Lower Bounds for the Spectra of Schrödinger

Operators with Magnetic Fields^{∗}

Naomasa Ueki

Department of Mathematics Faculty of Science Himeji Institute of Technology Shosha 2167, Himeji 671-22 Japan

Typeset by AMS-TEX

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

Running head: Lower bounds for spectra Author: Naomasa Ueki

Department of Mathematics Faculty of Science Himeji Institute of Technology Shosha 2167, Himeji 671-22 Japan

Abstract: An explicit representation of lower bounds for the spectra of Schrödinger operators with magnetic fields on σ-compact Riemannian manifolds is given, by using the positivity of Pauli Hamiltonian. This representation is applied to show some asymptotic properties of a stochastic oscillatory integral and transverse analyticity of the law of a stochastic line integral.

1. Introduction

In this paper, we concern with the infmum of the spectra of Schrödinger operators with magnetic fields. Our main contribution is to give an explicit representation of lower bounds for the spectra in terms of the curvature and the magnetic field (see Theorems 2.1 below).

On ℝ^{2}, the Schrödinger operator with a magnetic field db has the following form;

L(b) = \frac{1}{2} ∑_{j=1}^{2}(i\frac{∂}{∂x^{j}} + b^{j}(x))^{2} (i = \sqrt{−1}),

where b = ∑_{j=1}^{2} b^{j}(x)dx^{j}. We consider the Dirac operator,

Q = \frac{1}{\sqrt{2}} ( \begin{matrix}0 & −1 \\ −1 & 0 \end{matrix} ) (i\frac{∂}{∂x^{1}} + b^{1}(x)) + \frac{1}{\sqrt{2}} ( \begin{matrix}0 & i \\ −i & 0 \end{matrix} ) (i\frac{∂}{∂x^{2}} + b^{2}(x))

acting on ℂ^{2}-valued functions. Then the Pauli Hamiltonian Q^{2} has the following form;

Q^{2} = ( \begin{matrix}L(b) − db/2 & 0 \\ 0 & L(b) + db/2 \end{matrix} ) ,

where the 2-form db is identified with the function ∂b^{2}/∂x^{1} − ∂b^{1}/∂x^{2}. The positivity of Q^{2} leads to

inf spec L(b) ≥ inf_{x∈ℝ^{2}} \frac{|db(x)|}{2},

where spec L(b) is the spectrum of L(b).

In this paper, we apply the above method to the operators on a general σcompact Riemannian manifold. We define a Dirac operator Q (see (4.2) below) and a suitable operator P (see (4.9) and (4.10) below) on a space of forms, and show the following:

P^{∗}Q^{2}P = (L(b) + R)P_{0},

where P_{0} is a projection onto the space of 0-forms and R = ∑_{j=1}^{3} R_{j} − B is a function explicitly written as a sum of terms of the curvature and terms of the magnetic field (see (2.3), (2.4) and (3.1) below). Since P^{∗}Q^{2}P is a positive operator, we obtain

(1.1)  inf spec L(b) ≥ inf_{x∈M}{−R(x)}

(see Theorem 2.1 below). The operator P is an extension of the operator used by Shigekawa [20] (cf. also [3]). To define the operator P for our cases, we introduce an almost complex structure on a subbundle of the tangent bundle (see (2.2) below).

As an application of our estimate mentioned above, we show

(1.2)  \underline{lim}_{ξ→∞} \frac{inf spec L(ξb)}{ξ} ≥ \frac{1}{4} inf_{x∈M} ∥db(x)∥_{1}

under suitable assumptions (A.2) and (A.3) in Section 5 (see Theorem 4.1 below), where ∥·∥_{1} is the trace norm. This type of result was proven in Malliavin [13], [14], though the right hand side of (1.2) is slightly different. Consequently, as Mallivin also discussed, we can deduce results on asymptotics of an oscillatory integral over Wiener space and analyticity of the law of a stochastic line integral (see Corollaries 1 and 2 of Theorem 5.1 below). For asymptotics of oscillatory integrals, see [9] and references therein. The analyticity we prove is known as transverse analyticity, which has been studied by Malliavin [13], [14], [15] and Prat [18].

The organization of this paper is as follows. In Section 2, we state the main theorem, Theorem 2.1. In Section 3, we give several remarks and examples on Theorem 2.1. In Section 4, we prove Theorem 2.1: we define a Dirac operator Q and a suitable operator P and calculate P^{∗}Q^{2}P. This part is the key of this paper.

In Section 5, we apply the results in Section 2 to show (1.2). Furtheremore, we apply (1.2) to show results on asymptotics of an oscillatory integral and analyticity of the law of a stochastic line integral.

2. The statement of the main theorem

Throughout of this paper, M is a σ-compact C^{3} Riemannian manifold, which may have boundary, and b is a real C^{1} differential form on M. Let C_{0}^{2}(M^{∘}) be the set of all ℂ-valued C^{2} functions on the interior of M with compact support. For any φ ∈ C_{0}^{2}(M^{∘}), we set

(2.1)  L(b)φ = \frac{1}{2}(id + ext(b))^{∗}(id + ext(b))φ

where ext is the exterior multiplication, i.e., ext(b)φ = b ∧ φ, and (id + ext(b))^{∗} is the formal adjoint of id + ext(b). Let L^{2}(M) be the space of all ℂ-valued square integrable functions with respect to the Riemannian volume. We easily see that this L(b) is a symmetric operator in L^{2}(M). We take, accordingly, the Friedrichs extension of L(b), which we also denote by L(b): L(b) is the operator with Dirichlet boundary condition when the manifold M has boundary (see Remark 4.1 below). The infmum of the spectrum of this self-adjoint operator L(b) is our object.

Let U, V and W be C^{2} Riemannian vector bundles over M satisfying the following:

(2.2)

(i) T M = U ⊕ V ⊕ W (orthogonal direct sum),

(ii) There is a C^{2} isometry J on U such that J^{2} = −(identity), (iii) V is oriented.

As we allow any of these bundles to be zero, this formulation does not make any restrictions. J is an almost complex structure on U for which the Riemannian metric is Hermitian. According to a convention for complex structures, we set U′ = {v ∈ U ⊗ ℂ; Jv = iv} and U′′ = {v ∈ U ⊗ ℂ; Jv = −iv}. The orientation of V is needed to define the operator P_{1} in Section 4 (see (4.9) below).

Generally, for any complex (resp. real) subbundle E of T M ⊗ ℂ (resp. T M) and any complex (resp. real) subbundle F of E, let S(E, F) be the second fundamental form of F in E (cf. e.g. [7, p-78]). This is a section of the bundle F^{∗} ⊗ T^{∗}M ⊗ F^{⊥}, where F^{∗} is the dual of F and F^{⊥} is the orthogonal complement of F in E. Let ∥S(E, F)∥ be the fibre norm of S(E, F) on F^{∗} ⊗ T^{∗}M ⊗ F^{⊥}. On the other hand, let R(·, ·) be the curvature transformation of T M:

R(X, Y)Z = (∇_{X}∇_{Y} − ∇_{Y} ∇_{X} − ∇_{[X,Y]})Z for X, Y, Z ∈ T M,

where ∇ is the Levi-Civita connection of M and [X, Y] = XY − Y X is the Lie bracket. Let R(·, ·; F) be a trace on F of curvature tensors as follows;

R(X, Y; F) = ∑_{a=1}^{A}(R(\overline{δ_{a}}, X)Y, δ_{a}) for X, Y ∈ T M ⊗ ℂ

where {δ_{a}}_{a=1}^{A} is a unitary (resp. an orthonormal) basis of F and the metric (·, ·), the curvature transformation R(·, ·) are extended to complex vectors as complex linear in every variable. If F equals T M, R(·, ·; T M) is the Ricci tensor. Let R(E, F) be a trace on F^{⊥} of R(·, ·; F) as follows:

R(E, F) = ∑_{a′=1}^{A′} R(\overline{δ′_{a′}}, δ′_{a′}; F) = ∑_{a′=1}^{A′} ∑_{a=1}^{A}(R(\overline{δ_{a}}, \overline{δ′_{a′}})δ′_{a′}, δ_{a})

where {δ′_{a′}}_{a′=1}^{A′} is a unitary (resp. an orthonormal) basis of F^{⊥}. We remark that ∥S(E, F)∥ = ∥S(E, F^{⊥})∥ and R(E, F) = R(E, F^{⊥}).

We define functions R_{1}, R_{2} and R_{3} on M by

(2.3)  \begin{aligned}R_{1} & = \frac{1}{2}∥S(U ⊗ ℂ, U′)∥^{2} + \frac{1}{2}R(U ⊗ ℂ, U′), \\ R_{2} & = \frac{1}{4}∥S(T M, U)∥^{2} + \frac{1}{4}R(T M, U), \\ R_{3} & = \frac{1}{2}∥S(V ⊕ W, V)∥^{2} + \frac{1}{2}R(V ⊕ W, V). \end{aligned}

For the local expresseion, see Lemma 3.1 below.

We define a function B by a trace of db/2i on U′:

(2.4)  B = \frac{1}{2i} ∑_{ρ=1}^{l} db(\overline{ϵ_{ρ}}, ϵ_{ρ}),

where {ϵ_{ρ}}_{ρ=1}^{l} is a unitary basis of U′ and the 2-form db(·, ·) are extended to complex vectors as complex linear.

Then our main theorem on lower bounds for the spectra is stated as follows: Theorem 2.1. L(b) + ∑_{j=1}^{3} R_{j} − B is a positive operator.

In particular, we have

(2.5)  inf spec L(b) ≥ inf_{x∈M}{− ∑_{j=1}^{3} R_{j}(x) + B(x)}.

We prove this theorem at the end of Section 4.

3. Remarks and examples on Theorem 2.1

To use Theorem 2.1, the representation of R_{1}, R_{2}, R_{3} and B in terms of local frames is often useful. We summarize the representation: Lemma 3.1. Let 2l be the fibre dimension of U, m be that of U ⊕ V and d be the dimension of M (2l ≤ m ≤ d). Let {e_{j}}_{j=1}^{2l} be a local orthonormal frame of U such that \{ϵ_{ρ} := (e_{2ρ−1} − ie_{2ρ})/\sqrt{2}\}_{ρ=1}^{l} is a unitary frame of U′. Let \{e_{j}\}_{j=2l+1}^{m} and {e_{j}}_{j=m+1}^{d} be a local orthonormal frame of V and W, respectively.

