Asymptotics of the Infimum of the Spectrum of Schrödinger Operators with Magnetic Fields^{∗}

By

Naomasa Ueki

1. Introduction Let D be a domain in ℝ^{d} given a Riemannian metric and b be a 1-form on D. Let L(b) be the self-adjoint operator corresponding to the closed extension of the form

(1.1)  q(b)(φ) = \frac{1}{2}∥(id + ext(b))φ∥^{2},  i = \sqrt{−1},

for any φ ∈ C_{0}^{∞}(D^{∘}): L(b) is the Schrödinger operator with a magnetic field db and the Dirichlet boundary condition. For the notation, see Section 2 below.

In this paper we give some lower estimates of the asymptotics of the infimum, inf spec L(ξb), of the spectrum of the operator L(ξb) as the real parameter ξ tends to infinity. We intend particularly to its application to the study of the asymptotics of the function

(1.2)  I(ξ) := E[ exp ( − iξ ∫_{0}^{t} b(X(s, x)) ∘ dX(s, x))|X(t, x) = y],

as ξ tends to infinity, where X(s, x) is the absorbing barrier Brownian motion on a domain D, x, y are fixed points in D, and E[·|·] is the conditional expectation. This is called the stochastic oscillatory integral in Malliavin [11], Ikeda and Manabe [7] and so on. The

Typeset by AMS-TEX

∗ This work was supported by Grant-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan

connection between the operator L(ξb) and the function I(ξ) is given by the Feynman-Kac-Itô formula (see (2.17) below). By this formula and our estimate of inf spec L(ξb), we obtain some upper estimate of the absolute value |I(ξ)| of I(ξ). Accordingly, we obtain some results on the existence and the regularity of the density of the conditional probability with respect to the Lebesgue measure

(1.3)  P ( ∫_{0}^{t} b(X(s, x)) ∘ dX(s, x) ∈ dλ|X(t, x) = y)/dλ.

Moreover we consider the same problem for the operator L^{N}(b) with the Neumann boundary condition: L^{N}(b) is the self-adjoint operator corresponding to the closed extension of the form q(b)(φ) in (1.1) for any φ ∈ C_{0}^{∞}(\overline{D}). For the notation, see Section 4 below. For this operator, we should use the reflecting barrier Brownian motion X^{N}(s, x) instead of X(s, x) in (1.2) and (1.3). For this case, we consider only a half space as the domain D. However in both the Dirichlet and Neumann cases, our results are extended to suitable Riemannian manifolds easily, since we can consider the asymptotics locally by the IMS localization (see Lemma 2.1 below).

In particular, a lower bound of the spectrum for the uniform magnetic field and the Neumann boundary condition is obtained (see Theorem 4.1 below). Accordingly, the transverse analyticity, which was proved for the absorbing barrier Brownian motion in [22], is proved also for the reflecting barrier Brownian motion: the density in (1.3) where X(s, x) is replaced by X^{N}(s, x) is real analytic in λ when db is nondegenerate at everywhere (see Corollary 2 of Theorem 4.2 below).

The idea of considering the asymptotics of the spectrum to investigate the asymptotics of the function I(ξ) in (1.2) is appeared in Malliavin [12], [13] : he gives a lower bound of the asymptotics of the spectrum when the magnetic field db is non degenerate and the configulation space D is replaced by a manifold without boundary. For this case, we have more direct study of the function I(ξ) in (1.2) by Ikeda-Manabe [7] and of the density in (1.3) by Malliavin [14] and Plat [18]. The bound of the spectrum by Malliavin is sharpen and is extended to the operator with the Dirichlet boundary condition by the author [22].

For the case that the magnetic field db degenerates finitely on submanifolds, Montogomery [18] and Helffer-Mohamed [4] recently give the following estimate of the spectrum: if

U := {x ∈ D : db(x) = 0}

is a compact submanifold of D and

C_{1}d(x, U)^{ρ} ≤ ∥db(x)∥ ≤ C_{2}d(x, U)^{ρ}

on a neighborhood of U for some C_{1}, C_{2}, ρ > 0, then

(1.4)  C_{3}ξ^{2/(2+ρ)} ≤ inf specL(ξb) ≤ C_{4}ξ^{2/(2+ρ)}

for any ξ ≥ 1 and some C_{3}, C_{4} > 0, where d(x, U) is the distance of x from U and ∥ · ∥ is a fibre norm on the cotangent bundle (see Remark 2.1 below).

In this paper, we extend the lower estimate in (1.4) to the case that the magnetic field db may degenerate on some finite intersection of compact submanifolds finitely (see Theorem 2.1 below). The main tool is borrowed from Helffer-Mohamed [4]. From this result we will obtain exponential decay of the function I(ξ) in (1.2) and that the density in (1.3) belongs to some Gevrey class (see Corollaries 1 and 2 of Theorem 2.1 below).

On the other hand, we will consider also the case that the magnetic field db degenerates infinitely. For this case, we will use the idea of applying the Malliavin calculus to some problem with infinite degeneracy by Malliavin [10] and Kusuoka and Stroock [8] (see also [23]). In this case we show

lim_{ξ↑∞} \frac{inf specL(ξb)}{log ξ} = ∞,

from which we have

|I(ξ)| ≤ C_{k}ξ^{k}

for any k ∈ ℕ, and the boundedness of all derivatives of the density in (1.3) (see Theorem 3.1, Corollaries 1 and 2 of Theorem 3.1 below).

For the case of the operator L^{N}(b) with the Neumann boundary condition and the refrecting barrier Brownian motion, our basic method is to consider the double of D to reduce to the case without boundary condition. However, for the neccesary extension of L^{N}(b) to the double, the corresponding magnetic field db is not continuous. This is the difficulty and the particular point of the Neumann condition. To overcome this difficulty, we use the IMS localization (Lemma 2.1 below). For the operator L^{N}(b), we discuss only the case that the magnetic field db is nondegenerate. However our results are extended suitably to the case that db may be degenerate.

The organization of this paper is as follows. In Sections 2 and 3, we treat the operator with the Dirichlet boundary condition: in Section 2, we consider finitely degenerate cases and in Section 3, we consider infinitely degenerate cases. In Section 4, we treat the operator with the Neumann boundary condition.

2. The Dirichlet condition (I) finitely degenerate cases Let D be a domain in ℝ^{d} given a metric such that the metric tensors g_{jk}(x), j, k = 1, 2, ⋯, d, are C^{∞}, their derivatives are all bounded and

inf { ∑_{j,k} g_{jk}(x)ξ_{j}ξ_{k} : x ∈ D, ξ = (ξ_{1}, ξ_{2}, ⋯, ξ_{d}) ∈ ℝ^{d}, ∑_{j} ξ_{j}^{2} = 1} > 0.

Let b be an ℝ^{d} valued C^{∞} function on D, which is identified with the 1 form, and C_{0}^{∞}(D^{∘}) be the set of all ℂ-valued C^{∞} functions on the interior of D with compact support. For any φ ∈ C_{0}^{∞}(D^{∘}), we set

(2.1)  q(b)(φ) = \frac{1}{2}∥(id + ext(b))φ∥^{2},

where ∥ · ∥ is the L^{2} norm with respect to the above metric and ext is the exterior multiplication, i.e., ext(b)φ = b ∧ φ. Let Dom(q(b)) be the completion of C_{0}^{∞}(D^{∘}) by the norm \sqrt{q(b)(·) + ∥ · ∥^{2}}. We extend q(b)(·) naturally to Dom(q(b)) so that (q(b),Dom(q(b))) becomes a closed sesquilinear form. Let L(b) be the associated self-adjoint operator: L(b) is the operator with Dirichlet boundary condition. The 2 form db is regarded as the corresponding magnetic field. We will study the asymptotics of inf spec L(ξb) as the real parameter ξ tends to infinity, where spec L(ξb) is the spectral set of the self-adjoint operator L(ξb).

In this section, we consider the case that the magnetic field db has some finite degeneracy. We first introduce the conditions. For each x ∈ D, let ∥db(x)∥_{1} be the trace norm of db(x):

∥db(x)∥_{1} = tr\sqrt{B(x)^{t}B(x)},

where B(x) is the matrix (db(e_{j}(x), e_{k}(x)))_{1≤j,k≤d} and {e_{j}(x)}_{j=1}^{d} is an orthonormal basis of T_{x}D. Let (G) be the following condition:

(G)  sup_{x∈D} \frac{|∂_{ℓ}db_{jk}(x)|}{∥db(x)∥_{1} + 1} < ∞,  for any 1 ≤ j, k, ℓ ≤ d,

where ∂_{j} = ∂/∂x^{j} and db_{jk} = db(∂_{j}, ∂_{k}).

