Spectral Analysis of Schrödinger Operators with Magnetic Fields

Hiroyuki Matsumoto^{1} and Naomasa Ueki^{2}

Department of Mathematics, Faculty of General Education, Gifu University, Yanagido, Gifu 501–11, Japan, and Department of Mathematics, Faculty of Science, Himeji Institute of Technology, Shosha 2167, Himeji 671-22, Japan

Abstract. The spectra of quadratic Schrödinger operators in general dimensional Euclidean spaces are determined by considering the corresponding classical mechanics and the corresponding Dirac operators. The results are applied to the study on the semiclassical approximation of the eigenvalues of Schrödinger operators with magnetic fields.

1. Introduction

In this paper we will first be concerned with the Schrödinger operator

H_{(B,K)} = \frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}∂_{j} − (Bx)_{j}/2)^{2} + \frac{1}{2}⟨x, Kx⟩, ∂_{j} = \frac{∂}{∂x_{j}} , (1.1)

defined on the Hilbert space L^{2}(R^{d}) = L^{2}(R^{d}, dx), where B is a real skew-symmetric d×d matrix, (Bx)_{j} is the j-th component of Bx ∈ R^{d}, K is a real symmetric non-negative definite matrix, ⟨·, ·⟩ is the inner product of R^{d} and dx is the Lebesgue measure. H_{(B,K)} is quadratic in the sense that the corresponding classical Hamiltonian, which is obtained by setting p_{j} = ∂_{j}/\sqrt{−1}, is a homogeneous quadratic polynomial in (x, p) ∈ R^{d} × R^{d}.

1E-mail: matsu@cc.gifu-u.ac.jp

2E-mail: ueki@sci.himeji-tech.ac.jp

If B = 0, H_{(0,K)} is the usual harmonic oscillator and, if K = 0, H_{(B,0)} is the Schrödinger operator with a uniform magnetic field. The properties of these particular operators are well known.

For the general case, when d = 2 or 3 and K is nondegenerate, Davies [2] and the first author [17] determined the eigenvalues by using the Van Vleck-Pauli formula (cf. [4], [13] and [28]). Moreover, when d = 2, the first author [18] has given explicitly the eigenfunctions, which are related to the Hermite and the complex Hermite functions, by some direct calculations and he has also determined the spectral properties when K is degenerate.

In this paper we will study the operator H_{(B,K)} in the above general setting by two systematic methods. We will firstly represent it in terms of the creation and annihilation operators by referring to the corresponding Dirac operator. By this representation, we can give all of the eigenvalues and the eigenfunctions when K is nondegenerate. Secondly we will use the metaplectic representation of the real symplectic group (cf.[5]). We will show that the classical mechanics corresponding to H_{(B,K)} is equivalent to that corresponding to a harmonic oscillator by some symplectic transform. This implies the unitary equivalence of the Schrödinger operators and we can determine the spectra of H_{(B,K)} completely including the case when K is degenerate and the spectrum is not discrete.

We will apply the study of H_{(B,K)} to discuss the semiclassical approximation of the low lying eigenvalues. We will consider the Schrödinger operator with a magnetic field

H(ℏ) = \frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}ℏ∂_{j} − A_{j})^{2} + V + ℏv.  (1.2)

where V, v, A_{j}, j = 1, 2, ..., d, are the multiplication operators defined by the smooth functions and ℏ is the Planck constant. We will assume that V has nondegenerate or degenerate zeros (see Sections 7, 8 and 11). Then we will show that the asymptotics of the eigenvalues of H(ℏ) below the essential spectrum as ℏ ↓ 0 are determined by the quadratic operators, which is equivalent to the operator of the form H_{(B,K)} and is obtained by approximating the operator H(ℏ) locally at each zero of V.

When A ≡ 0, this type of asymptotics have been studied by several authors and many results are obtained (cf. Helffer [6], Simon [25] and the references therein). In the presence of magnetic fields, Helffer-Sjöstrand [11] have first studied this problem under the assumption that the magnetic field is weak. Moreover the first author [17] discussed this problem when d = 2 and V has only finite nondegenerate zeros.