Then the functions R_{1}, R_{2}, R_{3} and B are represented as

(3.1)  \begin{aligned}R_{1} & = \frac{1}{2} ∑_{\begin{matrix}1≤σ,ρ≤l \\ 1≤j≤d \end{matrix}} |(∇_{e_{j}}ϵ_{σ}, ϵ_{ρ})|^{2} + \frac{1}{2} ∑_{1≤σ,ρ≤l}(R(\overline{ϵ_{σ}}, \overline{ϵ_{ρ}})ϵ_{ρ}, ϵ_{σ}), \\ R_{2} & = \frac{1}{4} ∑_{\begin{matrix}1≤p≤2l \\ 2l<q≤d \\ 1≤j≤d \end{matrix}}(∇_{e_{j}}e_{q}, e_{p})^{2} + \frac{1}{4} ∑_{\begin{matrix}1≤p≤2l \\ 2l<q≤d \end{matrix}}(R(e_{p}, e_{q})e_{q}, e_{p}), \\ R_{3} & = \frac{1}{2} ∑_{\begin{matrix}2l<p≤m \\ m<q≤d \\ 1≤j≤d \end{matrix}}(∇_{e_{j}}e_{q}, e_{p})^{2} + \frac{1}{2} ∑_{\begin{matrix}2l<p≤m \\ m<q≤d \end{matrix}}(R(e_{p}, e_{q})e_{q}, e_{p}), \\ B & = \frac{1}{2i} ∑_{ρ=1}^{l} db(\overline{ϵ_{ρ}}, ϵ_{ρ}). \end{aligned}

On the other hand, Theorem 2.1 relates to the following, for example: Remark 3.1. In [10], [12] and [16], the long time asymptotics of the heat semigroup generated by L(b) is studied. Since the leading term of the asymptotics equals − inf spec L(b), those works relate to ours.

We now consider various assumptions to use Theorem 2.1 effectively. We first note the following: Proposition 3.1. If ∇J = 0, then R_{1} = 0.

This is easily seen from the definition of the function R_{1}.

We next assume the following:

(A.1)  For any v ∈ U, \frac{1}{i}db(\overline{v}, v) ≥ 0.

Under this assumption, we have the following: Proposition 3.2. Under the assumption (A.1), we have

(3.2)  B(x) = \frac{1}{4}∥db(x)|_{U}∥_{1},

where db(x)|_{U} is the matrix (db(e_{j}(x), e_{k}(x)))_{1≤j,k≤2l} for an orthonormal basis {e_{j}(x)}_{j=1}^{2l} of U_{x} and, for any matrix A, ∥A∥_{1} is its trace norm:

∥A∥_{1} = tr \sqrt{^{t}A · A}.

In particular, if db(v, w) = 0 for any v ∈ T M and any w ∈ V ⊕ W, we have

(3.3)  B(x) = \frac{1}{4}∥db(x)∥_{1}.

Proof. For any local orthonormal frame \{e_{j}\}_{j=1}^{d} taken as in Lemma 3.1, (db(\overline{ϵ_{ρ}}, ϵ_{σ}))_{1≤ρ,σ≤l} is a skew-Hermitian matrix. Therefore, we can take such a frame {e_{j}}_{j=1}^{d} satisfying

(db(\overline{ϵ_{ρ}}, ϵ_{σ}))_{1≤ρ,σ≤l} = ( \begin{matrix}iβ_{1} & & & & & 0 \\ & iβ_{2} \\ & & . & . & . \\ 0 & & & & & iβ_{l} \end{matrix} ) .

The assumption (A.1) implies β_{σ} ≥ 0, σ = 1, 2, ⋯, l. Then, by easy calculations, we have

B = \frac{1}{2} ∑_{σ=1}^{l} β_{σ} = \frac{1}{4}∥db(x)|_{U}∥_{1}.

About the assumption (A.1), we first note the following (cf. Darboux’s theorem [1, Theorem 3.2.2]):

Proposition 3.3. If the matrix db(x)|_{U} in Proposition 3.2 is nonsingular at any point x ∈ M, then there exists J satisfying (2.2) and (A.1).

Proof. Under the assumption, we can choose a local orthonormal frame {e_{j}}_{j=1}^{2l} such that

(db(e_{j}, e_{k}))_{1≤j,k≤2l} = ( \begin{matrix}0 & −β_{1} \\ β_{1} & 0 \end{matrix}  0 ; \begin{matrix}0 & −β_{2} \\ β_{2} & 0 \end{matrix} ; ... ; 0  \begin{matrix}0 & −β_{l} \\ β_{l} & 0 \end{matrix} ) .

where β_{σ}, σ = 1, 2, ⋯, l are positive functions defined locally. Then we can define J by

Je_{2σ−1} = e_{2σ} and Je_{2σ} = −e_{2σ−1}

for any σ = 1, 2, ⋯, l.

If the dimension of the manifold M is 2, to consider the effect of the magnetic field to the lower bound, we should assume that T M = U in (2.2). This implies R_{1} = R_{2} = R_{3} = 0 in (2.3). Then, in using Theorem 2.1, we may assume that the matrix db(x) is nonsingular at any point x ∈ M, since L(b) is positive. Therefore we have the following:

Proposition 3.4. If the dimension d of M is 2, we always have

(3.4)  inf spec L(b) ≥ inf_{x∈M} \frac{1}{4}∥db(x)∥_{1}.

However, if the dimension d of M is greater than 3, there are examples the assumption (A.1) is not satisfied:

Example 1. On ℝ^{d}, d ≥ 3, we consider the magnetic field db having the following form:

db(x) = ( \begin{matrix}0 & −x^{1} & 0 & … & 0 \\ x^{1} & 0 & 0 & … & 0 \\ 0 & 0 \\ ⋮ & ⋮ & & ∗ \\ 0 & 0 \end{matrix} ) .

Then, for the assumption (A.1), we should take J satisfying the following: on {x^{1} > 0},^{t}(1, −i, 0, ⋯, 0) ∈ U′ and^{t}(1, +i, 0, ⋯, 0) ∈ U′′; on {x^{1} < 0},^{t}(1, +i, 0, ⋯, 0) ∈ U′ and ^{t}(1, −i, 0, ⋯, 0) ∈ U′′. However we can not take such a J continuously.

For other remarks, see the end of the next section.

We next apply our theorem on Euclidean space: Example 2. On ℝ^{d} with the standard metric, the easiest way of applying Theorem 2.1 is the following: we put U =< ∂_{1}, ∂_{2}, ⋯, ∂_{2l} >, W =< ∂_{2l+1}, ∂_{2l+2}, ⋯, ∂_{d} >, 0 ≤ 2l ≤ d, and J∂_{2σ−1} = ∂_{2σ}, σ = 1, 2, ⋯, l, where ∂_{j} = ∂/∂x^{j}. Then we have R_{1} = R_{2} = R_{3} = 0 and

(3.5)  B = ∑_{σ=1}^{l}(db)_{2σ,2σ−1}(x).

Therefore, from Theorem 2.1, we have

(3.6)  inf spec L(b) ≥ inf_{x∈M} ∑_{σ=1}^{l}(db)_{2σ,2σ−1}(x).

However, we can obtain better estimates from Theorem 2.1. Let us suppose that d = 2l and db is written as follows: (3.7)

db(x) = e(x) ( \begin{matrix}0 & −β_{1}(x) \\ β_{1}(x) & 0 \end{matrix}  0 ; \begin{matrix}0 & −β_{2}(x) \\ β_{2}(x) & 0 \end{matrix} ; ... ; 0  \begin{matrix}0 & −β_{l}(x) \\ β_{l}(x) & 0 \end{matrix} )^{t}e(x),

where ℝ^{d} ∋ x ↦ e(x) = (e_{jk}(x))_{1≤j,k≤d} is an orthogonal matrix valued C^{2} function and β_{σ}, σ = 1, 2, ⋯, l, are nonnegative continuous functions. We define a global orthonormal frame {e_{j}(x)}_{j=1}^{d} by

e_{j}(x) = ∑_{k=1}^{d} e_{kj}(x)∂_{k}

and we put T ℝ^{d} = U and Je_{2σ−1} = e_{2σ}, σ = 1, 2, ⋯, l. Then we have

\begin{matrix}R_{1} = & \frac{1}{32} ∑_{\begin{matrix}1≤σ,ρ≤l \\ 1≤j≤d \end{matrix}} |∑_{k=1}^{d} ∂_{j}(e_{k,2σ−1} − ie_{k,2σ})^{2}|^{2} , \\ R_{2} = & R_{3} = 0, \\ B = & \frac{∥db(x)∥_{1}}{4} . \end{matrix}

Therefore, from Theorem 2.1, we have

(3.8) inf spec L(b) ≥ inf_{x∈M} (\frac{∥db(x)∥_{1}}{4} − \frac{1}{32} ∑_{\begin{matrix}1≤σ,ρ≤l \\ 1≤j≤d \end{matrix}} | ∑_{k=1}^{d} ∂_{j}(e_{k,2σ−1} − ie_{k,2σ})^{2}|^{2}) .

On the other hand, we are interested in our estimates on spaces of negative constant curvature, since sectional curvatures appear with positive coefficients in (2.3). We now consider on the hyperbolic space in the half-space (cf. [2], [4, Section 5.7]). Example 3. Let ℍ^{n+1} = {x = (x^{1}, x^{2}, ⋯, x^{n+1}) ∈ ℝ^{n+1}; 0 < x^{n+1} < ∞}. We give ℍ^{n+1} the metric (∂_{j}, ∂_{k}) = δ_{j,k}/y^{2}, j, k = 1, 2, ⋯, n + 1, where ∂_{j} = ∂/∂x^{j}, y = x^{j+1} and δ_{j,k} is Kronecker’s delta. Then the vector fields e_{j} = y∂_{j}, j = 1, 2, ⋯, n + 1, form a global orthonormal frame of T ℍ^{n+1}. For this frame, we have

(∇_{e_{j}}e_{k}, e_{h}) = \{ \begin{matrix}δ_{j,k}, & if h = n + 1, \\ −δ_{j,h}, & if k = n + 1, \\ 0, & otherwise, \end{matrix}
(R(e_{j}, e_{k})e_{q}, e_{p}) = δ_{j,q}δ_{k,p} − δ_{j,p}δ_{k,q}.

If we put U =< e_{1}, e_{2}, ⋯, e_{2l} >, V =< e_{2l+1}, e_{2l+2}, ⋯, e_{m} >, W =< e_{m+1}, e_{m+2}, ⋯, e_{n+1} >, 0 ≤ 2l ≤ m ≤ n, and Je_{2σ−1} = e_{2σ}, σ = 1, 2, ⋯, l, then we have

(3.9)  R_{1} = −\frac{l(l − 1)}{2}, R_{2} = −\frac{l(n − 2l)}{4}, R_{3} = −\frac{(m − 2l)(n − m)}{2}.

If n is even, ∑_{j=1}^{3} R_{j} takes on its minimum with respect to m at m = l + n/2 and its value is

(3.10)  ∑_{j=1}^{3} R_{j} = −\frac{n^{2}}{8} + \frac{l}{2}.

This value increases as l increases. Therefore, the effect of the term ∑_{j=1}^{3} R_{j} becomes worse as direction we estimate the magnetic field becomes wider. When l = 0 and m = n/2, we get ∑_{j=1}^{3} R_{j} = −n^{2}/8. Therefore we obtain

(3.11)  inf spec (−\frac{Δ}{2}) ≥ \frac{n^{2}}{8} ,

where Δ is the Laplace-Beltrami operator. This is the best estimate, since it is known that inf spec (−Δ/2) = n^{2}/8 for any n [4, Theorem 5.7.1].

If n is odd, ∑_{j=1}^{3} R_{j} takes on its minimum with respect to m at m = l+(n±1)/2 and its value is

(3.12)  ∑_{j=1}^{3} R_{j} = −\frac{n^{2}}{8} + \frac{l}{2} + \frac{1}{8} ≥ −\frac{n^{2}}{8} + \frac{1}{8}.