Taking n ∈ ℕ arbitrarily, we assume the following: (D, n) There are functions h_{μ}(s) : ℝ → [0, ∞) and ψ_{μ}(x) : D → ℝ, μ = 1, 2, ⋯, n,

satisfying the following:

(i) h_{μ} is even, non-decreasing on [0, ∞) and h_{μ}^{−1}(0) = {0}; (ii) ψ_{μ} is C^{∞}, the derivatives of ψ_{μ} are all bounded and

inf_{x∈D}(ψ_{μ}(x)^{2} + ∥dψ_{μ}(x)∥^{2}) > 0;

(iii) for any x ∈ D,

∥db(x)∥_{1} ≥ ∏_{μ=1}^{n} h_{μ}(ψ_{μ}(x));

(iv) inf_{x∉K} |ψ_{μ}(x)| > 0 for any μ = 1, 2, ⋯, n and some compact set K in \overline{D}. For \overline{ρ} = (ρ(1), ρ(2), ⋯, ρ(n)) ∈ [0, ∞)^{n}, let (P, \overline{ρ}) be the following condition:

(P, \overline{ρ})  \overline{lim}_{s↓0} \frac{s^{ρ(μ)}}{h_{μ}(s)} < ∞,  for any μ = 1, 2, ⋯, n.

Let (I) be the following condition:

(I) For each a ∈ N := ⋃_{μ=1}^{n} ψ_{μ}^{−1}(0), the system of the forms {dψ_{μ}(a)}_{μ∈Λ(a)} is linear independent, where Λ(a) = {μ : ψ_{μ}(a) = 0}.

Then the main theorem is the following: Theorem 2.1. We assume (G), (D, n) and (P, \overline{ρ}) for some \overline{ρ} ∈ [0, ∞)^{n}. Moreover we

assume (I) or \overline{ρ} ∈ (2ℤ_{+})^{n}. Then we have

(2.2)  \underline{lim}_{ξ↑∞} \frac{inf specL(ξb)}{ξ^{2/(|\overline{ρ}|+2)}} > 0,

where |\overline{ρ}| = ∑_{μ=1}^{n} ρ(μ).

Remark 2.1. When n = \overline{ρ} = 1 and d = 2, Theorem 2.1 is a part of the results of Montgomery [17]. When n = 1 and \overline{ρ}, d are general, Theorem 2.1 is a part of the results of Helffer-Mohamed [4]. In the case of n = 1, the condition (I) holds automatically.

We give simple examples:

Example 2.1. On D = {x ∈ ℝ^{2} : |x| < 1} with the Euclidean metric, we define b by b_{1} = (x^{1})^{ρ(1)}(x^{2})^{ρ(2)+1}, b_{2} = 0 and \overline{ρ} = (ρ(1), ρ(2)) ∈ (ℤ_{+})^{2}. Then we have db_{12} = −(ρ(2)+1)(x^{1})^{ρ(1)}(x^{2})^{ρ(2)}. This satisfies the conditions (D, 2), (P, \overline{ρ}) and (I) with ψ_{μ} = x^{μ} and h_{μ} = |s|^{ρ(μ)}. Thus we obtain the result of Theorem 2.1.

Example 2.2. On the same domain as that of Example 2.1, we take b so that db_{12} = (x^{1})^{ρ(1)}(ψ_{2}(x))^{ρ(2)} for some \overline{ρ} = (ρ(1), ρ(2)) ∈ (2ℤ_{+})^{2}, where ψ_{2} = (x^{2})^{2} + x^{1}. This does not satisfy the condition (I), since dψ_{1} = dψ_{2} at x = 0. However this satisfies the conditions (D, 2), (P, \overline{ρ}) and \overline{ρ} ∈ (2ℤ_{+})^{n}. Thus we still obtain the result of Theorem 2.1.

For the proof, we prepare several lemmas. The following lemma is one of the fundamental tools to consider the lower bound of the spectrum:

Lemma 2.1 (IMS localization). (i) Let {χ_{m}}_{m} ⊂ C^{∞}(D) satisfying ∑_{m} χ_{m}^{2} ≡ 1. Then, for any φ ∈ C_{0}^{∞}(D^{∘}), we have

(2.3)  L(b)φ = ∑_{m} χ_{m}L(b)χ_{m}φ − ∑_{m} \frac{∥dχ_{m}∥^{2}}{2} φ.

(ii) Let {χ_{m}}_{m} ⊂ H_{loc}^{1,2}(D) satisfying ∑_{m} χ_{m}^{2} ≡ 1, and b be an ℝ^{d} valued continuous function on D. Then, for any φ ∈ C_{0}^{∞}(D^{∘}), we have χ_{m}φ ∈ Dom(q(b)) and

(2.4)  q(b)(φ) = ∑_{m} q(b)(χ_{m}φ) − (∑_{m} \frac{∥dχ_{m}∥^{2}}{2} φ, φ).

In (ii), H_{loc}^{1,2}(D) = {φ : D → ℂ, φ ↾_{D′}∈ H^{1,2}(D′) for any relatively compact D′ ⊂ D}. For the definition of H^{1,2}(D′), see [1]. We use (i) in this section and use (ii) in later sections.

Proof. In the proof, we may assume that the domain D is relatively compact. For the proof of (i), see [21] for example. For that of (ii), we use an approximation arguement: for each m, there is a sequence {χ_{m}^{n}}_{n} ⊂ C^{∞}(D) such that χ_{m}^{n} → χ_{m}, dχ_{m}^{n} → dχ_{m} in L^{2}. Then for any φ ∈ C_{0}^{∞}(D^{∘}), we can show that χ_{m}^{n}φ → χ_{m}φ in L^{2} and {χ_{m}^{n}φ}_{n} is a Cauchy sequence in Dom(q(b)). Therefore we have χ_{m}φ ∈Dom(q(b)) and χ_{m}^{n}φ → χ_{m}φ in Dom(q(b)). For each m and n, we have

q(b)(χ_{m}^{n}φ) = \frac{1}{2}∥χ_{m}^{n}(id + ext(b))φ∥^{2} + Re(iφχ_{m}^{n}dχ_{m}^{n}, (id + ext(b))φ) + \frac{1}{2}∥φdχ_{m}^{n}∥^{2}.

By taking the limit in n and then taking the sum in m, we obtain (2.4).

The following lemma is due to Helffer-Nourrigat [3], [5], [16]:

Lemma 2.2. Let b_{j}^{0}(x), j = 1, 2, ⋯, d, and V_{ℓ}, ℓ = 1, 2, ⋯, m, be real polynomials on ℝ^{d} of degree ≤ r. Let

(2.5)  Ψ(x) = ∑_{1≤j<k≤d} ∑_{\begin{matrix}α∈ℤ_{+}^{d} \\ |α|≤r−1 \end{matrix}} |∂^{α}db_{jk}^{0}(x)|^{1/(|α|+2)} + ∑_{ℓ=1}^{m} ∑_{\begin{matrix}β∈ℤ_{+}^{d} \\ |β|≤r \end{matrix}} |∂^{β}V_{ℓ}(x)|^{1/(|β|+1)}.

Then there exists a constant C depending only on d, m and r such that

(2.6)  ∫_{ℝ^{d}}(∑_{j=1}^{d} |(i∂_{j} + b_{j}^{0})φ|^{2} + ∑_{ℓ=1}^{m} |V_{ℓ}φ|^{2})dx ≥ C ∫_{ℝ^{d}} |Ψφ|^{2}dx

for any φ ∈ C_{0}^{∞}(ℝ^{d}).

In [4], the following is deduced from Lemmas 2.1 and 2.2:

Lemma 2.3. Under the assumption (G), for any ε ∈ (0, 1), there exist constants C_{ε}, C′_{ε} > 0 such that

(2.7)  q(ξb)(φ) ≥ C_{ε} ∫_{D} ξ∥db∥_{1}|φ|^{2}dvol − C′_{ε}ξ^{ε}∥φ∥^{2}

for any φ ∈ C_{0}^{∞}(D^{∘}) and ξ > 1.