We now describe the contents of this paper.

As is mentioned above, we have two purposes in this paper. Firstly we will study on the quadratic Hamiltonian defined by (1.1) and, secondly, on the semiclassical approximation of the eigenvalues for Schrödinger operator with a magnetic field defined by (1.2).

In Section 2, we will rewrite the quadratic Hamiltonian H_{(B,K)} by using the creation and the annihilation operators which satisfy the canonical commutation relation. For this purpose we will introduce the Dirac operators and use the techniques in the Clifford algebras. In Section 3, we will give the exact forms of the eigenfunctions and determine the spectrum of H_{(B,K)} under the assumption that K is non-degenerate. The results in these two sections will be useful in the study of the semiclassical approximation of the eigenvalues.

Sections 4, 5 and 6 will be devoted to the more detailed study on the quadratic Hamiltonian H_{(B,K)}. We obtain a system of orthogonal polynomials from the eigenfunctions for H_{(B,K)} and we will study on them in Section 4. In Section 5, we will calculate an index of the Pauli operator given in Section 2. The calculation will give us an alternative proof for the completeness in L^{2}(R^{d}) of the set of the eigenfunctions given in Section 3.

In Section 6, we will consider the classical mechanics corresponding to H_{(B,K)} and will show that the classical paths are mapped into classical paths for a harmonic oscillator by a symplectic transformation. This enables us to determine completely the spectrum of H_{(B,K)} involving the case when K is degenerate and H_{(B,K)} is not discrete by virtue of the theory of the metaplectic representations.

At Section 7, we will start the study on the semiclassical approximation of the eigenvalues of H(ℏ) defined by (1.2) and we will consider the problem in that section under the assumption that the Hessian matrices of the scalar potential V at its zeros are all non-degenerate. The case when V has degenerate zeros will be discussed in Section 8. The following two sections, Sections 9 and 10, will be devoted to the proof of the main theorem in Section 8. A similar degenerate asymptotics to that mentioned in Section 8 is seen for the eigenvalues of the Schrödinger operator given by H_{A}(ℏ) = (\sqrt{−1}ℏ∇ − A)^{2} if H_{A}(ℏ) has some eigenvalues below the essential spectrum. It will be given in the final Section 11.

2. Quadratic Hamiltonians

In this section, for a given quadratic Hamiltonian, we construct concretely the creation and the annihilation operators satisfying the canonical commutation relations. It enables us to identify the spectrum as a set if the spectrum is discrete.

Let K = (K_{i,j})_{i,j=1,2,...,d} be a real non-negative definite symmetric d × d matrix and B = (B_{i,j}) be a real skew-symmetric one. We consider the Schrödinger operator H_{(B,K)} defined by

H_{(B,K)} = \frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}∂_{j} − (Bx)_{j}/2)^{2} + \frac{1}{2} ∑_{i,j=1}^{d} K_{i,j}x_{i}x_{j},  (2.1)

where (Bx)_{j} is the j-th component of Bx ∈ R^{d}, x ∈ R^{d}. We denote also by H_{(B,K)} its unique self-adjoint realization on L^{2}(R^{d}).

This operator is quadratic in the sense that the corresponding classical Hamiltonian H_{cl} is given by a homogeneous quadratic polynomial in the position and the momenta (x, p) ∈ R^{d} × R^{d} :

H_{cl}(x, p) = \frac{1}{2} ∑_{j=1}^{d}(p_{j} + (Bx)_{j}/2)^{2} + \frac{1}{2} ∑_{i,j=1}^{d} K_{i,j}x_{i}x_{j}.  (2.2)

The Hamilton equation is written as

\frac{d}{dt} (\begin{matrix}x \\ p \end{matrix}) = Ω (\begin{matrix}x \\ p \end{matrix}) ,  Ω = Ω_{(B,K)} = (\begin{matrix}B/2 & E_{d} \\ B^{2}/4 − K & B/2 \end{matrix}) ,  (2.3)

where E_{d} is the d × d unit matrix.