We next put U =< e′_{1}, e′_{2}, ⋯, e′_{2l} >, V =< e′_{2l+1}, e′_{2l+2}, ⋯, e′_{m} >, W =< e′_{m+1}, e′_{m+2}, ⋯, e′_{n+1} >, 2 ≤ 2l ≤ m ≤ n + 1, and Je′_{2σ−1} = e′_{2σ}, σ = 1, 2, ⋯, l, where e′_{j} = e_{n+1−j}: we consider the case e_{n+1} ∈ U. If n is odd, ∑_{j=1}^{3} R_{j} takes on its minimum with respect to m at m = l + (n + 1)/2 and its value is

(3.13)  ∑_{j=1}^{3} R_{j} = −\frac{n^{2}}{8} + \frac{l}{2} − \frac{3}{8} ≥ −\frac{n^{2}}{8} + \frac{1}{8}, l ≥ 1.

If n is even, ∑_{j=1}^{3} R_{j} takes on its minimum with respect to m at m = l+(n+1±1)/2 and its value is

(3.14)  ∑_{j=1}^{3} R_{j} = −\frac{n^{2}}{8} + \frac{l}{2} − \frac{1}{4} ≥ −\frac{n^{2}}{8} + \frac{1}{4}, l ≥ 1.

4. Proof of Theorem 2.1

To prove Theorem 2.1 in the previous section, we define a Dirac operator Q and an operator P and calculate P^{∗}Q^{2}P. This calculation is the most important part of this paper. The result is summarized in Theorem 4.1 below.

In the following, we do local calculation for the most part. Let {e_{j}}_{j=1}^{d} be a local orthonormal frame of T M taken as in Lemma 3.1.

We first prepare the following formula for L(b), which is easily shown: Lemma 4.1. L(b) is locally represented as

(4.1)  L(b) = −\frac{1}{2} ∑_{j=1}^{d}(e_{j}e_{j} − ∑_{k=1}^{d} Γ_{jjk}e_{k}) + i ∑_{j=1}^{d} b(e_{j})e_{j} − \frac{id^{∗}b}{2} + \frac{∥b∥^{2}}{2}

where

Γ_{jkh} = (∇_{e_{j}}e_{k}, e_{h}), j, k, h = 1, 2, ⋯, d,

and ∥b∥ is the usual norm on cotangent space of b.

We define a Dirac operator Q as follows: let Γ_{0}^{1}(M^{∘}; Λ(T^{∗}M) ⊗ ℂ) be the set of all C^{1} forms on the interior of M with compact support. Let Q be an operator defined on Γ_{0}^{1}(M^{∘}; Λ(T^{∗}M) ⊗ ℂ) by

(4.2)  Q = \frac{1}{\sqrt{2}}\{(id + ext(b)) + (id + ext(b))^{∗}\}.

This Q is locally represented as

(4.3)  Q = \frac{1}{\sqrt{2}} ∑_{j=1}^{d} γ^{j}(i∇_{e_{j}} + b(e_{j}))

(cf. [6, p-149]), where γ^{j}, j = 1, 2, ⋯, d, are local sections to the bundle of Hermitian matrices of Λ(T^{∗}M) ⊗ ℂ defined by

(4.4)  γ^{j} = ext(e^{j}) + int(e^{j}).

We remark that {γ^{j}}_{j=1}^{d} satisfies the following commutation relation:

(4.5)  γ^{j}γ^{k} + γ^{k}γ^{j} = 2δ_{j,k}, j, k = 1, 2, ⋯, d.

When b = 0, the corresponding Pauli Hamiltonian Q^{2} becomes the de Rham-Hodge-Kodaira Laplacian.

The Pauli Hamiltonian Q^{2} has the following representation:

Lemma 4.2. Q^{2} is locally represented as

(4.6)

Q^{2} =\underline{L(b)} + ∑_{1≤j,p,q≤d} Γ_{jqp}ext(e^{q})int(e^{p})\underline{e_{j}}

+ ∑_{1≤j,p,q≤d}\{\frac{1}{2}(e_{j}Γ_{jqp}) − \frac{1}{2}Γ_{jjk}Γ_{kqp} − ib(e_{j})Γ_{jqp}\}ext(e^{q})int(e^{p})

− \frac{1}{2} ∑_{1≤j,p,q,r,s≤d} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})

+ \frac{1}{2} ∑_{\begin{matrix}1≤j<k≤d \\ 1≤p,q≤d \end{matrix}} R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})

+ \frac{1}{2} ∑_{1≤j<k≤d} iγ^{j}γ^{k}db(e_{j}, e_{k})

where

R_{jkqp} = (R(e_{j}, e_{k})e_{q}, e_{p}), j, k, q, p = 1, 2, ⋯, d,

and, for any differential operator A on space of functions, \underline{A} is the operator acting only on the coefficients of {e^{j_{1}} ∧ e^{j_{2}} ∧ ⋯ ∧ e^{j_{p}}}_{1≤j_{1}<j_{2}<···<j_{p}≤d}, i.e.,

\begin{matrix}\underline{A} ∑_{1≤j_{1}<j_{2}<···<j_{p}≤d} φ_{j_{1}j_{2}···j_{p}}e^{j_{1}} ∧ e^{j_{2}} ∧ ⋯ ∧ e^{j_{p}} \\ = ∑_{1≤j_{1}<j_{2}<···<j_{p}≤d} φ_{j_{1}j_{2}···j_{p}}(Aφ_{j_{1}j_{2}···j_{p}})e^{j_{1}} ∧ e^{j_{2}} ∧ ⋯ ∧ e^{j_{p}}. \end{matrix}

Proof. We take any x_{0} ∈ M and fix it. We take {e_{j}} such that ∇e_{j} = 0 at x_{0}. At x_{0}, we can calculate as follows:

Q^{2} =\frac{1}{2} ∑_{j=1}^{d} γ^{j}γ^{j}(i∇_{e_{j}} + b(e_{j}))(i∇_{e_{j}} + b(e_{j}))

+ \frac{1}{2} ∑_{j<k} γ^{j}γ^{k}[i∇_{e_{j}} + b(e_{j}), i∇_{e_{k}} + b(e_{k})]

=\frac{1}{2} ∑_{j=1}^{d}(−∇_{e_{j}}∇_{e_{j}} + i(e_{j}b(e_{j})) + 2ib(e_{j})∇_{e_{j}} + b(e_{j})^{2})

+ \frac{1}{2} ∑_{j<k} γ^{j}γ^{k}\{−[∇_{e_{j}}, ∇_{e_{k}}] + i(e_{j}b(e_{k}) − e_{k}b(e_{j}))\}.

We introduce the curvature operator R^{T^{∗}M}(·, ·) on space of forms by

R^{T^{∗}M}(X, Y)ω = (∇_{X}∇_{Y} − ∇_{Y} ∇_{X} − ∇_{[X,Y]})ω, X, Y ∈ T M, ω ∈ T^{∗}M,

and we rewrite the above equation as follows:

(4.7)  \begin{aligned}Q^{2} = & − \frac{1}{2} ∑_{j=1}^{d} ∇_{e_{j}e_{j}}^{2} + i ∑_{j=1}^{d} b(e_{j})∇_{e_{j}} − \frac{id^{∗}b}{2} + \frac{∥b∥^{2}}{2} \\ & − \frac{1}{2} ∑_{j<k} γ^{j}γ^{k}R^{T^{∗}M}(e_{j}, e_{k}) + \frac{1}{2} ∑_{j<k} iγ^{j}γ^{k}db(e_{j}, e_{k}). \end{aligned}

Both sides of (4.7) are independent of the choice of {e_{j}} and (4.7) holds at any point. Furthermore, we use Lemma 4.1 and the following to obtain (4.6):

(4.8)

∇_{e_{j}} = \underline{e_{j}} − ∑_{p,q} Γ_{jqp}ext (e^{q})int (e^{p}),

∇_{e_{j}e_{j}}^{2} = \underline{e_{j}}\underline{e_{j}} − ∑_{p,q}(e_{j}Γ_{jqp})ext (e^{q})int (e^{p})

− 2 ∑_{p,q} Γ_{jqp}ext (e^{q})int (e^{p})\underline{e_{j}}

+ ∑_{p,q,r,s} Γ_{jsr}Γ_{jqp}ext (e^{s})int (e^{r})ext (e^{q})int (e^{p})

− ∑_{k} Γ_{jjk}\underline{e_{k}} + ∑_{k,p,q} Γ_{jjk}Γ_{kqp}ext (e^{q})int (e^{p}),

R^{T^{∗}M}(e_{j}, e_{k}) = − ∑_{p,q} R_{jkqp}ext (e^{q})int (e^{p}).

We introduce the following operators:

(4.9)

P_{0} := int(e^{d}) ⋯ int(e^{2})int(e^{1})ext(e^{1})ext(e^{2}) ⋯ ext(e^{d}),
P_{1} := ext(e^{2l+1})ext(e^{2l+2}) ⋯ ext(e^{m}),
P_{2} := ∏_{σ=1}^{l} \frac{1 + iγ^{2σ−1}γ^{2σ}}{2} = ∏_{σ=1}^{l} \frac{γ^{2σ−1} − iγ^{2σ}}{2} · \frac{γ^{2σ−1} + iγ^{2σ}}{2} .

To define P_{1} globally, in the following, we assume that the frame {e_{j}}_{j=2l+1}^{m} is positively oriented. Then P_{0}, P_{1} and P_{2} are globally defined elements of Γ^{2}(M; End(Λ(T^{∗}M)⊗ ℂ)). P_{0} is the orthogonal projection onto the space of zero forms. P_{2} is the orthogonal projection onto the space {ω ∈ Λ(T^{∗}M)⊗ℂ; iγ^{2σ−1}γ^{2σ}ω = ω, σ = 1, 2, ⋯, l}. We use the operator P defined by

(4.10)  P = 2^{\frac{l}{2}} P_{2}P_{1}P_{0}.

This operator also can be writen as follows:

(4.11)  P = 2^{− \frac{l}{2}} ext( exp (∑_{σ=1}^{l} e^{2σ−1} ∧ e^{2σ}) )P_{1}P_{0}.

The form ∑_{σ=1}^{l} e^{2σ−1} ∧ e^{2σ} is analogous to symplectic form, and P_{1} is analogous to the Hodge ∗-operator on V.

Then we can state the following important formula: Theorem 4.1.

(4.12)  P^{∗}Q^{2}P = {L(b) + ∑_{j=1}^{3} R_{j} − B} P_{0}

where R_{j}, j = 1, 2, 3 and B are given in (2.3) and (2.4). Proof. We consider each term of the right hand side of (4.6) separately. We devide the calculation into several steps. In Steps 0-4, we use the following

(4.13)  P1 = 2^{− \frac{l}{2}} ⋀_{σ=1}^{l}(1 + ie^{2σ−1} ∧ e^{2σ}) ∧ e^{2l+1} ∧ e^{2l+2} ∧ ⋯ ∧ e^{m},

(4.14)  (1 + iγ^{2σ−1}γ^{2σ})1 = 1 + ie^{2σ−1} ∧ e^{2σ},
(4.15)  (1 + iγ^{2σ−1}γ^{2σ})e^{2σ−1} = e^{2σ−1} − ie^{2σ},
(4.16)  (1 + iγ^{2σ−1}γ^{2σ})e^{2σ} = i(e^{2σ−1} − ie^{2σ}),
(4.17)  (1 + iγ^{2σ−1}γ^{2σ})(e^{2σ−1} ∧ e^{2σ}) = −i(1 + ie^{2σ−1} ∧ e^{2σ}),
(4.18)  P^{∗} = 2^{− \frac{l}{2}} P_{0}int(e^{m}) ⋯ int(e^{2l+2})int(e^{2l+1})P_{2},
(4.19)  P^{∗}P = P_{0}.