Proof. Let χ^{(1)} and χ^{(2)} be smooth functions on \overline{D} such that (χ^{(1)})^{2} + (χ^{(2)})^{2} ≡ 1, supp χ^{(1)} is compact, supp χ^{(2)} ⊂ K^{c} and dχ^{(2)} is bounded, where K is the compact set in the condition (D, n) (iv). By Lemma 2.1, we have

q(ξb)(φ) = ∑_{ν=1}^{2}\{q(ξb)(χ^{(ν)}φ) − \frac{1}{2}∥φdχ^{(ν)}∥^{2}\}.

Then Theorem 4.5 of [4] leads to the following:

(2.8)  q(ξb)(χ^{(1)}φ) ≥ C_{ε} ∫_{D} ξ∥db∥_{1}|χ^{(1)}φ|^{2}dvol − C′_{ε}ξ^{ε}∥χ^{(1)}φ∥^{2}.

Since inf_{x∉K} ∥db(x)∥_{1} > 0, Theorem 3.1 of [4] leads to the following:

q(ξb)(χ^{(2)}φ) ≥ C ∫_{D} ξ∥db∥_{1}|χ^{(2)}φ|^{2}dvol

for some C > 0. By these, we obtain (2.7).

The following lemma is well known (see e.g., [6], [20]):

Lemma 2.4 (Diamagnetic estimate). For any ℝ^{d} valued continuous function b and any real continuous function V on D which is bounded below, we have

(2.9)  inf spec(L(b) + V) ≥ inf spec(L(0) + V),

where L(b) + V is the Friedrichs extension of the corresponding operator on C_{0}^{∞}(D^{∘}). We now prove Theorem 2.1.

Proof of Theorem 2.1.

We take 0 < ε < 2/(2 + |\overline{ρ}|) and fix it. By Lemmas 2.3 and 2.4, and the assumptions of this theorem, we have

(2.10)  inf specL(ξb) ≥ C_{1} inf spec(−Δ + ξ ∏_{μ=1}^{n} |ψ_{μ}|^{ρ(μ)}) − C_{2}ξ^{ε}

for some constants C_{1} and C_{2}, where Δ is the Laplace-Beltrami operator with Dirichlet boundary condition.

We first assume the condition (I). Then, for each a ∈ N, there exists a coordinate neighborhood (ν(a), (y^{1}, y^{2}, ⋯, y^{d})) of D around a such that ψ_{μ}(y) = y^{n(μ,a)} for some 1 ≤ n(μ, a) ≤ d and any μ ∈ Λ(a) and ψ_{μ}(y) ≠ 0 for any y ∈ ν(a) and any μ ∈ Λ(a)^{c}. Let ν_{0}(a) be a neighborhood of a so that \overline{ν_{0}(a)} ⊂ ν(a). Since N is compact, we can take a finite set N_{0} ⊂ N so that

(2.11)  ⋃_{a∈N_{0}} ν_{0}(a) ⊃ N .

By using Lemma 2.1, we have

(2.12)

inf spec(−Δ + ξ ∏_{μ=1}^{n} |ψ_{μ}|^{ρ(μ)})

≥ inf_{a∈N_{0}} inf spec(−Δ + ξ ∏_{μ=1}^{n} |ψ_{μ}|^{ρ(μ)})_{ν(a)}

∧ inf spec(−Δ + ξ ∏_{μ=1}^{n} |ψ_{μ}|^{ρ(μ)})_{ν} − C_{3}

for some C_{3} > 0, where (−Δ + ξ ∏_{μ=1}^{n} |ψ_{μ}|^{ρ(μ)})_{ν(a)} is the Friedrichs extension of the restriction of the corresponding operator to C_{0}^{∞}(ν(a)) and

(2.13)  ν = (\overline{⋃_{a∈N_{0}} ν_{0}(a)})^{c} .

We easily see that

(2.14)  inf spec(−Δ + ξ ∏_{μ=1}^{n} |ψ_{μ}|^{ρ(μ)})_{ν} ≥ C_{4}ξ

for some C_{4} > 0. For each a ∈ N_{0}, by using the coordinate and the ellipticity of the metric, we have

(2.15)

inf spec(−Δ + ξ ∏_{μ=1}^{n} |ψ_{μ}|^{ρ(μ)})_{ν(a)}

≥C_{5} inf { ∫_{ℝ^{d}} ( ∑_{j=1}^{d} |∂_{j}φ(y)|^{2} + ξ ∏_{μ∈Λ(a)} |y^{n(μ,a)}|^{ρ(μ)}|φ(y)|^{2})dy
|φ ∈ C_{0}^{∞}(ℝ^{d}), ∫ |φ(y)|^{2}dy = 1}

for some C_{5}. By making the change of vasriables (y → ξ^{−1/(2+ρ[a])}y where ρ[a] = ∑_{μ∈Λ(a)} ρ(μ)), we can rewrite the right hand side of (2.15) as

\begin{matrix}C_{5}ξ^{2/(2+ρ[a])} inf \{ ∫_{ℝ^{d}} ( ∑_{j=1}^{d} |∂_{j}φ(y)|^{2} + ∏_{μ∈Λ(a)} |y^{n(μ,a)}|^{ρ(μ)}|φ(y)|^{2})dy \\ : φ ∈ C_{0}^{∞}(ℝ^{d}), ∫ |φ(y)|^{2}dy = 1\}. \end{matrix}

Since this is positive, we obtain (2.2).

We next assume \overline{ρ} ∈ (2ℤ_{+})^{n} for any μ = 1, 2, ⋯, n. For each a ∈ N, let (ν(a), (y^{1}, y^{2}, ⋯, y^{d})) be a coordinate neighborhood of D around a, such that

∑_{|α|=ρ[a]/2+1} |∂^{α} ∏_{μ∈Λ(a)} ψ_{μ}(y)^{ρ(μ)/2}| ≤ C_{a} ∑_{|α|≤ρ[a]/2} |∂^{α} ∏_{μ∈Λ(a)} ψ_{μ}(y)^{ρ(μ)/2}| ≠ 0

for any y ∈ ν(a) and some C_{a} > 0 depending only on a, and ψ_{μ}(y) ≠ 0 for any y ∈ ν(a) and μ ∈ Λ(a)^{c}. Let ν_{0}(a) be a neighborhood of a so that \overline{ν_{0}(a)} ⊂ ν(a), N_{0} ⊂ N be a finite set satisfying (2.11) and ν be the set as in (2.13). Then we have (2.12) and (2.14). For each a ∈ N_{0}, by using the ellipticity of the metric, we have

(2.16)

inf spec(−Δ + ξ ∏_{μ=1}^{n} |ψ_{μ}|^{ρ(μ)})_{ν(a)}

≥C_{6} inf { ∫_{ℝ^{d}} ( ∑_{j=1}^{d} |∂_{j}φ(y)|^{2} + ξ ∏_{μ∈Λ(a)} |ψ_{μ}(y)|^{ρ(μ)}|φ(y)|^{2})dy
: φ ∈ C_{0}^{∞}(ν(a)), ∫ |φ(y)|^{2}dy = 1}

By Theorem (1.1) of [16], this is dominated from below by

C_{7} inf { ∫_{ℝ^{d}} | ∑_{k=0}^{ρ[a]/2} (1 + ∑_{|α|=k} |ξ^{1/2}∂^{α} ∏_{μ∈Λ(a)} ψ_{μ}^{ρ(μ)/2}|)^{1/(k+1)}φ|^{2}dy

: φ ∈ C_{0}^{∞}(ν(a)), ∫ |φ(y)|^{2}dy = 1} − C_{8}

≥ C_{9}ξ^{1/(ρ[a]/2+1)} − C_{10}.

By all these, we obtain (2.2).

We next apply Theorem 2.1 to the asymptotics of heat kernels as in Malliavin [12], [13] and Ueki [22]. In the rest of this section, for simplicity, we assume

(P G)  sup_{x∈D} \frac{|∇^{2}b(x)|}{(1 + |x|)^{k}} < ∞  for some k ∈ ℕ.

