||μ|| = μ(B[0, R]) ⩽ c_{μ}R^{n−α}  (6.2)

and hence

κ_{0}^{2} − κ_{1}^{2} = λ_{1} − λ_{0} ⩾ 2ω_{n−1}|λ_{1}|c_{μ}^{−1}R^{α−n−1}ε^{2}δ^{n−1} ⩾ γκ_{1}^{2},

where γ := 2ω_{n−1}(c_{μ} + 1)^{−1}(d + 1)^{α−n−1}ε^{2}δ^{n−1}. This implies κ_{0}^{2} ⩾ (1 + γ)κ_{1}^{2}, and therefore κ_{0}^{2} − κ_{1}^{2} ⩾ κ_{0}^{2} − \frac{1}{1+γ} κ_{0}^{2} = \frac{γ}{1+γ} κ_{0}^{2}. Since γ ⩽ 2ω_{n−1}(\frac{1}{4})^{2} ⩽ 1, we conclude that

λ_{1} − λ_{0} ⩾ \frac{γ}{1+γ} κ_{0}^{2} ⩾ \frac{γ}{2}κ_{0}^{2} = ω_{n−1}|λ_{0}|(c_{μ} + 1)^{−1}(d + 1)^{α−n−1}ε^{2}δ^{n−1}.  (6.3)

In the case n ⩾ 5 we have ||φ_{1}||_{2}^{2} ⩽ 6ω_{n}R^{n} by Proposition 2.5(c); then (6.1) yields

λ_{1} − λ_{0} ⩾ \frac{2ω_{n−1}}{6ω_{n}} (d + 1)^{−n−1}ε^{2}δ^{n−1}.  (6.4)

We are going to show that

ε^{2}δ^{n−1} ⩾ C_{0}(c_{μ} + 1)^{−p_{0}}(d + 1)^{−q_{0}}e^{−βκ_{0}(d+1)},  (6.5)

with β as in the assertion of the theorem and constants C_{0}, p_{0}, q_{0} > 0 depending only on n and α. Then (6.3) implies (1.2) with C = ω_{n−1}C_{0}, p = p_{0} + 1, q = q_{0} + n + 1 − α, and the additional assertion for n ⩾ 5 follows from (6.4).

Recall from Proposition 2.2 that (5.1) holds with θ = 1− \frac{α}{2} and c = \frac{2}{2−α}2^{−α}π^{− \frac{n}{2}} Γ(\frac{n−α}{2} + 1)c_{μ}. Hence, by Theorem 5.2 and the definition of ε we have

ε ⩾ \begin{cases}\frac{1}{8e}(4c + 1)^{− \frac{1}{2−α}} (2R + 1)^{−1}e^{−κ_{0}(2R+1)} & if n = 2, \\ \frac{c_{n}}{4} (4c)^{− \frac{n−2}{2−α}} (2R + 1)^{2−n}e^{−κ_{0}(2R+1)} & if n ⩾ 3. \end{cases}

We infer that there exists a constant c_{n,α} > 0 (depending only on n and α) such that

ε ⩾ c_{n,α}(c_{μ} + 1)^{− \frac{n−1}{2−α}} (d + 1)^{−(n−1)}e^{−κ_{0}(d+1)}.  (6.6)

Assume now that α < 1. Then ||μ||^{− \frac{1−α}{n−α}} c_{μ}^{− \frac{n−1}{n−α}} ⩾ R^{−(1−α)}c_{μ}^{−1} by (6.2). According to Theorem 4.3 we thus have δ̃ ⩾ c̃_{n,α}(c_{μ} + 1)^{−1}(d + 1)^{−(1−α)}ε =: δ_{1}, with c̃_{n,α} > 0 depending only on n and α. Without loss of generality c̃_{n,α} ⩽ 4; then δ_{1} ⩽ 1 since ε ⩽ \frac{1}{4}. We obtain that

ε^{2}δ^{n−1} ⩾ c̃_{n,α}^{n−1}(c_{μ} + 1)^{−(n−1)}(d + 1)^{−(n−1)(1−α)}ε^{n+1},

so from (6.6) we deduce that (6.5) holds with β = n + 1, p_{0} = \frac{n^{2}−1}{2−α} + n − 1, q_{0} = n^{2} − 1 + (n − 1)(1 − α) and C_{0} = c̃_{n,α}^{n−1}c_{n,α}^{n+1}.

In the case α > 1 we have δ̃ ⩾ c̃_{n,α}(\frac{ε}{c_{μ}+1})^{\frac{1}{2−α}} =: δ_{2}. Again, a suitable choice of c̃_{n,α} ensures that δ_{2} ⩽ 1, and in the same way as above we deduce (6.5), now with β = \frac{n−1}{2−α} + 2, p_{0} = β^{2} − β − 2, q_{0} = (n − 1)β and C_{0} = c̃_{n,α}^{n−1}c_{n,α}^{β}.