Step 0. Calculation of P^{∗}\underline{L(b)}P.

From (4.19), we have

(4.20)  P^{∗}\underline{L(b)}P = L(b)P_{0}.

Step 1. Calculation of P^{∗}Γ_{jqp}ext(e^{q})int(e^{p})\underline{e_{j}}P and P^{∗}\{\frac{1}{2}Γ_{jjk}Γ_{kqp} − ib(e_{j})Γ_{jqp}\}ext(e^{q})int(e^{p})P.

When p = 2ρ − 1 (ρ ≤ l), we have

int(e^{p})P1 = 2^{− \frac{l}{2}} ie^{2ρ} ⋀_{σ≠ρ}(1 + ie^{2σ−1} ∧ e^{2σ}) ∧ e^{2l+1} ∧ e^{2l+2} ∧ ⋯ ∧ e^{m}.

From (4.16) and (4.18), we know that q = p is necessary for P^{∗}ext(e^{q})int(e^{p})P1 ≠ 0. Similarly, for p = 2ρ (ρ ≤ l), we know the same thing, from (4.15) instead of (4.16). For p > 2l, it is quite easy to see the same thing, from (4.13) and (4.18). On the other hand, Γ_{jqp} = 0 for q = p. Therefore we conclude that

(4.21)

P^{∗}Γ_{jqp}ext(e^{q})int(e^{p})\underline{e_{j}}P

= P^{∗}\{\frac{1}{2}Γ_{jjk}Γ_{kqp} − ib(e_{j})Γ_{jqp}\}ext(e^{q})int(e^{p})P

= 0.

Step 2. Calculation of P^{∗} ∑_{j<k} iγ^{j}γ^{k}db(e_{j}, e_{k})P. When k = 2ρ − 1 (ρ ≤ l), we have

γ^{k}P1 = 2^{− \frac{l}{2}} (e^{2ρ−1} + ie^{2ρ}) ∧ ⋀_{σ≠ρ}(1 + ie^{2σ−1} ∧ e^{2σ}) ∧ e^{2l+1} ∧ e^{2l+2} ∧ ⋯ ∧ e^{m}.

From (4.15), (4.16) and (4.18), we know that P^{∗}γ^{j}γ^{k}P = 0 for any j < k. When k = 2ρ (ρ ≤ l), we have

γ^{k}P1 = 2^{− \frac{l}{2}} \frac{e^{2ρ−1} + ie^{2ρ}}{i} ∧ ⋀_{σ≠ρ}(1 + ie^{2σ−1} ∧ e^{2σ}) ∧ e^{2l+1} ∧ ⋯ ∧ e^{m}.

From (4.15), (4.16) and (4.18), we know that P^{∗}γ^{j}γ^{k}P = 0 for any j < 2ρ − 1 and

(4.22) \begin{matrix}P^{∗}γ^{2ρ−1}γ^{2ρ}P1 \\ = P^{∗}2^{− \frac{l}{2}} \frac{1 + ie^{2ρ−1} ∧ e^{2ρ}}{i} ∧ ⋀_{σ≠ρ}(1 + ie^{2σ−1} ∧ e^{2σ}) ∧ e^{2l+1} ∧ ⋯ ∧ e^{m} \\ = \frac{1}{i}. \end{matrix}

If 2l < k ≤ m, then

γ^{k}P1 = (−1)^{k−1}2^{− \frac{l}{2}} ⋀_{σ=1}^{l}(1 + ie^{2σ−1} ∧ e^{2σ}) ∧ e^{2l+1} ∧ ⋯ ∧ e^{k−1} ∧ e^{k+1} ∧ ⋯ ∧ e^{m}

and so P^{∗}γ^{j}γ^{k}P = 0 for any j < k.

If m < k, then γ^{k}P1 = e^{k} ∧ P1 and so P^{∗}γ^{j}γ^{k}P = 0 for any j < k. Therefore we have

(4.23)  \begin{aligned}P^{∗} ∑_{j<k} iγ^{j}γ^{k}db(e_{j}, e_{k})P & = ∑_{σ=1}^{l} db(e_{2σ−1}, e_{2σ}) \\ & = − ∑_{σ=1}^{l} \frac{db(\overline{ϵ_{σ}}, ϵ_{σ})}{i}. \end{aligned}

Step 3. Calculation of P^{∗}Γ_{jqp}Γ_{jrs}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P (I). If the number of elements of the set {p, q, r, s} ∩ {1, 2, ⋯, 2l} is odd, then

(4.24)  P^{∗}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P = 0,

from (4.15), (4.16) and (4.18). Therefore, we decompose as follows:

(4.25)

P^{∗} ∑_{1≤p,q,r,s≤d} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P

= P^{∗} ∑_{p,q,r,s≤2l} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{p,q≤2l<r,s} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{p,r≤2l<q,s} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{p,s≤2l<q,r} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{q,r≤2l<p,s} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{q,s≤2l<p,r} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{r,s≤2l<p,q} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{2l<p,q,r,s} Γ_{jqp}Γ_{jsr}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P

=: I_{1} + I_{2} + I_{3} + I_{4} + I_{5} + I_{6} + I_{7} + I_{8}.

In I_{3}, the suffices for the exterior multiplication are greater than 2l but the suffices for the interior multiplication are less than or equal to 2l. Therefore I_{3} = 0.

Similarly, in I_{6}, the suffices for the interior multiplication are greater than 2l but the suffices for the exterior multiplication are less than or equal to 2l. Therefore I_{6} = 0.

In I_{2} and I_{7}, we may consider the operations of ext(e^{s})int(e^{r}) and ext(e^{q})int(e^{p}) separately. Therefore, by the same reason in Step 1, we have I_{2} = I_{7} = 0.

In I_{4}, from (4.13) and (4.18), we have

I_{4} = P^{∗} ∑_{\begin{matrix}p,s≤2l \\ q>m \end{matrix}} Γ_{jqp}Γ_{jsq}ext(e^{s})int(e^{p})P.

The same reason in Step 1 leads to

I_{4} = P^{∗} ∑_{\begin{matrix}p≤2l \\ q>m \end{matrix}} Γ_{jqp}Γ_{jpq}ext(e^{p})int(e^{p})P.

From (4.13), (4.17) and (4.18), we see that

(4.26)  P^{∗}ext(e^{p})int(e^{p})P = \frac{1}{2}

for p ≤ 2l. Therefore, we obtain

(4.27)  I_{4} = \frac{1}{2} ∑_{\begin{matrix}p≤2l \\ q>m \end{matrix}} Γ_{jqp}Γ_{jpq} = −\frac{1}{2} ∑_{\begin{matrix}p≤2l \\ q>m \end{matrix}}(Γ_{jqp})^{2}.

In I_{5}, from (4.13) and (4.18), we have

I_{5} = P^{∗} ∑_{\begin{matrix}q,r≤2l \\ 2l<p≤m \end{matrix}} Γ_{jqp}Γ_{jpr}int(e^{r})ext(e^{q})P.

As in Step 1, we have

I_{5} = P^{∗} ∑_{\begin{matrix}q≤2l \\ 2l<p≤m \end{matrix}} Γ_{jqp}Γ_{jpq}int(e^{q})ext(e^{q})P.

From (4.13), (4.14) and (4.18), we see that

(4.28)  P^{∗}int(e^{q})ext(e^{q})P = \frac{1}{2}

for p ≤ 2l. Therefore, we obtain

(4.29)  I_{5} = \frac{1}{2} ∑_{\begin{matrix}q≤2l \\ m<p≤m \end{matrix}} Γ_{jqp}Γ_{jpq} = −\frac{1}{2} ∑_{\begin{matrix}p≤2l \\ 2l<q≤m \end{matrix}}(Γ_{jqp})^{2}.

In I_{8}, by using (4.13), (4.18) and the fact that Γ_{jqp} = 0 for q = p, we have

(4.30)  I_{8} = ∑_{\begin{matrix}2l<p≤m \\ q>m \end{matrix}} Γ_{jqp}Γ_{jpq} = − ∑_{\begin{matrix}2l<p≤m \\ q>m \end{matrix}}(Γ_{jqp})^{2}.

We consider I_{1} in Step 6 below.

Step 4. Calculation of P^{∗}R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})P (I).

If the number of elements of the set {j, k, q, p} ∩ {1, 2, ⋯, 2l} is odd, then

(4.31)  P^{∗}γ^{j}γ^{k}ext(e^{q})int(e^{p})P = 0,

from (4.15), (4.16) and (4.18). Therefore, we decompose as follows:

(4.31)

P^{∗} ∑_{\begin{matrix}1≤j<k≤d \\ 1≤p,q≤d \end{matrix}} R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})P

= P^{∗} ∑_{\begin{matrix}j,k,q,p≤2l \\ j<k \end{matrix}} R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{j<k≤2l<q,p} R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{j,q≤2l<k,p} R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{j,p≤2l<k,q} R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{p,q≤2l<j<k} R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})P

+ P^{∗} ∑_{\begin{matrix}2l<j,k,q,p \\ j<k \end{matrix}} R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})P

=: J_{1} + J_{2} + J_{3} + J_{4} + J_{5} + J_{6}.

In J_{2} and J_{5}, we may consider the operations of γ^{j}γ^{k} and ext(e^{q})int(e^{p}) separately. Therefore, by the same reason in Step 1, we have J_{2} = J_{5} = 0.

In J_{3}, from (4.13) and (4.18), we have

(4.33)  J_{3} = −P^{∗} ∑_{\begin{matrix}j,q≤2l \\ 2l<p≤m \end{matrix}} R_{jpqp}γ^{j}ext(e^{q})P.

For q = 2ρ − 1 (ρ ≤ l), we have (4.34)

P^{∗}γ^{j}ext(e^{2ρ−1})P1 = \{ \begin{matrix}\frac{1}{2}, & if j = 2ρ − 1, from (4.14) and (4.18), \\ \frac{i}{2}, & if j = 2ρ, from (4.17) and (4.18), \\ 0, & otherwise , from (4.15), (4.16) and (4.18). \end{matrix}

For q = 2ρ (ρ ≤ l), we have (4.35)

P^{∗}γ^{j}ext(e^{2ρ})P1 = \{ \begin{matrix}−\frac{i}{2}, & if j = 2ρ − 1, from (4.17) and (4.18), \\ \frac{1}{2}, & if j = 2ρ, from (4.14) and (4.18), \\ 0, & otherwise , from (4.15), (4.16) and (4.18). \end{matrix}

Therefore, we obtain (4.36)

\begin{matrix}J_{3} = & − ∑_{\begin{matrix}ρ≤l \\ 2l<p≤m \end{matrix}} (\frac{1}{2}R_{2ρ−1,p,2ρ−1,p} + \frac{i}{2}R_{2ρ,p,2ρ−1,p} − \frac{i}{2}R_{2ρ−1,p,2ρ,p} + \frac{1}{2}R_{2ρ,p,2ρ,p}) \\ = & \frac{1}{2} ∑_{\begin{matrix}p≤2l \\ 2l<q≤m \end{matrix}} R_{pqqp}, \end{matrix}

since R_{2ρ,p,2ρ−1,p} = R_{2ρ−1,p,2ρ,p}.