Let e^{−tL(ξb)}(x, y), (t, x, y) ∈ (0, ∞)×D ×D be the integral kernel of the semigroup e^{−tL(ξb)} generated by the operator L(ξb), which is called the heat kernel: for any φ ∈ L^{2}(D),

(e^{−tL(ξb)}φ)(x) =< e^{−tL(ξb)}(x, ·), φ >,

where < ·, · > is the inner product in L^{2} which is complex linear in the both variables. The heat kernel has the following representation: let X(t, x), t > 0, x ∈ D be the absorbing barrier Brownian motion on D starting at x: X(t, x) is a diffusion process generated by Δ/2, where Δ is the Laplace-Beltrami operator with Dirichlet boundary condition. Let ∫_{0}^{t} b(X(s, x)) ∘ dX(s, x) be the stochastic line integral (see e.g. [9] §VI-6) and e^{tΔ/2}(x, y) be the integral kernel of the semigroup e^{tΔ/2}. Then we have

(2.17)  \begin{matrix}e^{−tL(ξb)}(x, y) \\ = & E[ exp ( − iξ ∫_{0}^{t} b(X(s, x)) ∘ dX(s, x))|X(t, x) = y]e^{tΔ/2}(x, y), \end{matrix}

where E[·|X(t, x) = y] is the conditional expectation with respect to the Brownian motion X(·, x) conditioned by X(t, x) = y.

From Theorem 2.1, we obtain the following:

Corollary 1. Under the assumption of Theorem 2.1 and (P G), it holds that

(2.18)  \overline{lim}_{ξ→±∞} sup_{(t,x,y)∈[t_{0},∞)×D×D} \frac{1}{t|ξ|^{2/(|\overline{ρ}|+2)}} log |e^{−tL(ξb)}(x, y)| < 0

for any t_{0} > 0.

Corollary 1 is an upper estimate of asymptotics of a stochastic oscillatory integral for which the phase function is a stochastic line integral (see e.g. [7] and [11]). Moreover, since e^{−tL(ξb)}(x, y) is the Fourier transform of the distribution of ∫_{0}^{t} b(X(s, x)) ∘ dX(s, x) under the conditional probability P (·|X(t, x) = y), we have the following:

Corollary 2. Under the assumption of Theorem 2.1 and (P G), for any t > 0, x, y ∈ D, the distribution of ∫_{0}^{t} b(X(s, x))∘dX(s, x) under the conditional probability P (·|X(t, x) = y) has a smooth density function belonging to the Gevrey class of order (|\overline{ρ}| + 2)/2: there is a function p(t, x, y; λ), t > 0, x, y ∈ D, λ ∈ ℝ such that

(2.19)  P ( ∫_{0}^{t} b(X(s, x)) ∘ dX(s, x) ∈ A|X(t, x) = y) = ∫_{A} p(t, x, y; λ)dλ

for any Borel set A in ℝ, p(t, x, y; λ) is smooth in λ, and

(2.20)  |\frac{∂^{k}}{∂λ^{k}} p(t, x, y; λ)| ≤ \frac{1}{παC^{(k+1)/α}} Γ (\frac{n + 1}{α})

for any λ ∈ ℝ, k ∈ ℤ_{+} and some C > 0, where α = 2/(|\overline{ρ}| + 2) and Γ(·) is the gamma function.

3. The Dirichlet condition (II) infinitely degenerate cases In this section, we consider the case that the magnetic field db has some infinite degeneracy: for ρ > 0, let (E, ρ) be the following condition:

(E, ρ)  lim_{s↓0} s^{ρ} log \frac{1}{h_{μ}(s)} = 0 for any μ = 1, 2, ⋯, n.

Let (H) be the following condition:

(H)  Δψ_{μ} = 0 and dΔψ_{μ} = 0 on ψ_{μ}^{−1}(0) for any μ = 1, 2, ⋯, n,

where Δ is the Laplace-Beltrami operator.

Then the main theorem is the following:

Theorem 3.1. We assume (D, n) and (E, ρ) for some ρ < 2. Moreover we assume (H) or ρ < 1. Then we have

(3.1)  lim_{ξ↑∞} \frac{inf specL(ξb)}{log ξ} = ∞.

We give simple examples:

Example 3.1. On D = {x ∈ ℝ^{2} : |x| < 1} with the Euclidean metric, we define b by

b_{1} = exp(−|x^{1}|^{−ρ}) ∫_{0}^{x^{2}} exp(−|t|^{−ρ})dt,

b_{2} = 0 and ρ ∈ (0, 2). Then we have db_{12} = − exp(−|x^{1}|^{−ρ} − |x^{2}|^{−ρ}). This satisfies the conditions (D, 2), (H) and (E, ρ′) for any ρ′ ∈ (0, 2) with ψ_{μ} = x^{μ} and h_{μ} = exp(−|s|^{−ρ}). Thus we obtain the result of Theorem 3.1.

Example 3.2. On D = {x ∈ ℝ^{2} : |x − e| < 1} with the Euclidean metric, we take b so that db_{12} = exp(−|ψ_{1}|^{−ρ}) for some ρ ∈ (0, 2), where ψ_{1} = log |x| − 1. This db vanishes on the arc {|x| = e}. However, since Δψ_{1} = 0, the condition (H) is satisfied. Moreover the conditions (D, 1) and (E, ρ′) for any ρ′ ∈ (0, 2) are satisfied. Thus we still obtain the result of Theorem 3.1.

For the proof, we prepare several lemmas. We first show the following:

Lemma 3.1. Under the assumption (D, n), there exist constants C_{1} and C_{2} independent of ξ such that

(3.2)  q(ξb)(φ) ≥ C_{1} ∫_{ℝ^{d}} ξ(∏_{μ=1}^{n} |ψ_{μ}|h_{μ} ∘ ψ_{μ})|φ|^{2}dx − C_{2}∥φ∥^{2}

for any φ ∈ C_{0}^{∞}(D^{∘}) and ξ ≥ 0.

Proof. For each a ∈ D, there exist a neighborhood ν(a) of a and an orthonormal frame e_{1}, e_{2}, ⋯, e_{d} on ν(a) such that

\frac{∥db(x)∥_{1}}{4} = |B(x)|,

where

B(x) = \frac{1}{2} ∑_{σ=1}^{[d/2]} db(e_{2σ−1}(x), e_{2σ}(x)).

By Theorem 2.1 of [22], we have

q(ξb)(φ) ≥ ((±ξB − R)φ, φ)

for any φ ∈ C_{0}^{∞}(ν(a)), where R is some bounded function independent of ξ.

For each σ = (σ(1), σ(2), ⋯, σ(n)) ∈ {0, 1}^{n}, let {ν(a, σ, ι) : ι = 1, 2, ⋯, n(a, σ)} be the arcwise connected components of the set {x ∈ ν(a) : (−1)^{σ(μ)}ψ_{μ}(x) > 0 for μ = 1, 2, ⋯, n}. Then for each σ and ι, there exists ε(a, σ, ι) ∈ {0, 1} such that ν(a, σ, ι) ⊂ {x ∈ ν(a) : (−1)^{ε(a,σ,ι)}B(x) > 0}. We now introduce a function η(x) on ν(a) by

η(x) = \{ \begin{matrix}(−1)^{ε(a,σ,ι)}| ∏_{μ=1}^{n} ψ_{μ}(x)| & if x ∈ ν(a, σ, ι), \\ 0 & otherwise. \end{matrix}

Let ζ_{±} be smooth functions on ℝ satisfying ζ_{±}(s) = \sqrt{(1 ± s)/2} for s ∈ [−1/2, 1/2], 0 ≤ ζ_{±} ≤ 1 and ζ_{+}^{2} + ζ_{−}^{2} ≡ 1. We set χ_{±}(x) := ζ_{±}(η(x)). These belong to H_{loc}^{1,2}(ν(a)). Therefore by Lemma 2.1 (ii), we have

q(ξb)(φ) =q(ξb)(χ_{+}φ) + q(ξb)(χ_{−}φ)

− \frac{1}{2}((|dχ_{+}|^{2} + |dχ_{−}|^{2})φ, φ)

≥ξ((χ_{+}^{2} − χ_{−}^{2})Bφ, φ) − (R_{1}φ, φ)

for any φ ∈ C_{0}^{∞}(ν(a)) and some bounded function R_{1}. It is easy to see that

(χ_{+}^{2} − χ_{−}^{2})B ≥ C ∏_{μ=1}^{n} |ψ_{μ}|h_{μ} ∘ ψ_{μ}

for some C > 0. By the IMS localization (Lemma 2.1), we obtain (3.2).