Finally assume that α = 1. Given β > n + 1, there exists α̃ ∈ (1, 2) such that β = \frac{n−1}{2−α̃} + 2. Then μ(B(x, r)) ⩽ c_{μ}(r ∧ R)^{n−1} ⩽ c_{μ}R^{α̃−1}r^{n−α̃} for all x ∈ ℝ^{n}, r > 0, and from the preceding paragraph we obtain that

ε^{2}δ^{n−1} ⩾ c̃_{n,α̃}^{n−1}c_{n,α̃}^{β}(R^{α̃−1}c_{μ} + 1)^{−(β^{2}−β−2)}(d + 1)^{−(n−1)β}e^{−βκ_{0}(d+1)}.

Using the inequality R^{α̃−1}c_{μ} + 1 ⩽ (R + 1)^{α̃−1}(c_{μ} + 1), we deduce by straightforward computation that (6.5) holds with p_{0} = β^{2} − β − 2, q_{0} = β^{2} − β − n − 1 and C_{0} = c̃_{n,α̃}^{n−1}c_{n,α̃}^{β}.

Proof of Theorem 1.3. Normalising μ, φ_{0} and φ_{1} as in the proof of Theorem 1.1, we obtain from (3.5) and Theorem 5.2(a) that

λ_{1} − λ_{0} ⩾ \frac{1}{d} inf_{[−d/2,d/2]} φ_{0}^{2}||φ_{1}||_{2}^{−2} ⩾ \frac{1}{d}e^{−2κ_{0}d}||φ_{1}||_{2}^{−2}.

The second assertions thus follows from Proposition 2.5(b). Proposition 2.5(a) implies that λ_{1} − λ_{0} ⩾ \frac{|λ_{1}|}{d||μ||}e^{−2κ_{0}d}, so the first assertion follows as in the argument leading to equation (6.3).

A Appendix

Here we prove the estimate of G_{κ} from below that is needed in the proof of Theorem 5.2. We assume that n ⩾ 2 and recall that σ_{n−1} = \frac{2π^{n/2}}{Γ(n/2)} is the (n−1)-dimensional volume of the unit sphere in ℝ^{n}.

A.1 Proposition. Let κ > 0, and let G_{κ} be the convolution kernel of (κ^{2} − Δ)^{−1} on L_{2}(ℝ^{n}).

(a) If n ⩾ 3 then G_{κ}(x) ⩾ \frac{1}{(n−2)σ_{n−1}} |x|^{2−n}e^{−κ|x|} for all x ∈ ℝ^{n} \ \{0\}.

(b) If n = 2 then G_{κ}(x) ⩾ \frac{1}{2π} ln(1 + \frac{1}{κ|x|})e^{−κ|x|} for all x ∈ ℝ^{2} \ \{0\}.

Proof. By [Sch66; eq. (VII,10;15)] we have

G_{κ}(x) = (2π)^{− \frac{n}{2}} κ^{\frac{n−2}{2}} |x|^{− \frac{n−2}{2}} K_{\frac{n−2}{2}} (κ|x|) (x ∈ ℝ^{n} \ \{0\}),  (A.1)

where K_{ν} is the modified Bessel function of the third kind of order ν ⩾ 0. In particular we see that G_{κ}(x) = κ^{n−2}G_{1}(κx), and that it therefore suffices to show the assertion for κ = 1.

(a) By [AbSt72; 9.6.9] we obtain from (A.1) that

G_{1}(x) = \frac{1}{(n − 2)σ_{n−1}} |x|^{2−n} + o(|x|^{2−n}) as x → 0.

Let ε > 0,

φ_{ε}(x) := \frac{1 − ε}{(n − 2)σ_{n−1}} |x|^{2−n}e^{−|x|} − G_{1}(x) (x ∈ ℝ^{n} \ \{0\}).

Then there exists δ > 0 such that φ_{ε} < 0 on B(0, δ) \ {0}, and φ_{ε}(x) → 0 as |x| → ∞. Moreover, a straightforward computation shows that (1 − Δ)φ_{ε}(x) = −\frac{1−ε}{(n−2)σ_{n−1}} (n − 3)|x|^{1−n}e^{−|x|} ⩽ 0 for x ≠ 0. Thus φ_{ε} has no positive maxima. We conclude that φ_{ε} < 0 on ℝ^{n} \ {0} for all ε > 0, and for ε → 0 we obtain (a).

(b) By (A.1) we have G_{1}(x) = \frac{1}{2π}K_{0}(|x|), so we must show that K_{0}(r) ⩾ ln(1 + \frac{1}{r})e^{−r} for all r > 0. Let a := 2e^{−γ}, where γ = 0.577 … is the Euler-Mascheroni constant. (Then 1 < a < 2.) From [AbSt72; 9.6.13] it follows that K_{0}(r) = ln \frac{a}{r} + o(r) as r → 0.