In J_{4}, from (4.13) and (4.18), we have

J_{4} = P^{∗} ∑_{\begin{matrix}j,p≤2l \\ q>m \end{matrix}} R_{jqqp}γ^{j}int(e^{p})P.

For p = 2ρ − 1 (ρ ≤ l), we have (4.37)

P^{∗}γ^{j}int(e^{2ρ−1})P1 = \{ \begin{matrix}\frac{1}{2}, & if j = 2ρ − 1, from (4.17) and (4.18), \\ \frac{i}{2}, & if j = 2ρ, from (4.14) and (4.18), \\ 0, & otherwise , from (4.15), (4.16) and (4.18). \end{matrix}

For p = 2ρ (ρ ≤ l), we have

(4.38) P^{∗}γ^{j}int(e^{2ρ})P1 = \{ \begin{matrix}−\frac{i}{2}, & if j = 2ρ − 1, from (4.14) and (4.18), \\ \frac{1}{2}, & if j = 2ρ, from (4.17) and (4.18), \\ 0, & otherwise , from (4.15), (4.16) and (4.18). \end{matrix}

Therefore, we obtain

(4.39) \begin{matrix}J_{4} = & ∑_{\begin{matrix}ρ≤l \\ q>m \end{matrix}} (\frac{1}{2}R_{2ρ−1,q,q,2ρ−1} + \frac{i}{2}R_{2ρ,q,q,2ρ−1} − \frac{i}{2}R_{2ρ−1,q,q,2ρ} + \frac{1}{2}R_{2ρ,q,q,2ρ}) \\ = & \frac{1}{2} ∑_{\begin{matrix}p≤2l \\ q>m \end{matrix}} R_{pqqp}, \end{matrix}

since R_{2ρ,q,q,2ρ−1} = R_{2ρ−1,q,q,2ρ}.

In J_{6}, by using (4.13), (4.18) and the fact that R_{jkqp} = 0 for q = p, we have

(4.40)  J_{6} = ∑_{\begin{matrix}2l<p≤m \\ q>m \end{matrix}} R_{pqqp}.

We consider J_{1} in Step 7 below.

Step 5. Complex expression.

From now on, we may assume that d = 2l.

To calculate I_{1} and J_{1} in Steps 3 and 4, the unitary frame {ϵ_{σ} = (e_{2σ−1} − ie_{2σ})/\sqrt{2}\}_{σ=1}^{l} is useful. Let \{ϵ^{σ}\} and \{\overline{ϵ^{σ}}\} be the dual and conjugate dual basis, respectively. In this step, we express Γ_{jqp}ext(e^{q})int(e^{p}) in terms of {ϵ_{σ}} and give fundamental formulas with respect to the exterior and interior multiplication of \{ϵ^{σ}\} and \{\overline{ϵ^{σ}}\}. These provide us with useful tools in the next two steps.

Since {e_{j}} is orthonormal, Γ_{jqp} = 0 for q = p. For any j ∈ {1, 2, ⋯, d} and A, B ∈ \{1, 2, ⋯, l, ¯1, ¯2, ⋯, l̄\}, let Γ_{jAB}^{c} = (∇_{e_{j}}ϵ_{A}, ϵ_{B}), where ϵσ̄ := \overline{ϵ_{σ}}, σ = 1, 2, ⋯, d. Since

e_{2σ−1} = \frac{ϵ_{σ} + \overline{ϵ_{σ}}}{\sqrt{2}} and e_{2σ} = i\frac{ϵ_{σ} − \overline{ϵ_{σ}}}{\sqrt{2}} ,

we have

(4.41)

Γ_{j,2σ,2σ−1} =iΓ_{jσσ̄}^{c},
Γ_{j,2σ−1,2σ} = − iΓ_{jσσ̄}^{c},
Γ_{j,2ρ−1,2σ−1} =\frac{1}{2}(Γ_{jρσ}^{c} + Γ_{jρ̄σ}^{c} + Γ_{jρσ̄}^{c} + Γ_{jρ̄σ̄}^{c}),
Γ_{j,2ρ,2σ} =\frac{1}{2}(−Γ_{jρσ}^{c} + Γ_{jρ̄σ}^{c} + Γ_{jρσ̄}^{c} − Γ_{jρ̄σ̄}^{c}),
Γ_{j,2ρ,2σ−1} =\frac{i}{2}(Γ_{jρσ}^{c} − Γ_{jρ̄σ}^{c} + Γ_{jρσ̄}^{c} − Γ_{jρ̄σ̄}^{c}),
Γ_{j,2ρ−1,2σ} =\frac{i}{2}(Γ_{jρσ}^{c} + Γ_{jρ̄σ}^{c} − Γ_{jρσ̄}^{c} − Γ_{jρ̄σ̄}^{c}),

for σ ≠ ρ. Therefore, we have

∑_{p,q} Γ_{jqp}ext(e^{q})int(e^{p})

= ∑_{1≤σ≤l} Γ_{jσσ̄}^{c}i(ext(e^{2σ})int(e^{2σ−1}) − ext(e^{2σ−1})int(e^{2σ}))

+ ∑_{\begin{matrix}1≤σ,ρ≤l \\ σ≠ρ \end{matrix}}\{\frac{1}{2}Γ_{jρσ}^{c}(ext(e^{2ρ−1})int(e^{2σ−1}) − ext(e^{2ρ})int(e^{2σ})

+ iext(e^{2ρ})int(e^{2σ−1}) + iext(e^{2ρ−1})int(e^{2σ})) + \frac{1}{2}Γ_{jρ̄σ}^{c}(ext(e^{2ρ−1})int(e^{2σ−1}) + ext(e^{2ρ})int(e^{2σ})

− iext(e^{2ρ})int(e^{2σ−1}) + iext(e^{2ρ−1})int(e^{2σ})) + \frac{1}{2}Γ_{jρσ̄}^{c}(ext(e^{2ρ−1})int(e^{2σ−1}) + ext(e^{2ρ})int(e^{2σ})

+ iext(e^{2ρ})int(e^{2σ−1}) − iext(e^{2ρ−1})int(e^{2σ}))

+ \frac{1}{2}Γ_{jρ̄σ̄}^{c}(ext(e^{2ρ−1})int(e^{2σ−1}) − ext(e^{2ρ})int(e^{2σ})

− iext(e^{2ρ})int(e^{2σ−1}) − iext(e^{2ρ−1})int(e^{2σ}))}.

Since

e^{2σ−1} = \frac{ϵ^{σ} + \overline{ϵ^{σ}}}{\sqrt{2}} and e^{2σ} = \frac{ϵ^{σ} − \overline{ϵ^{σ}}}{\sqrt{2}i} ,

we have (4.42)

∑_{p,q} Γ_{jqp}ext(e^{q})int(e^{p})

= ∑_{1≤σ≤l} Γ_{jσσ̄}^{c}(ext(ϵ^{σ})int(\overline{ϵ^{σ}}) − ext(\overline{ϵ^{σ}})int(ϵ^{σ}))

+ ∑_{\begin{matrix}1≤σ,ρ≤l \\ σ≠ρ \end{matrix}} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}} Γ_{jBA}^{c}ext(ϵ^{B})int(ϵ^{A}),

where

ext(ω + iη) =ext(ω) + i ext(η)
int(ω + iη) =int(ω) + i int(η)

for ω, η ∈ T^{∗}M.

In the next two steps, the following is fundamental:

(4.43)  P = ∏_{σ=1}^{l} \frac{ext(\overline{ϵ^{σ}})ext(ϵ^{σ}) + 1}{\sqrt{2}}P_{0}

and

(4.44)  P^{∗} = P_{0} ∏_{σ=1}^{l} \frac{int(\overline{ϵ^{σ}})int(ϵ^{σ}) + 1}{\sqrt{2}}.

Furtermore, the operation of int(ϵ^{σ}) and int(\overline{ϵ^{σ}}) is as follows:

(4.45)  \begin{aligned}int(ϵ^{σ})ϵ^{ρ} & = 0, \\ int(\overline{ϵ^{σ}})ϵ^{ρ} & = δ_{σρ}, \end{aligned} \begin{aligned}int(ϵ^{σ})\overline{ϵ^{ρ}} & = δ_{σρ}, \\ int(\overline{ϵ^{σ}})\overline{ϵ^{ρ}} & = 0. \end{aligned}

Step 6. Calculation of P^{∗}Γ_{jqp}Γ_{jrs}ext(e^{s})int(e^{r})ext(e^{q})int(e^{p})P (II). We consider I_{1} in Step 3. I_{1} is written as

(4.46)  I_{1} = P^{∗} (∑_{p,q} Γ_{jqp}ext(e^{q})int(e^{p}))^{2} P.

Since we have

ext(ϵ^{σ})int(\overline{ϵ^{σ}})P1 = ext(\overline{ϵ^{σ}})int(ϵ^{σ})P1

from (4.43), we obtain

(4.47)  (ext(ϵ^{σ})int(\overline{ϵ^{σ}}) − ext(\overline{ϵ^{σ}})int(ϵ^{σ}))P = 0

and

(4.48)  P^{∗}(ext(ϵ^{σ})int(\overline{ϵ^{σ}}) − ext(\overline{ϵ^{σ}})int(ϵ^{σ})) = 0.

Therefore we have

(4.49)  I_{1} = P^{∗} ( ∑_{\begin{matrix}1≤σ,ρ≤l \\ σ≠ρ \end{matrix}} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}} Γ_{jBA}^{c}ext(ϵ^{B})int(ϵ^{A}))^{2} P,

from (4.42).

By using (4.43)-(4.45), we see that, if {σ, ρ} ≠ {σ′, ρ′},

(4.50)  P^{∗}ext(ϵ^{D})int(ϵ^{C})ext(ϵ^{B})int(ϵ^{A})P = 0

for A ∈ {σ, σ̄}, B ∈ {ρ, ρ̄}, C ∈ {σ′, σ̄′}, D ∈ {ρ′, ρ̄′}. From this, we have

(4.51) I_{1} = P^{∗} ∑_{1≤σ<ρ≤l} \{ ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}} Γ_{jBA}^{c}(ext(ϵ^{B})int(ϵ^{A}) − ext(ϵ^{B})int(ϵ^{A}))\}^{2} P.