Lemma 3.2. Let L(ξ), ξ > 0 be a family of self-adjoint operators on a Hilbert space. Then the following (i) and (ii) are equivalent:

(i)  lim_{ξ↑∞} \frac{inf specL(ξ)}{log ξ} = ∞.
(ii)  lim_{ξ↑∞} ξ^{k}∥e^{−tL(ξ)}∥_{o} = 0 for any k ∈ ℕ and t > 0,

where e^{−tL(ξ)}, t > 0, is the semigroup generated by L(ξ) and ∥ · ∥_{o} is the operator norm on the Hilbert space.

We can prove this easily by using the relation

(3.3)  ∥e^{−tL(ξ)}∥_{o} = exp(−t inf specL(ξ)).

We omit the detail.

Lemma 3.3. Let V be a nonnegative continuous function on D. Then, for any k ∈ ℕ, there is a constant C_{k} > 0 such that

(3.4)  \begin{matrix}ξ^{k}∥ exp \{ − t( − \frac{Δ}{2} + ξV )\}∥_{o} \\ ≤ & C_{k} sup_{x∈D} E[( ∫_{0}^{t} V (X(s, x))ds)^{−2k} : τ > t], \end{matrix}

where τ = inf{t > 0 : X(t, x) ∈ ∂D}.

Proof. For any φ ∈ C_{0}^{∞}(D^{∘}), we have

|ξ^{k}[ exp \{ − t( − \frac{Δ}{2} + ξV )\}φ](x)|

=|E[ξ^{k} exp ( − ξ ∫_{0}^{t} V (X(s, x))ds)φ(X(t, x)) : τ > t]|
≤C_{k}E[( ∫_{0}^{t} V (X(s, x))ds)^{−2k} : τ > t]^{1/2}E[|φ(X(t, x))|^{2} : τ > t]^{1/2}

and

ξ^{k}∥ exp \{ − t( − \frac{Δ}{2} + ξV )\}∥_{o}

≤C_{k} sup_{x∈D} E[( ∫_{0}^{t} V (X(s, x))ds)^{−2k} : τ > t]^{1/2}∥e^{tΔ/2}∥_{(1)}^{1/2},

where ∥ · ∥_{(1)} is the operator norm on the L^{1} space. Since

∥e^{tΔ/2}∥_{(1)} ≤ 1,

we have (3.4)

Lemma 3.4. Under the assumption (H), there exist constants C_{1} and C_{2} such that |ψ_{μ}(x)| ≤ C_{1} implies |Δψ_{μ}(x)| ≤ C_{2}|ψ_{μ}(x)|^{2}. Proof. By virtue of the condition (D, n) (ii), for each a ∈ ψ_{μ}^{−1}(0), there exists a coordinate neighborhood (ν(a), (y^{1}, y^{2}, ⋯, y^{d})) of D around a such that y^{1} = ψ_{μ}(y) and (y^{2}, ⋯, y^{d}) is a coordinate of the submanifold ψ_{μ}^{−1}(0). By the Taylor theorem, we estimate as

|Δψ_{μ}(y^{1}, y^{2}, ⋯, y^{d}) − Δψ_{μ}(0, y^{2}, ⋯, y^{d})| ≤ C_{a}|y^{1}|^{2}

for some constant C_{a} depending on a. This implies

|Δψ_{μ}(y)| ≤ C_{a}|ψ_{μ}(y)|^{2}

on ν(a). Since ψ_{μ}^{−1}(0) is compact, we can take constants C_{1} and C_{2} satisfying the statement of this lemma.

The following is the key estimate to prove Theorem 3.1 (cf. Kusuoka-Stroock [8] and Proposition 3.2 in [23]).

Proposition 3.1. Under the assumptions of Theorem 3.1, we have

(3.5)  sup_{x∈D} E[( ∫_{0}^{t} ∏_{μ=1}^{n} h_{μ}(ψ_{μ}(X(s, x)))ds)^{−p} : τ > t] < ∞

for any p ≥ 1.

Proof. We give the proof only under the condition (H). The proof under the condition ρ < 1 can be given by referring the following proof and the proof of Proposition 3.3 (ii) in [23]. Let (w^{1}(t), w^{2}(t), ⋯, w^{d}(t)) be the d-dimensional Brownian motion, σ(x) = (σ^{jk}(x))_{1≤j,k≤d} be the square root of (g^{jk}(x))_{1≤j,k≤d} which is the inverse of the metric tensor (g_{jk}(x))_{1≤j,k≤d}, β(x) = (β^{j}(x))_{1≤j≤d} = ((1/2g) ∑_{k=1}^{d} ∂(gg^{jk})/∂x^{k})_{1≤j≤d} and g(x) = \sqrt{det(g_{jk}(x))}. Then we can regard the Brownian motion X(t, x) as the solution of the following stochastic differential equation:

X(0) = x,
dX^{j}(t) = ∑_{k=1}^{d} σ^{jk}(X(t))dw^{k}(t) + β^{j}(X(t))dt for τ > t,
X(t) is killed at t = τ,

where τ = inf{t > 0 : X(t) ∈ ∂D}. In particular we have

(3.6)  \begin{matrix}ψ_{μ}(X(t ∧ τ, x, w)) − ψ_{μ}(x) \\ = & ∑_{j,k=1}^{d} ∫_{0}^{t∧τ} (σ^{jk}∂_{k}ψ_{μ})(X(s, x, w))dw^{j}(s) + ∫_{0}^{t∧τ} \frac{1}{2}(Δψ_{μ})(X(s, x, w))ds \end{matrix}

for any μ = 1, 2, ⋯, n. As in [23], we estimate the probability

P (∫_{0}^{t} ∏_{μ=1}^{n} h_{μ}(ψ_{μ}(X(s, x)))ds < \frac{1}{R}, τ > t)

=P (∫_{0}^{t} ∏_{μ=1}^{n} h_{μ}(ψ_{μ}(X(s, x)))ds < \frac{1}{R}, τ_{λ}(x) ≤ \frac{t}{2}, τ > t)

+ P (∫_{0}^{t} ∏_{μ=1}^{n} h_{μ}(ψ_{μ}(X(s, x)))ds < \frac{1}{R}, τ_{λ}(x) > \frac{t}{2}, τ > t)

=:I_{1} + I_{2},

where λ ∈ (0, 1] is an additional variable and τ_{λ}(x) = inf{s ≥ 0 : |ψ_{μ}(X(s, x))| ≥ λ for any μ = 1, 2, ⋯, n}.

By using the strong Markov property as in [23], we have

I_{1} ≤ sup \{P (\frac{t}{2} ∧ σ_{λ/2}(x) < \frac{1}{R ∏_{μ=1}^{n} h_{μ}(\frac{λ}{2})})
: x ∈ D, |ψ_{μ}(x)| > λ for any μ = 1, 2, ⋯, n},

where σ_{λ/2}(x) = inf{s ≥ 0 : |ψ_{μ}(X(s, x))| < λ/2 for some μ ∈ {1, 2, ⋯, n}}. If R ∏_{μ=1}^{n} h_{μ}(λ/2) ≥ 2/t, then we have

I_{1} ≤ sup_{x∈D} P ( sup ; 0<s<1/\{R ∏_{μ=1}^{n} h_{μ}(λ/2)\} |ψ_{ν}(X(s, x)) − ψ_{ν}(x)| > \frac{λ}{2}
for some ν ∈ {1, 2, ⋯, n})
≤ ∑_{ν=1}^{n} sup_{x∈D} P ( sup ; 0<s<1/\{R ∏_{μ=1}^{n} h_{μ}(λ/2)\} | ∑_{j,k=1}^{d} ∫_{0}^{s}(σ^{jk}∂_{k}ψ_{μ})(X(u, x, w))dw^{j}(u)| > \frac{λ}{4},

τ > \frac{1}{R ∏_{μ=1}^{n} h_{μ}(λ/2)})

+ ∑_{ν=1}^{n} sup_{x∈D} P ( sup ; 0<s<1/\{R ∏_{μ=1}^{n} h_{μ}(λ/2)\} | ∫_{0}^{s} \frac{Δψ_{ν}}{2} (X(u, x))du| > \frac{λ}{4},

τ > \frac{1}{R ∏_{μ=1}^{n} h_{μ}(λ/2)})

=:I_{11} + I_{12}.