Let f(r) := ln(1 + \frac{a}{r})e^{−r} for all r > 0. One easily sees that f(r) = (1 − r) ln \frac{a}{r} + O(r) as r → 0, so g := f − K_{0} < 0 on (0, ε) for some ε > 0. We will prove that g < 0 on (0, ∞); then the claim follows. A straightforward computation shows that

g′′(r) + \frac{1}{r}g′(r) − g(r) = f′′(r) + \frac{1}{r}f′(r) − f(r) = \frac{a}{r(a+r)}h(\frac{r}{a})e^{−r}  (A.2)

for all r > 0, with h(s) := \frac{1}{a} \frac{1}{1+s} + 2 − (1 + s) ln(1 + \frac{1}{s}), and that

h′′(s) = \frac{1}{(1 + s)^{3}} (\frac{2}{a} − (1 + \frac{1}{s})^{2}) (s > 0).

We obtain that h is concave on (0, s_{0}) and convex on (s_{0}, ∞), where s_{0} = (\sqrt{\frac{2}{a}} − 1)^{−1}. Since h(s) → −∞ as s → 0 and h(s) → 1 as s → ∞, we infer that there exists s_{1} ∈ (0, s_{0}) such that h < 0 on (0, s_{1}) and h > 0 on (s_{1}, ∞).

Now, if g has a positive maximum at r > 0 then the left hand side of (A.2) is negative, and hence \frac{r}{a} ∈ (0, s_{1}). Similarly, if g has a negative minimum at r > 0 then \frac{r}{a} ∈ (s_{1}, ∞). Thus, to the left of a positive maximum there cannot be any negative minimum. Since g < 0 on (0, ε) and g(r) → 0 as r → 0, we conclude that g has no positive maximum. This implies that g < 0 on (0, ∞) since g(r) → 0 as r → ∞.

Acknowledgement. The author thanks Ivan Veselić for introducing him to the subject and for some valuable discussions.

References

[AbSt72] M. Abramovitz and I. A. Stegun: Handbook of mathematical functions. Dover, New York, 1972.

[BlMa90] Ph. Blanchard and Z. M. Ma: Semigroup of Schrödinger operators with potentials given by Radon measures. In Stochastic processes, physics and geometry (Ascona and Locarno, 1988), pp. 160–195, World Sci. Publ., Teaneck, NJ, 1990.

[BEKŠ94] J. F. Brasche, P. Exner, Yu. A. Kuperin and P. Šeba: Schrödinger operators with singular interactions. J. Math. Anal. Appl. 184 (1994), no. 1, 112– 139.

[DaSi84] E. B. Davies and B. Simon: Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59 (1984), no. 2, 335–395.

[Dav95] E. B. Davies: L^{p} spectral independence and L^{1} analyticity. J. London Math. Soc. (2) 52 (1995), no. 1, 177–184.

[Har78] E. M. Harrell: On the rate of asymptotic eigenvalue degeneracy. Comm. Math. Phys. 60 (1978), no. 1, 73–95.

[Har80] E. M. Harrell: Double wells. Comm. Math. Phys. 75 (1980), no. 3, 239–261.

[KiSi87] W. Kirsch and B. Simon: Comparison theorems for the gap of Schrödinger operators. J. Funct. Anal. 75 (1987), no. 2, 396–410.

[KoVe07] S. Kondej and I. Veselić: Lower bounds on the lowest spectral gap of singular potential Hamiltonians. Ann. Henry Poincaré 8 (2007), no. 1, 109– 134.

[Sch66] L. Schwartz: Théorie des distributions. Nouvelle éd. Hermann, Paris, 1966.

[Sim82] B. Simon: Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526.

[Sim84] B. Simon: Semiclassical analysis of low lying eigenvalues. II. Tunneling. Ann. of Math. (2) 120 (1984), no. 1, 89–118.

[SWYY85] I. M. Singer, B. Wong, S.-T. Yau and S. S.-T. Yau: An estimate of the gap of the first two eigenvalues in the Schrödinger operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 2, 319–333.

[Smi96] R. G. Smits: Spectral gaps and rates to equilibrium for diffusions in convex domains. Michigan Math. J. 43 (1996), no. 1, 141–157.

[StVo96] P. Stollmann and J. Voigt: Perturbation of Dirichlet forms by measures. Potential Anal. 5 (1996), no. 2, 109–138.

[Voi77] J. Voigt: On the perturbation theory for strongly continuous semigroups. Math. Ann. 229 (1977), 163–171.

[Voi86] J. Voigt: Absorption semigroups, their generators and Schrödinger semigroups. J. Funct. Anal. 67 (1986), no. 2, 167–205.