Since we have

ext(ϵ^{ρ})int(ϵ^{σ})P1 = −ext(ϵ^{σ})int(ϵ^{ρ})P1,
ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}})P1 = −ext(\overline{ϵ^{σ}})int(\overline{ϵ^{ρ}})P1,
ext(\overline{ϵ^{ρ}})int(ϵ^{σ})P1 = ext(ϵ^{σ})int(\overline{ϵ^{ρ}})P1,
ext(ϵ^{ρ})int(\overline{ϵ^{σ}})P1 = ext(\overline{ϵ^{σ}})int(ϵ^{ρ})P1,

from (4.43)-(4.45), we obtain

(4.52) ext(ϵ^{ρ})int(ϵ^{σ})P = −ext(ϵ^{σ})int(ϵ^{ρ})P, ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}})P = −ext(\overline{ϵ^{σ}})int(\overline{ϵ^{ρ}})P, P^{∗}ext(ϵ^{ρ})int(ϵ^{σ}) = −P^{∗}ext(ϵ^{σ})int(ϵ^{ρ}), P^{∗}ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}}) = −P^{∗}ext(\overline{ϵ^{σ}})int(\overline{ϵ^{ρ}}), (ext(\overline{ϵ^{ρ}})int(ϵ^{σ}) − ext(ϵ^{σ})int(\overline{ϵ^{ρ}}))P = 0, (ext(ϵ^{ρ})int(\overline{ϵ^{σ}}) − ext(\overline{ϵ^{σ}})int(ϵ^{ρ}))P = 0, P^{∗}(ext(\overline{ϵ^{ρ}})int(ϵ^{σ}) − ext(ϵ^{σ})int(\overline{ϵ^{ρ}})) = 0, P^{∗}(ext(ϵ^{ρ})int(\overline{ϵ^{σ}}) − ext(\overline{ϵ^{σ}})int(ϵ^{ρ})) = 0.

Therefore, we have

(4.53)  I_{1} = P^{∗}4 ∑_{1≤σ<ρ≤l}\{Γ_{jρσ}^{c}ext(ϵ^{ρ})int(ϵ^{σ}) + Γ_{jρ̄σ̄}^{c}ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}})\}^{2}P.

Since

(ext(ϵ^{ρ})int(ϵ^{σ}))^{2} = (ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}}))^{2} = 0,

we have (4.54)

I_{1} =P^{∗}4 ∑_{1≤σ<ρ≤l} |Γ_{jρσ}^{c}|^{2}(ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}})ext(ϵ^{ρ})int(ϵ^{σ})
+ ext(ϵ^{ρ})int(ϵ^{σ})ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}}))P.

Finally, since we have

(4.55)

P^{∗}ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}})ext(ϵ^{ρ})int(ϵ^{σ})P = P^{∗}ext(ϵ^{ρ})int(ϵ^{σ})ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}})P

= −\frac{1}{4}

from (4.43)-(4.45), we obtain

(4.56)  I_{1} = −2 ∑_{1≤σ<ρ≤l} |Γ_{jρσ}^{c}|^{2}.

Step 7. Calculation of P^{∗}R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})P (II).

We consider J_{1} in Step 4. By the same procedure in Step 5, we have

(4.57)

∑_{\begin{matrix}j<k≤2l \\ q,p≤l \end{matrix}} R_{jkqp}γ^{j}γ^{k}ext(e^{q})int(e^{p})

= ∑_{\begin{matrix}j<k≤2l \\ 1≤σ≤l \end{matrix}}(R(e_{j}, e_{k})ϵ_{σ}, \overline{ϵ_{σ}})γ^{j}γ^{k}(ext(ϵ^{σ})int(\overline{ϵ^{σ}}) − ext(\overline{ϵ^{σ}})int(ϵ^{σ}))

+ ∑_{\begin{matrix}j<k≤2l \\ 1≤σ,ρ≤l \\ σ≠ρ \end{matrix}} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}}(R(e_{j}, e_{k})ϵ_{B}, ϵ_{A})γ^{j}γ^{k}ext(ϵ^{B})int(ϵ^{A}).

From (4.47), we have

(4.58)

J_{1} = P^{∗} ∑_{\begin{matrix}j<k≤2l \\ 1≤σ,ρ≤l \\ σ≠ρ \end{matrix}} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}}(R(e_{j}, e_{k})ϵ_{B}, ϵ_{A})γ^{j}γ^{k}ext(ϵ^{B})int(ϵ^{A})P

= P^{∗} \frac{1}{2} ∑_{\begin{matrix}j,k≤2l \\ 1≤σ,ρ≤l \\ σ≠ρ \end{matrix}} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}}(R(e_{j}, e_{k})ϵ_{B}, ϵ_{A})γ^{j}γ^{k}ext(ϵ^{B})int(ϵ^{A})P.

We now define δ^{σ} and \overline{δ^{σ}} by

δ^{σ} = ext(ϵ^{σ}) + int(ϵ^{σ}) = \frac{γ^{2σ−1} − iγ^{2σ}}{\sqrt{2}} ,
\overline{δ^{σ}} = ext(\overline{ϵ^{σ}}) + int(\overline{ϵ^{σ}}) = \frac{γ^{2σ−1} + iγ^{2σ}}{\sqrt{2}} .

For any A, B, C, D ∈ {1, 2, ⋯, l, ¯1, ¯2, ⋯, l̄}, let R_{ABCD}^{c} = (R(ϵ_{A}, ϵ_{B})ϵ_{C}, ϵ_{D}). Then, by the same procedure in Step 5, we have

(4.59)

J_{1} =P^{∗} \frac{1}{2} ∑_{\begin{matrix}σ,ρ,σ′≤l \\ σ≠ρ \end{matrix}} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}} R_{σ′σ′BA}^{c}(δ^{σ′} \overline{δ^{σ′}} − \overline{δ^{σ′}}δ^{σ′})ext(ϵ^{B})int(ϵ^{A})P

+ P^{∗} \frac{1}{2} ∑_{\begin{matrix}σ,ρ,σ′,ρ′≤l \\ σ≠ρ \\ σ′≠ρ′ \end{matrix}} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \\ C∈\{σ′,σ̄′\} \\ D∈\{ρ′,ρ̄′\} \end{matrix}} R_{DCBA}^{c}δ^{D}δ^{C}ext(ϵ^{B})int(ϵ^{A})P

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

By using (4.43) and (4.45), we have

(4.60)  \overline{δ^{σ}}δ^{σ}P = 2P, \overline{δ^{σ}}P = 0,

and so

(4.61)  P^{∗}\overline{δ^{σ}}δ^{σ} = 2P^{∗}, P^{∗}δ^{σ} = 0.

From this, we have

(4.62)  J_{11} = −P^{∗} ∑_{σ,ρ,σ′≤l} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}} R_{σ′σ̄′BA}^{c}ext(ϵ^{B})int(ϵ^{A})P.

Then, as in Step 1, we see that

(4.63)  J_{11} = 0.

By using (4.61) and the fact that \overline{δ^{σ}}δ^{ρ} = −δ^{ρ}\overline{δ^{σ}} for σ ≠ ρ, we have

(4.64)  J_{12} = P^{∗} \frac{1}{2} ∑_{\begin{matrix}σ,ρ,σ′,ρ′≤l \\ σ≠ρ \\ σ′≠ρ′ \end{matrix}} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}} R_{ρ̄′σ̄′BA}^{c}\overline{δ^{ρ′}} \overline{δ^{σ′}}ext(ϵ^{B})int(ϵ^{A})P.

As in (4.50), we see that, if {σ, ρ} ≠ {σ′, ρ′} as sets,

(4.65)  P^{∗}δ^{D}δ^{C}ext(ϵ^{B})int(ϵ^{A})P = 0

for A ∈ {σ, σ̄}, B ∈ {ρ, ρ̄}, C ∈ {σ′, σ̄′} and D ∈ {ρ′, ρ̄′}. From this and symmetricity of R(\overline{ϵ_{σ′}}, \overline{ϵ_{ρ′}})\overline{δ^{ρ′}} \overline{δ^{σ′}} in (ρ′, σ′), we have

(4.66)  J_{12} = P^{∗} ∑_{\begin{matrix}1≤σ,ρ≤l \\ σ≠ρ \end{matrix}} ∑_{\begin{matrix}A∈\{σ,σ̄\} \\ B∈\{ρ,ρ̄\} \end{matrix}} R_{σ̄ρ̄BA}^{c}\overline{δ^{σ}} \overline{δ^{ρ}}ext(ϵ^{B})int(ϵ^{A})P.

By using antisymmetricity of curvature tensors, this is rewritten as

(4.67)

J_{12} =P^{∗}2 ∑_{1≤σ,ρ≤l} \overline{δ^{σ}} \overline{δ^{ρ}}\{R_{σ̄ρ̄ρσ}^{c}(ext(ϵ^{ρ})int(ϵ^{σ}) − ext(ϵ^{σ})int(ϵ^{ρ}))
+ R_{σ̄ρ̄ρ̄σ̄}^{c}(ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}}) − ext(\overline{ϵ^{σ}})int(\overline{ϵ^{ρ}}))
+ R_{σ̄ρ̄ρ̄σ}^{c}(ext(\overline{ϵ^{ρ}})int(ϵ^{σ}) − ext(ϵ^{σ})int(\overline{ϵ^{ρ}}))
+ R_{σ̄ρ̄ρσ̄}^{c}(ext(ϵ^{ρ})int(\overline{ϵ^{σ}}) − ext(\overline{ϵ^{σ}})int(ϵ^{ρ}))\}P.

By using (4.52), we have

(4.68)

J_{12} =P^{∗}4 ∑_{1≤σ,ρ≤l} \overline{δ^{σ}} \overline{δ^{ρ}}\{R_{σ̄ρ̄ρσ}^{c}ext(ϵ^{ρ})int(ϵ^{σ})
+ R_{σ̄ρ̄ρ̄σ̄}^{c}ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}})\}P.

Finally, by using (4.43)-(4.45), we see that

(4.69)

P^{∗}\overline{δ^{σ}} \overline{δ^{ρ}}ext(ϵ^{ρ})int(ϵ^{σ})P = 1,
P^{∗}\overline{δ^{σ}} \overline{δ^{ρ}}ext(\overline{ϵ^{ρ}})int(\overline{ϵ^{σ}})P = 0.

Therefore we obtain

(4.70)  J_{12} = 4 ∑_{1≤σ,ρ≤l} R_{σ̄ρ̄ρσ}^{c}.

Finally, we prove Theorem 2.1 by using Theorem 4.1.

Proof of Theorem 2.1. We set R = ∑_{j=1}^{3} R_{j}−B. Let (·, ·)_{L^{2}(M)} and (·, ·)_{L^{2}(M;Λ(T^{∗}M)⊗ℂ)} be the inner product of L^{2}(M) and L^{2}(M; Λ(T^{∗}M) ⊗ ℂ), respectively, where L^{2}(M; Λ(T^{∗}M) ⊗ ℂ) is the space of all complex coefficient square integrable forms on M. For any φ ∈ C_{0}^{2}(M^{∘}), we have

(4.71)  \begin{matrix}((L(b) + R)φ, φ)_{L^{2}(M)} \\ = ((L(b) + R)φ, φ)_{L^{2}(M;Λ(T^{∗}M)⊗ℂ)} \\ = (P^{∗}Q^{2}Pφ, φ)_{L^{2}(M;Λ(T^{∗}M)⊗ℂ)} \\ = ∥QPφ∥_{L^{2}(M;Λ(T^{∗}M)⊗ℂ)}^{2} \\ ≥ 0, \end{matrix}

where ∥ · ∥_{L^{2}(M;Λ(T^{∗}M)⊗ℂ)}^{2} is the norm of L^{2}(M; Λ(T^{∗}M) ⊗ ℂ). Since C_{0}^{2}(M^{∘}) is dence in L^{2}(M), we see that L(b) + R is a positive operator. Remark 4.1. In (4.71), it is essential that the support of φ does not intersect the boundary of M. Therefore our proof works only for the operator with Dirichlet boundary condition. Remark 4.2. In the proof of Theorem 2.1, we use only the positivity of the Pauli Hamiltonian Q^{2}. However, if inf spec Q^{2} is greater than zero, then Theorem 2.1 is improved as follows: L(b) + ∑_{j=1}^{3} R_{j} − B − inf spec Q^{2} is a positive operator. In particular, we have

(4.72)  inf spec L(b) ≥ inf_{x∈M}{− ∑_{j=1}^{3} R_{j}(x) + B(x)} + inf spec Q^{2}.