By Theorem II-7.2’ of [9], for each ν = 1, 2, ⋯, n and each x ∈ D, there exists a 1dimensional Brownian motion B(t) such that

(3.7)  \begin{matrix}∑_{j,k=1}^{d} ∫_{0}^{s}(σ^{jk}∂_{k}ψ_{ν})(X(u, x, w))dw^{j}(u) \\ = & B (∫_{0}^{s} \frac{1}{2}∥dψ_{ν}(X(u, x, w))∥^{2}du) . \end{matrix}

Since ∥dψ_{ν}∥ is bounded, we have

\begin{matrix}I_{11} ≤ & nP ( sup ; 0<s<C_{1}/\{R ∏_{μ=1}^{n} h_{μ}(λ/2)\} |B(s)| > \frac{λ}{4}) \\ ≤ & C_{2} exp ( − C_{3}λ^{2}R ∏_{μ=1}^{n} h_{μ}(\frac{λ}{2})) \end{matrix}

from (8.29) in [8]. On the other hand, since Δψ_{ν} is bounded, there is some C_{4} > 0 such that I_{12} = 0 if λR ∏_{μ=1}^{n} h_{μ}(λ/2) ≥ C_{4}.

On the other hand, for each ν = 1, 2, ⋯, n, we set

σ_{λ,ν}^{1}(x) := inf{s ≥ 0 : |ψ_{ν}(X(s, x))| < λ},
τ_{λ,ν}^{1}(x) := inf{s ≥ σ_{λ,ν}^{1}(x) : |ψ_{ν}(X(s, x))| > λ},
⋮
σ_{λ,ν}^{n}(x) := inf{s ≥ τ_{λ,ν}^{n−1}(x) : |ψ_{ν}(X(s, x))| < λ},
τ_{λ,ν}^{n}(x) := inf{s ≥ σ_{λ,ν}^{n}(x) : |ψ_{ν}(X(s, x))| > λ},
⋮

Then we have

I_{2} ≤ P ( ∑_{m=1}^{∞} |(τ_{λ,ν}^{m}(x) ∧ \frac{t}{2}) − (σ_{λ,ν}^{m}(x) ∧ \frac{t}{2})| > \frac{t}{2n} for some ν ∈ \{1, 2, ⋯, n\})

and, for each ν = 1, 2, ⋯, n,

(3.8)  sup_{0≤s≤t/2} | ∑_{m=1}^{∞}(ψ_{ν}(X(τ_{λ,ν}^{m}(x) ∧ s)) − ψ_{ν}(X(σ_{λ,ν}^{m}(x) ∧ s)))| ≤ 2λ.

By using (3.6), we have either

sup_{0≤s≤t/2} | ∑_{m=1}^{∞} ∑_{j,k=1}^{d} ∫_{σ_{λ,ν}^{m}(x)∧s}^{τ_{λ,ν}^{m}(x)∧s}(σ^{jk}∂_{k}ψ_{ν})(X(u, x, w))dw^{j}(u)| ≤ 4λ

or

sup_{0≤s≤t/2} | ∑_{m=1}^{∞} ∫_{σ_{λ,ν}^{m}(x)∧s}^{τ_{λ,ν}^{m}(x)∧s} \frac{Δψ_{ν}}{2} (X(u, x, w))du| ≥ 2λ

from (3.8). Accordingly, we estimate as

I_{2} ≤ ∑_{ν=1}^{n} P ( ∑_{m=1}^{∞} |(τ_{λ,ν}^{m}(x) ∧ \frac{t}{2}) − (σ_{λ,ν}^{m}(x) ∧ \frac{t}{2})| > \frac{t}{2n},

sup_{0≤s≤t/2} | ∑_{m=1}^{∞} ∑_{j,k=1}^{d} ∫_{σ_{λ,ν}^{m}(x)∧s}^{τ_{λ,ν}^{m}(x)∧s}(σ^{jk}∂_{k}ψ_{ν})(X(u, x, w))dw^{j}(u)| ≤ 4λ, τ > \frac{t}{2})

+ ∑_{ν=1}^{n} P ( sup_{0≤s≤t/2} | ∑_{m=1}^{∞} ∫_{σ_{λ,ν}^{m}(x)∧s}^{τ_{λ,ν}^{m}(x)∧s} \frac{Δψ_{ν}}{2} (X(u, x, w))du| ≥ 2λ, τ > \frac{t}{2})

=:I_{21} + I_{22}.

As in (3.7), there exists a 1-dimensional Brownian motion B(t) such that

\begin{matrix}∑_{m=1}^{∞} ∑_{j,k=1}^{d} ∫_{σ_{λ,ν}^{m}(x)∧s}^{τ_{λ,ν}^{m}(x)∧s}(σ^{jk}∂_{k}ψ_{ν})(X(u, x, w))dw^{j}(u) \\ = & B ( ∑_{m=1}^{∞} ∫_{σ_{λ,ν}^{m}(x)∧s}^{τ_{λ,ν}^{m}(x)∧s} ∥dψ_{ν}(X(u.x, w))∥^{2}du) . \end{matrix}

By the condition (D, n) (ii), we have

inf{∥dψ_{ν}(x)∥ : x ∈ D, |ψ_{ν}(x)| ≤ λ} ≥ C − C′λ^{2}

for some C, C′ > 0. Therefore, from (8.27) in [8], we obtain

I_{21} ≤ C′′ exp (−\frac{t(C − C′λ^{2})}{λ^{2}}) .

On the other hand, by Lemma 3.4, we see that

I_{22} = 0  if  λ < C (1 ∧ \frac{1}{t})

for some C > 0.

As in [23], we now note the condition (E, ρ). For an arbitrary fixed 0 < ε < 1/n, we take λ_{ε} > 0 such that λ^{ρ} log(1/h_{μ}(λ)) < ε/2 for any 0 < λ < λ_{ε} and any μ = 1, 2, ⋯, n. If we set λ = 2(log R)^{−1/ρ}, then we see that λ^{2}R ∏_{μ=1}^{n} h_{μ}(λ/2) > R^{1/2} and t(C − C′λ^{2})/λ^{2} > C′′t(log R)^{2/ρ} for large enough R. Thus there are C_{1}, C_{2}, ⋯, C_{6} > 0 such that

P (∫_{0}^{t} ∏_{μ=1}^{n} h_{μ}(ψ_{μ}(X(s, x))ds < \frac{1}{R}, τ > t)

≤C_{1} exp(−C_{2}R^{1/2}) + C_{3} exp(−C_{4}t(log R)^{2/ρ})

for any R ≥ exp(C_{5}t^{ρ}) ∨ (2/t)^{2} ∨ C_{6}. Then as in (3.49) of [23], we obtain (3.5) under the condition (E, ρ) for some ρ < 2.

Proof of Theorem 3.1. By Lemma 3.1 and the diamagnetic estimate (Lemma 2.4), we have

inf specL(ξb) ≥ C_{1} inf spec(−Δ + ξ ∏_{μ=1}^{n} |ψ_{μ}|h_{μ} ∘ ψ_{μ}) − C_{2}

for some C_{1}, C_{2} > 0. Then, by Lemmas 3.2 and 3.3, it is enough to show that

sup_{x∈D} E[( ∫_{0}^{t} ∏_{μ=1}^{n}(|ψ_{μ}|h_{μ} ∘ ψ_{μ})(X(s, x))ds)^{−p} : τ > t] < ∞

for any p ≥ 1. This follows from Proposition 3.1.

From Theorem 3.1, we obtain the following as in Section 2:

Corollary 1. Under the assumption of Theorem 3.1 and (P G), it holds that

sup_{\begin{matrix}ξ∈ℝ \\ x,y∈D \end{matrix}} |ξ|^{k}|e^{−tL(ξb)}(x, y)| < ∞

for any k ∈ ℕ and t > 0.

Moreover Corollary 1 leads to the following:

Corollary 2. Under the assumption of Theorem 3.1 and (P G), for any t > 0, x, y ∈ D, the distribution of ∫_{0}^{t} b(X(s, x)) ∘ dX(s, x) under the conditional probability P (·|X(t, x) = y) has a smooth density function whose derivatives are all bounded: there is a function p(t, x, y; λ), t > 0, x, y ∈ D, λ ∈ ℝ such that (2.19) holds, p(t, x, y; λ) is smooth in λ, and ∂^{k}p(t, x, y; λ)/∂λ^{k} are all bounded in λ for any k ∈ ℤ_{+}.