5. Applications to asymptotics of a stochastic oscillatory integral and transverse analyticity

In this section, we apply Theorem 2.1 to studies of asymptotic behavior of an oscillatory integral and transverse analyticity.

For any x ∈ M, there is an orthonormal basis [e_{1,x}, e_{2,x}, ⋯, e_{d,x}] of T_{x}M such that

(5.1)  \begin{matrix}(db(e_{j,x}, e_{k,x}))_{1≤j,k≤2l} \\ = ( \begin{matrix}0 & −β_{1}(x) \\ β_{1}(x) & 0 \end{matrix}  0 ; ... ; \begin{matrix}0 & −β_{l(x)}(x) \\ β_{l(x)}(x) & 0 \end{matrix} ; 0 ; ... ; 0  0 ) , \end{matrix}

where β_{σ}(x) > 0 for 1 ≤ σ ≤ l(x). We define a homomorphism J_{x} : T_{x}M → T_{x}M by

(5.2)  \begin{aligned}J_{x}e_{2σ−1,x} & = e_{2σ,x}, \\ J_{x}e_{2σ,x} & = −e_{2σ−1,x}, \\ J_{x}e_{j,x} & = 0, \end{aligned} \begin{aligned}σ & ≤ l(x), \\ σ & ≤ l(x), \\ j & ≥ l(x). \end{aligned}

This definition of J_{x} is independent of the choice of the basis {e_{j,x}} as above. We need not consider the continuity of J_{x} in x ∈ M.

In this section, we assume the following:

(i) sup_{r=(x,e)∈O(M)}(R(e_{1}, e_{2})e_{2}, e_{1}) < ∞, (A.2)

(ii) sup_{r∈O(M)}(R(J_{x}e_{1}, e_{2})J_{x}e_{2}, e_{1}) < ∞,

(A.3) O(M) ∋ r ⟼ db(e_{1}, e_{2}) ∈ ℝ is uniformly continuous, where O(M) is the orthonormal frame bundle of T M, i.e.,

O(M) = {r = (x, e); e = [e_{1}, e_{2}, ⋯, e_{d}] is an orthonormal basis of T_{x}M}.

We can give O(M) a natural Riemannian structure.

Then we obtain the following: Theorem 5.1. Under (A.2) and (A.3), it holds that

(5.3) \underline{lim}_{ξ→∞} \frac{inf spec L(ξb)}{ξ} ≥ \frac{1}{4} inf_{x∈M} ∥db(x)∥_{1}.

This type of theorem is proven by Malliavin ([13] for compact manifolds, [14] for Euclidean spaces. cf. also [5]). In [13] and [14], the right hand side of (5.3) is slightly different: the constant and the norm are different. However, the estimate (5.3) is best possible when we consider a uniform magnetic field on ℝ^{d}.

The conclusion (5.3) of this theorem is weaker than the conclusions of Section 2 in some cases. In fact, in the circumstances of Section 2, we assume that sup_{x∈M} R_{j}(x) < ∞ for j = 1, 2, 3. Then we easily obtain

(5.4) \underline{lim}_{ξ→∞} \frac{inf spec L(ξb)}{ξ} ≥ inf_{x∈M} B(x)

from Theorem 2.1, without (A.3). Proof of Theorem 5.1. It is enough to show that

(5.5) \underline{lim}_{ℕ∋n↗∞} \frac{inf spec L(nb)}{n} ≥ \frac{1}{4} inf_{x∈M} ∥db(x)∥_{1}.

For any bounded operator A on any Banach space B, we denote the operator norm by ∥A∥_{End(B)}. Then we have

(5.6)  inf spec L(nb) = − log ∥e^{−L(nb)}∥_{End(L^{2}(M))},

where e^{−tL(nb)}, t > 0, is the semigroup generated by L(nb). We put L^{n}(b) := L(nb)/n. Then we have

(5.7)  \frac{1}{n} log ∥e^{−L(nb)}∥_{End(L^{2}(M))} ≤ log ∥e^{−L^{n}(b)}∥_{End(L^{2}(M))}.

Therefore the proof of Thorem 5.1 is reduced to Lemma 5.1 below.

Lemma 5.1. Under (A.2) and (A.3), it holds that

(5.8)  \overline{lim}_{ℕ∋n↗∞} log ∥e^{−L^{n}(b)}∥_{End(L^{2}(M))} ≤ −\frac{1}{4} inf_{x∈M} ∥db(x)∥_{1}.

Proof. Since M is σ-compact, the set C_{0}(M) of all continuous functions with compact support is dence in L^{2}(M). Therefore, we have

(5.9)

\overline{lim}_{n↗∞} log ∥e^{−L^{n}(b)}∥_{End(L^{2}(M))}

= \overline{lim}_{n↗∞} log inf_{\begin{matrix}φ∈C_{0}(M) \\ ∥φ∥_{L^{2}(M)}=1 \end{matrix}} ∥e^{−L^{n}(b)}φ∥_{L^{2}(M)}

≤ inf_{\begin{matrix}φ∈C_{0}(M) \\ ∥φ∥_{L^{2}(M)}=1 \end{matrix}} log \overline{lim}_{n↗∞} ∥e^{−L^{n}(b)}φ∥_{L^{2}(M)}.

Let X(t, x), t > 0, x ∈ M be the absorbing barrier Brownian motion on M, 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. It is well known that e^{−L^{n}(b)}φ has the folowing representation:

(5.10) (e^{−L^{n}(b)}φ)(x) = E[ exp (−in ∫0^{\frac{t}{n}} b(X(s, x)) ∘ dX(s, x)) φ(X(\frac{t}{n}, x))],

where E[·] is the expectation with respect to X(·, x) and ∫0^{\frac{t}{n}} b(X(s, x)) ∘ dX(s, x) is a stochastic line integral of the 1-form b (see e.g. [11] and [19]).

For any δ > 0 and x ∈ M, there is an open set U_{δ}(x) of M satisfying the following:

(i) there is an orthonormal frame [e_{1,x}(y), e_{2,x}(y), ⋯, e_{d,x}(y)]_{y∈U_{δ}(x)} of T M

on U_{δ}(x) such that
(5.11)  \begin{matrix}[e_{1,x}(x), e_{2,x}(x), ⋯, e_{d,x}(x)] = [e_{1,x}, e_{2,x}, ⋯, e_{d,x}], \\ |(R(e_{p,x}(y), e_{q,x}(y))e_{r,x}(y), e_{s,x}(y)) − (R(e_{p,x}, e_{q,x})e_{r,x}, e_{s,x})| < 1, \\ |(∇_{e_{p,x}(y)}e_{q,x}(y), e_{r,x}(y))| < 1, \end{matrix}

for any 1 ≤ p, q, r, s ≤ d and y ∈ U_{δ}(x), where [e_{1,x}, e_{2,x}, ⋯, e_{d,x}] is the base taken at (5.1).

(ii) d(x, y) < δ for any y ∈ U_{δ}(x), where d(x, y) is the Riemannian

distance between x and y. (iii) U_{δ}(x) is a domain surrounded by finite number of C^{2} hypersurfaces.

Since M is σ-compact, there is a sequence {x_{ι}}_{ι=1}^{∞} of points of M such that ⋃_{ι=1}^{∞} U_{δ}(x_{ι}) = M. For any ι ∈ ℕ, we set

(5.12)  V_{δ,ι} := U_{δ}(x_{ι}) − ⋃_{κ=1}^{ι} U_{δ}(x_{κ}).

Then V_{δ,ι} is a finite union of sets surrounded by finite number of C^{2} hypersurfaces and M is a disjoint union of V_{δ,ι}, ι ∈ ℕ. We devide the integral (5.10) as follows: when x ∈ V_{δ,ι},

(5.13)

(e^{−L^{n}(b)}φ)(x)

= E[ exp (−in ∫0^{\frac{t}{n}} b(X(s, x)) ∘ dX(s, x)) φ(X(\frac{t}{n}, x))

; X(s, x) ∈ V_{δ,ι} for any 0 ≤ s ≤ \frac{t}{n}]

+ E[ exp (−in ∫0^{\frac{t}{n}} b(X(s, x)) ∘ dX(s, x)) φ(X(\frac{t}{n}, x))

; X(s, x) ∉ V_{δ,ι} for some 0 ≤ s ≤ \frac{t}{n}].

The first term is a representation of

(e^{−L^{n}(b)_{V_{δ,ι}}} φ)(x),

where L^{n}(b)_{V_{δ,ι}} is the Friedrichs extension of the restriction of L^{n}(b) to the set C_{0}^{2}(V_{δ,ι}^{∘}) of all C^{2} functions on the interior V_{δ,ι}^{∘} with compact support. The second term is dominated by

E[|φ(X(\frac{t}{n}, x))|; X(s, x) ∉ V_{δ,ι} for some 0 ≤ s ≤ \frac{t}{n}].

Then we have

(5.14)

\overline{lim}_{n↗∞} ∥e^{−L^{n}(b)}φ∥_{L^{2}(M)}

= \overline{lim}_{n↗∞} (∑_{ι=1}^{∞} ∥e^{−L^{n}(b)_{V_{δ,ι}}} φ∥_{L^{2}(V_{δ,ι})}^{2})^{1/2}

+ ( \overline{lim}_{n↗∞} ∑_{ι=1}^{∞} ∫_{V_{δ,ι}} E[|φ(X(\frac{t}{n}, x))|

; X(s, x) ∉ V_{δ,ι} for some 0 ≤ s ≤ \frac{t}{n}]^{2}dvol(x))^{1/2},

where dvol(x) is the Riemannian volume.

We now consider the second term of the right hand side of (5.14). By using Hölder’s inequality,

(5.15)

∑_{ι=1}^{∞} ∫_{V_{δ,ι}} E[|φ(X(\frac{t}{n}, x))|

; X(s, x) ∉ V_{δ,ι} for some 0 ≤ s ≤ \frac{t}{n}]^{2}dvol(x)

≤ ∑_{ι=1}^{∞} ∫_{V_{δ,ι}} E[|φ(X(\frac{t}{n}, x))|^{2}]

× P (X(s, x) ∉ V_{δ,ι} for some 0 ≤ s ≤ \frac{t}{n}) dvol(x)

≤ ∥φ∥_{∞}^{2}\{ ∫_{M}(e^{\frac{tΔ}{2n}} χ_{supp φ})(x)dvol(x)

− ∑_{ι=1}^{∞} ∫_{V_{δ,ι}}(e^{\frac{tΔ_{V_{δ,ι}}}{2n}} χ_{supp φ}|_{V_{δ,ι}})(x)dvol(x)\},

where P (·) is the probability with respect to X(·, x), ∥φ∥_{∞} = sup_{x∈M} |φ(x)|, χ_{supp φ} is the defining function of the support supp φ of the function φ, and Δ_{V_{δ,ι}} = L(0)_{V_{δ,ι}}.

The second factor of the right hand side of (5.15) is rewritten as follows:

(5.16)

∫_{M}(e^{\frac{tΔ}{2n}} χ_{supp φ})(x)dvol(x)

− ∑_{ι=1}^{∞} ∫_{V_{δ,ι}}(e^{\frac{tΔ_{V_{δ,ι}}}{2n}} χ_{supp φ}|_{V_{δ,ι}})(x)dvol(x)

= ∑_{ι=1}^{∞} ∫_{V_{δ,ι}∩ supp φ} P (X(s, x) ∉ V_{δ,ι} for some 0 ≤ s ≤ \frac{t}{n}

and X(\frac{t}{n}, x) ∈ M)dvol(x).

In fact, since V_{δ,ι} is relatively compact, we have

(5.17)

∫_{V_{δ,ι}}(e^{\frac{tΔ_{V_{δ,ι}}}{2n}} χ_{supp φ}|_{V_{δ,ι}})(x)dvol(x) = ∫_{V_{δ,ι}} χ_{supp φ}(x)(e^{\frac{tΔ_{V_{δ,ι}}}{2n}} 1)(x)dvol(x)

= ∫_{V_{δ,ι}∩ supp φ} P (X(s, x) ∈ V_{δ,ι} for any 0 ≤ s ≤ \frac{t}{n})dvol(x).

On the other hand, though M is not assumed to be compact, we take an increasing sequence {K_{j}}_{j=1}^{∞} of compact sets of M such that ⋃_{j=1}^{∞} K_{j} = M. Then we use the fact that χ_{K_{j}} ∈ L^{2}(M) and the monotone converjence theorem to obtain

(5.18)

∫_{M}(e^{\frac{tΔ}{2n}} χ_{supp φ})(x)dvol(x)

= lim_{j→∞} ∫_{M}(e^{\frac{tΔ}{2n}} χ_{supp φ})(x)χ_{K_{j}}(x)dvol(x) = lim_{j→∞} ∫_{M} χ_{supp φ}(x)(e^{\frac{tΔ}{2n}} χ_{K_{j}})(x)dvol(x)

= lim_{j→∞} ∫_{supp φ} P (X(\frac{t}{n}, x) ∈ K_{j})dvol(x)

= ∫_{supp φ} P (X(\frac{t}{n}, x) ∈ M)dvol(x).

Form (5.17) and (5.18), we have (5.16).

For the right hand side of (5.16), we easily have

(5.19)

\overline{lim}_{n→∞} ∑_{ι=1}^{∞} ∫_{V_{δ,ι}∩ supp φ} P (X(s, x) ∉ V_{δ,ι} for some 0 ≤ s ≤ \frac{t}{n}

and X(\frac{t}{n}, x) ∈ M)dvol(x)

= 0, by using bounded convergence theorem, since supp φ is compact. Therefore we conclude that

(5.20)

\overline{lim}_{n↗∞} ∑_{ι=1}^{∞} ∫_{V_{δ,ι}} E[|φ(X(\frac{t}{n}, x))|

; X(s, x) ∉ V_{δ,ι} for some 0 ≤ s ≤ \frac{t}{n}]^{2}dvol(x)

= 0.

On the other hand, for any φ ∈ C_{0}(M) such that ∥φ∥_{L^{2}(M)} = 1, we have

(5.21)

(∑_{ι=1}^{∞} ∥e^{−L^{n}(b)_{V_{δ,ι}}} φ∥_{L^{2}(V_{δ,ι})}^{2})^{1/2}

≤ (∑_{ι=1}^{∞} exp ( − 2 inf spec L^{n}(b)_{V_{δ,ι}})∥φ∥_{L^{2}(V_{δ,ι})}^{2})^{1/2}

≤ exp ( inf_{1≤ι<∞} inf spec L^{n}(b)_{V_{δ,ι}})

≤ exp ( inf_{1≤ι<∞} inf spec L^{n}(b)_{U_{δ}(x_{ι})}).

On U_{δ}(x_{ι}), let U_{ι} (resp. W_{ι}) be the Riemannian vector bundle such that [e_{1,x_{ι}}(y), e_{2,x_{ι}}(y), ⋯, e_{2l,x_{ι}}(y)] (resp. [e_{2l+1,x_{ι}}(y), e_{2l+2,x_{ι}}(y), ⋯, e_{d,x_{ι}}(y)]) is the global orthonormal frame. We define J_{ι} : U_{ι} → U_{ι} by J_{ι}e_{2σ−1,x} = e_{2σ,x}, J_{ι}e_{2σ,x} = −e_{2σ−1,x}, 1 ≤ σ ≤ l. In this situation, we apply Theorem 2.1 and we have

(5.22) inf spec L^{n}(b)_{U_{δ}(x_{ι})}

≥ \frac{1}{n} inf_{y∈U_{ι}} (− ∑_{j=1}^{2} R_{j}^{ι}(y) + nB^{ι}(y))

≥ \frac{−1}{n} sup_{y∈U_{ι}} ∑_{j=1}^{2} R_{j}^{ι}(y) + inf_{y∈U_{ι}} B^{ι}(y),

where R_{j}^{ι} and B^{ι} are the corresponding R_{j} and B. From the definition of U_{ι}, we have

(5.23)  sup_{1≤ι<∞} sup_{y∈U_{ι}} ∑_{j=1}^{2} R_{j}^{ι}(y) < ∞.

Therefore, for any φ ∈ C_{0}(M) such that ∥φ∥_{L^{2}(M)} = 1, we have

(5.24) \overline{lim}_{n↗∞} (∑_{ι=1}^{∞} ∥e^{−L^{n}(b)_{V_{δ,ι}}} φ∥_{L^{2}(V_{δ,ι})}^{2})^{1/2} ≤ exp (− inf_{1≤ι<∞} inf_{y∈U_{δ}(x_{ι})} B^{ι}(y)) .

From (5.9), (5.14), (5.20) and (5.24), we obtain

(5.25)  \overline{lim}_{n↗∞} log ∥e^{−L^{n}(b)}∥_{End(L^{2}(M))} ≤ − inf_{1≤ι<∞} inf_{y∈U_{δ}(x_{ι})} B^{ι}(y).

Since δ > 0 is arbitrary, by using (A.3) and the fact B^{ι}(x_{ι}) = ∥db(x_{ι})∥_{1}/4, we obtain (5.8).

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

(e^{−tL(ξb)}φ)(x) = ∫_{M} e^{−tL(ξb)}(x, y)φ(y)dvol(y).

From Theorem 5.1, we obtain a result on asymptotic behavior of e^{−tL(ξb)}(x, y): Corollary 1. Under (A.2) and (A.3), it holds that

(5.26)  \overline{lim}_{ξ↗∞} \frac{1}{ξ} log |e^{−tL(ξb)}(x, y)| ≤ −\frac{t}{4} inf_{x∈M} ∥db(x)∥_{1}

for any x, y ∈ M.

As in (5.10), the heat kernel has the following representation:

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

where e^{\frac{tΔ}{2}} (x, y) is the integral kernel of the semigroup generated by Δ/2. Therefore Corollary 1 is a general statement on the upper estimate of an asymptotic behaviour of a stochastic oscillatory integral for which the phase function is a stochastic line integral (see e.g. [9]).

Proof of Corollary 1. We take δ > 0 sufficiently small. By using the semigroup property, we have

(5.28)

e^{−tL(ξb)}(x, y)
= ∫_{M} ∫_{M} e^{−δL(ξb)}(x, x′)e^{−(t−2δ)L(ξb)}(x′, y′)
× e^{−δL(ξb)}(y′, y)dvol(x′)dvol(y′)
= (e^{−δL(ξb)}(x, ·), \overline{e^{−(t−2δ)L(ξb)}(e^{−δL(ξb)}(·, y))(·)})_{L^{2}(M)} .

This is estimated as follows:

(5.29)  \begin{matrix}|e^{−tL(ξb)}(x, y)| \\ ≤ ∥e^{−δL(ξb)}(x, ·)∥_{L^{2}(M)} \\ × ∥e^{−(t−2δ)L(ξb)}(e^{−δL(ξb)}(·, y))(·)∥_{L^{2}(M)} \\ ≤ ∥e^{−δL(ξb)}(x, ·)∥_{L^{2}(M)} ∥e^{−(t−2δ)L(ξb)}∥_{End(L^{2}(M))} \\ × ∥e^{−δL(ξb)}(·, y)∥_{L^{2}(M)} . \end{matrix}

By using (5.27) and the symmetricity and the semigroup property of e^{\frac{tΔ}{2}} (x, y), we have

(5.30)  \begin{matrix}∥e^{−δL(ξb)}(x, ·)∥_{L^{2}(M)}^{2} = ∥e^{−δL(ξb)}(·, x)∥_{L^{2}(M)}^{2} \\ ≤ ∫_{M} e^{\frac{δΔ}{2}} (x, x′)^{2}dvol(x′) \\ = e^{δΔ}(x, x). \end{matrix}

Using also

∥e^{−(t−2δ)L(ξb)}∥_{End(L^{2}(M))} = exp (−(t − 2δ) inf spec L(ξb)) ,

we have

(5.31)

log |e^{−tL(ξb)}(x, y)|

≤ \frac{1}{2} log e^{δΔ}(x, x) + \frac{1}{2} log e^{δΔ}(y, y)

− (t − 2δ) inf spec L(ξb).

If we take δ = 1/ξ, we have

(5.32)

\overline{lim}_{ξ↗∞} \frac{1}{ξ} log |e^{−tL(ξb)}(x, y)|

≤ \frac{1}{2} \overline{lim}_{ξ↗∞} \frac{1}{ξ} log e^{\frac{Δ}{ξ}} (x, x) + \frac{1}{2} \overline{lim}_{ξ↗∞} \frac{1}{ξ} log e^{\frac{Δ}{ξ}} (y, y)

− \underline{lim}_{ξ↗∞} \frac{1}{ξ} (t − \frac{2}{ξ}) inf spec L(ξb).

It is well known that the first and the second terms are zero (cf.e.g. [17]). Therefore, we can conclude the proof by using Theorem 5.1.

The formula (5.27) says that 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). Consequently, Corollary 1 leads the following: Corollary 2. We assume that (A.2) and (A.3) holds and

inf_{x∈M} ∥db(x)∥_{1} > 0.

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

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.

The radius of convergence of its Taylor series around any point is greater than or equal to t inf_{x∈M} ∥db(x)∥_{1}/4.

This type of result is known as transverse analyticity, which has been studied by Malliavin [13],[14],[15], and Prat [18].

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