4. The Neumann condition In this section, we consider the Neumann boundary condition. We consider only nondegenerate magnetic fields and we take only a half space as the domain D : let D = {x = (x^{1}, x^{2}, ⋯, x^{d}) ∈ ℝ^{d} : x^{1} > 0} given a metric as before, and b be a real C^{∞} differential form as before. Let C_{0}^{∞}(\overline{D}) be the set of all ℂ-valued functions on D which can be extended to C^{∞} functions with compact supports on ℝ^{d}. For any φ ∈ C_{0}^{∞}(\overline{D}), we set q(b)(φ) as in (2.1) and Dom(q^{N}(b)) as the completion of C_{0}^{∞}(\overline{D}) by the norm \sqrt{q(b)(·) + ∥ · ∥^{2}}. We extend q(b)(·) naturally to Dom(q^{N}(b)) so that (q(b),Dom(q^{N}(b))) becomes a closed sesquilinear form. Let L^{N}(b) be the associated self-adjoint operator: L^{N}(b) is the operator with the Neumann boundary condition. We will study the asymptotics of inf spec L^{N}(ξb) as ξ tends to infinity, where spec L^{N}(ξb) is the spectral set of the self adjoint operator L^{N}(ξb).

The main theorems are the following:

Theorem 4.1. We assume that the metric on D is standard and that the magnetic field is uniform: g_{jk}(x) ≡ δ_{jk} and db_{jk}(x), j, k = 1, 2, ⋯, d are independent of x. Then there exists a universal positive constant K such that

(4.1)  inf specL^{N}(b) ≥ K∥db∥_{∞},

where ∥db∥_{∞} is the maximum norm of db as

∥db∥_{∞} = max_{1≤j,k≤d} |db_{jk}|.

Theorem 4.2. Under the condition (G), it holds that

(4.2) \underline{lim}_{ξ↑∞} \frac{inf specL^{N}(ξb)}{ξ} ≥ inf_{x∈D} K∥db(x)∥_{∞},

where K is the constant in Theorem 4.1.

Remark 4.1. (i) In the circumstances of Theorem 4.1, by Weyl’s criterion (see e.g.[19]), we can show

ess specL^{N}(b) ⊃ specL(b),

where ess specL^{N}(b) is the essential spectrum of the operator L^{N}(b) and L(b) is the operator with the uniform magnetic field on ℝ^{d}. If we consider the Dirichlet condition instead of the Neumann condition, then, by the same criterion, we can show

specL(b) = ess specL(b) = specL(b).

In particular we have

(4.1)’  inf specL(b) = \frac{1}{4}∥db∥_{1}.

(ii) In Theorem 4.2, if we consider the Dirichlet condition instead of the Neumann condition, then we have

(4.2)’  lim_{ξ↑∞} \frac{inf specL(ξb)}{ξ} = inf_{x∈D} \frac{1}{4}∥db(x)∥_{1}

(cf. Theorem 11.1 in [15], Theorem 4.1 in [22]).

For the proof, we first prepare a fundamental lemma. For any x = (x^{1}, x^{2}, ⋯, x^{d}) ∈ ℝ^{d}, we write \overline{x} = (x^{2}, ⋯, x^{d}). For any function φ on D, let φ̂ and φ̃ be functions on ℝ^{d}, defined by

\begin{aligned}φ̂(x) & = sgn(x^{1})φ(|x^{1}|, \overline{x}), \\ φ̃(x) & = φ(|x^{1}|, \overline{x}). \end{aligned}

We extend the metric to ℝ^{d} by

(ǧ_{jk}(x))_{1≤j,k≤d} = ( \begin{matrix}g̃_{11}(x) & ĝ_{12}(x) & ⋯ & ĝ_{1d}(x) \\ ĝ_{21}(x) & g̃_{22}(x) & ⋯ & g̃_{2d}(x) \\ ⋮ & ⋮ & & ⋮ \\ ĝ_{d1}(x) & g̃_{d2}(x) & ⋯ & g̃_{dd}(x) \end{matrix} )

and extend the form b to ℝ^{d} by

b̌(x) = b̂_{1}(x)dx^{1} + b̃_{2}(x)dx^{2} + ⋯ + b̃_{d}(x)dx^{d}.

In the following, for simplicity, we take b so that b_{1} = 0 on {x^{1} = 0}. This is always possible by the gauge invariance. Then b̌ is continuous on ℝ^{d}. For any ψ ∈ C_{0}^{∞}(ℝ^{d}), we define Q(b̌)(ψ) as in (2.1) where D is replaced by ℝ^{d}. Then we have the following:

Lemma 4.1.

(4.3)  inf\{q(b)(φ) : φ ∈ C_{0}^{∞}(\overline{D}), ∥φ∥ = 1\} = inf\{Q(b̌)(ψ) : ψ ∈ C_{0}^{∞}(ℝ^{d}), ∥ψ∥ = 1\}.

Proof. It is easy to see that

(4.4)  \begin{aligned}& ∫_{D} ∑_{j,k=1}^{d} g^{jk}(x)\{(i∂_{j} + b_{j}(x))φ(x)\}\overline{\{(i∂_{k} + b_{k}(x))φ(x)\}}g(x)dx \\ = & ∫_{D^{∗}} ∑_{j,k=1}^{d} ǧ^{jk}(x)\{(i∂_{j} + b̌_{j}(x))φ̃(x)\}\overline{\{(i∂_{k} + b̌_{k}(x))φ̃(x)\}}ǧ(x)dx \end{aligned}

for any φ ∈ C_{0}^{∞}(\overline{D}), where D^{∗} = (−∞, 0) × ℝ^{d−1} and ǧ(x) = \sqrt{det(ǧ_{jk}(x))}. From this, we have

q(b)(φ) = \frac{1}{2}Q(b̌)(φ̃).

Since φ̃ can be approximated by elements of C_{0}^{∞}(ℝ^{d}) in Dom(Q(b̌)), we have

inf\{q(b)(φ) : φ ∈ C_{0}^{∞}(\overline{D}), ∥φ∥ = 1\} ≥ inf\{Q(b̌)(ψ) : ψ ∈ C_{0}^{∞}(ℝ^{d}), ∥ψ∥ = 1\}.

On the other hand, from (4.4), we have

Q(b̌)(ψ) = q(b)(ψ|_{D}) + q(b)(ψ(−x^{1}, \overline{x})|_{D})

for any ψ ∈ C_{0}^{∞}(ℝ^{d}). From this, we have

inf\{Q(b̌)(ψ) : ψ ∈ C_{0}^{∞}(ℝ^{d}), ∥ψ∥ = 1\} ≥ inf\{q(b)(φ) : φ ∈ C_{0}^{∞}(\overline{D}), ∥φ∥ = 1\}.

Proof of Theorem 4.1.

We may assume that the dimension d is 2. In fact, for the general dimensional case, we choose 1 ≤ j, k ≤ d so that

db_{jk} = ∥db∥_{∞},

and estimate as

q(b)(φ) ≥ \frac{1}{2}(∥(i∂_{j} + b_{j}(x))φ(x)∥^{2} + ∥(i∂_{k} + b_{k}(x))φ(x)∥^{2}).

The right hand side is estimated by the result of the 2 dimensional case.

From Lemma 4.1, we have

inf specL^{N}(b) = inf{Q(b̌)(ψ) : ψ ∈ C_{0}^{∞}(ℝ^{2}), ∥ψ∥ = 1}.

We can show that

(4.5)  Q(b̌)(ψ) ≥ ±\frac{|db|}{2}((sgn x^{1})ψ, ψ),

where db is identified with the component db_{12}. In fact, we have

\begin{matrix}0 ≤ & \frac{1}{2} ∫_{ℝ^{2}} dx |\{(i∂_{1} + b̌_{1}) ± (i∂_{2} + b̌_{2})\}ψ|^{2} \\ = & Q(b̌)(ψ) ± \frac{1}{2} ∫_{ℝ^{2}} dx sgn(x^{1})(db)|ψ|^{2}. \end{matrix}

For an arbitrarily fixed 0 < α < 1, we set η(t) = (t ∧ α) ∨ (−α) for t ∈ ℝ and

χ_{±}(x) = \sqrt{\frac{1 ± η(x^{1})}{2}}.

It is easy to see that χ_{±} ∈ H_{loc}^{1,2}(ℝ^{2}). Then, by the IMS localization (Lemma 2.1), we have

Q(b̌)(ψ) ≥ ((\frac{|db|}{2}|η(x^{1})| − ∑_{σ∈\{+,−\}} \frac{|dχ_{σ}|^{2}}{2} )ψ, ψ).

By an easy calculation, we have

ess sup ∑_{σ∈\{+,−\}} |dχ_{σ}|^{2} = \frac{1}{4(1 − α^{2})}.

Moreover, by using also the diamagnetic estimate (Lemma 2.4), we have

inf specL^{N}(b) ≥ inf spec (−\frac{Δ}{4} + \frac{|db|}{4}|η(x^{1})|) − \frac{1}{16(1 − α^{2})},

where the operator on the right hand side is the Friedrichs extension of the corresponding operator on C_{0}^{∞}(ℝ^{2}). However, by the min-max principle, we easily see that

inf spec (−\frac{Δ}{4} + \frac{|db|}{4}η(x^{1})) ≥ \frac{1}{4} inf spec (−\frac{d^{2}}{dt^{2}} + |db|t)_{[0,α]}^{N} ,

where the operator on the right hand side is the operator on the interval [0, α] with the Neumann boundary condition.

On the other hand, by using the min-max principle and changing the variables, we easily see that

(4.7)  inf specL^{N}(b) = \frac{inf specL^{N}(ξb)}{ξ}

for any ξ > 0. Therefore we have

inf specL^{N}(b) ≥ \frac{1}{4ξ} inf spec (−\frac{d^{2}}{dt^{2}} + ξ|db|t)_{[0,α]}^{N} − \frac{1}{16ξ(1 − α^{2})}.

By using the min-max principle and changing the variables, we have

\frac{1}{ξ} inf spec (−\frac{d^{2}}{dt^{2}} + |db|t)_{[0,α]}^{N} = \frac{|db|^{2/3}}{ξ^{1/3}} N((ξ|db|)^{1/3}),

where

N(R) = inf spec (−\frac{d^{2}}{dt^{2}} + t)_{[0,R]}^{N}

for R > 0. By the min-max principle, we see that N(R) is an increasing function in R. Therefore for arbitrarily fixed R > 0, we have

inf specL^{N}(b) ≥ \frac{|db|^{2/3}}{4ξ^{1/3}} N(R) − \frac{1}{16ξ(1 − α^{2})}

for any ξ ≥ R^{3}/|db|. If we regard the right hand side of this inequality as a function of all ξ > 0, then the maximum is

\frac{|db|N(R)^{3/2}\sqrt{1 − α^{2}}}{3^{3/2}} =: A

and this is attained at

\{\frac{3}{4(1 − α^{2})N(R)}\}^{3/2} \frac{1}{|db|} =: ξ_{0}.

Therefore, if ξ_{0} ≥ R^{3}/|db|, then we have inf specL^{N}(b) ≥ A. This condition is rewritten as

\frac{3}{4} ≥ N(R)R^{2}(1 − α^{2}).

Now we take α and R so that this holds as an equality. Then we have

A = \frac{|db|N(R)}{6R}.

Therefore we have

(4.8)  inf specL^{N}(b) ≥ \frac{|db|}{6} sup_{R>0} \frac{N(R)}{R},

which completes the proof.

The proof of Theorem 4.2 is reduced to that of the following modification of Theorem 3.1 of [4]:

Lemma 4.2. Under the condition (G), there exist constants C_{1}, C_{2} such that

(4.9)  (1 + \frac{C_{1}}{ξ^{1/4}} ) q(ξb)(φ) ≥ ξ ∫_{D} (K∥db(x)∥_{∞} − \frac{C_{2}}{ξ^{1/4}} ) |φ(x)|^{2}dvol(x)

for any φ ∈ C_{0}^{∞}(\overline{D}) and large enough ξ > 0.

The proof is almost the same with that of Theorem 3.1 of [4] except for using Theorem 4.1.

We next consider the asymptotics of the heat kernel as in Sections 2 and 3. We assume (BO) g_{1j}(x) ≡ δ_{1j}.

We first give the heat kernel e^{−tL^{N} (ξb)}(x, y) : for any φ ∈ L^{2}(D),

(e^{−tL^{N} (ξb)}φ)(x) =< e^{−tL^{N} (ξb)}(x, ·), φ >,

where < ·, · > is the inner product in L^{2} which is complex linear in the both variables. Then we have the following representation:

Proposition 4.1. We assume the conditions (BO) and (P G). Let X^{N}(t, x), t > 0, x ∈ \overline{D}, be the reflecting barrier Brownian motion on \overline{D}, starting at x: X^{N}(t, x) is a diffusion process generated by Δ^{N}/2, where Δ^{N} is the Laplace-Beltrami operator with the Neumann boundary condition. Let e^{tΔ^{N} /2}(x, y) be the integral kernel of the semigroup e^{tΔ^{N} /2}. Then we have

(4.10)  \begin{matrix}e^{−tL^{N} (b)}(x, y) \\ = & E[ exp ( − i ∫_{0}^{t} b(X^{N}(s, x)) ∘ dX^{N}(s, x))|X^{N}(t, x) = y]e^{tΔ^{N} /2}(x, y), \end{matrix}

for (t, x, y) ∈ (0, ∞) × \overline{D} × \overline{D}.

Proof. If b_{1} = 0 on ∂D, then the proof is the same with that for the case b = 0 (cf.[2], [9]). For the general case, we choose f ∈ C_{b}^{∞}(\overline{D}) so that ∂f/∂x^{1} = b_{1} on ∂D and set b^{∘} := b − df, which satisfies b^{∘}_{1} = 0 on ∂D. Then, by the gauge invariance, we can show (4.10) as follows:

E[ exp ( − i ∫_{0}^{t} b(X^{N}(s, x)) ∘ dX^{N}(s, x))|X^{N}(t, x) = y]e^{tΔ^{N} /2}(x, y)

=e^{if(x)}E[ exp ( − i ∫_{0}^{t} b^{∘}(X^{N}(s, x)) ∘ dX^{N}(s, x))|X^{N}(t, x) = y]
× e^{tΔ^{N} /2}(x, y)e^{−if(y)}
=e^{if(x)}e^{−tL^{N} (b^{∘})}(x, y)e^{−if(y)}
=e^{−tL^{N} (b)}(x, y).

From Theorem 4.2, we obtain the following as in [22]:

Corollary 1. Under the conditions (P G) and (BO), it holds that

(4.11) \overline{lim}_{ξ→±∞} sup_{(t,x,y)∈[t_{0},∞)×\overline{D}×\overline{D}} \frac{1}{t|ξ|} log |e^{−tL^{N} (ξb)}(x, y)| ≤ − inf_{x∈D} K∥db(x)∥_{∞}

for any t_{0} > 0.

Moreover Corollary 1 leads to the following:

Corollary 2. We assume the conditions (P G), (BO) and

inf_{x∈D} ∥db(x)∥_{∞} > 0.

Then for any t > 0, x, y ∈ \overline{D}, the distribution of ∫_{0}^{t} b(X^{N}(s, x)) ∘ dX^{N}(s, x) under the conditional probability P (·|X^{N}(t, x) = y) has a real analytic density function: there is a function p^{N}(t, x, y; λ), t > 0, x, y ∈ \overline{D}, λ ∈ ℝ such that p^{N}(t, x, y; λ) is real analytic in λ and

(4.12)  P ( ∫_{0}^{t} b(X^{N}(s, x)) ∘ dX^{N}(s, x) ∈ A|X^{N}(t, x) = y) = ∫_{A} p^{N}(t, x, y; λ)dλ

for any Borel set A in ℝ. The radius of convergence of its Taylor series around any point is greater than or equal to t inf_{x∈D} K∥db(x)∥_{∞}.

Remark 4.2. Corollary 2 is known as transverse analyticity. This is proven in Malliavin [12], [13], [14] and Prat [18] for the case without boundary and in Ueki [22] for the absorbing barrier Brownian motion.

Acknowledgements. The author would like to express his gratitude to Professor David Elworthy for the invitation to Mathematics Research Center, University of Warwick, where some of this work was done.

Department of Mathematics Himeji Institute of Technology

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