We see that the eigenvalues of Ω are purely imaginary or zero and we denote them by ±\sqrt{−1}s_{j}, s_{j} ≧ 0, j = 1, 2, ..., d. In fact, if we denote by \sqrt{K} the non-negative symmetric square root of K and introduce the skew-symmetric matrix

Ω̃ = (\begin{matrix}B & \sqrt{K} \\ −\sqrt{K} & 0 \end{matrix}) ,

then we can show the following easily :

Lemma 2.1. For any nonzero eigenvalue α of Ω,

T (\begin{matrix}x \\ p \end{matrix}) = (\begin{matrix}x \\ −\sqrt{K}x/α \end{matrix})

is a homomorphism from the eigenspace of Ω onto the eigenspace of Ω̃ corresponding to the same eigenvalue α. Its inverse is written as

T^{−1} (\begin{matrix}x \\ p \end{matrix}) = (\begin{matrix}x \\ (α − B/2)x \end{matrix}) .

In particular, ±\sqrt{−1}s_{j}, s_{j} ≧ 0, j = 1, 2, ..., d are also the eigenvalues of Ω̃ and we have

∏_{j=1}^{d} s_{j}^{2} = det Ω̃ = det K.  (2.4)

In our analysis, the matrix Ω̃ plays important roles. This matrix Ω̃ appears naturally if we consider the Dirac operator and the Pauli Hamiltonian (see Lemma 2.4 below).

Now we are in the position to mention the first result. Theorem 2.2. If K is strictly positive definite, then the spectrum σ(H_{(B,K)}) of the quadratic operator H_{(B,K)} is given by

σ(H_{(B,K)}) = {∑_{j=1}^{d}(n_{j} + 1/2)s_{j}; n_{j} ∈ Z_{+}, j = 1, 2, ..., d},  (2.5)

including the multiplicities.

In the rest of this section, we prove that (2.5) holds as a set by rewriting H_{(B,K)} in a canonical way. The proof will be completed if we show the completeness in L^{2}(R^{d}) of the set of the corresponding eigenfunctions in the next section or if we show the nondegeneracy of the ground state in Section 5.

Let U = (U_{α,β}) be a 2d × 2d orthogonal matrix such that

U^{t} Ω̃ U = (\begin{matrix}0 & Λ \\ −Λ & 0 \end{matrix}) ,  (2.6)

where U^{t} is the transpose of U and Λ is the diagonal matrix whose diagonal elements are s_{j}, j = 1, 2, ..., d. We define the first order differential operators R_{α}, α = 1, 2, ..., 2d, and A_{j}, j = 1, 2, ..., d, by

R_{α} = ∑_{k=1}^{d}\{U_{k,α}(\sqrt{−1}∂_{k} − (Bx)_{k}/2) + U_{d+k,α}(\sqrt{K}x)_{k}\}

and

A_{j} = \frac{1}{\sqrt{2}}R_{j} + \begin{matrix}√ & \overline{−1} \\ √ & \overline{2} \end{matrix}R_{d+j},

respectively.

Then, by the following theorem and the standard commutator calculus, we can prove that (2.5) holds as a set. Note that we do not have to assume the non-degeneracy of K.

Theorem 2.3. (i) [A_{j}, A_{k}] = 0, [A_{j}, A_{k}^{∗}] = δ_{j,k}s_{j}, (ii)

H_{(B,K)} = ∑_{j=1}^{d}(A_{j}^{∗}A_{j} + s_{j}/2),

where A_{k}^{∗} is the formal adjoint of A_{k} and [ , ] is the commutator defined by [A, A′] = AA′ − A′A.

For the proof we introduce the Dirac matrices {γ_{α}}_{α=1}^{2d} and the Dirac operator. γ_{α}’s are the 2^{d} × 2^{d} Hermitian matrices satisfying

{γ_{α}, γ_{β}} ≡ γ_{α}γ_{β} + γ_{β}γ_{α} = 2δ_{α,β}E_{2^{d}}  (2.7)

and the Dirac operator Q is defined by

Q = \frac{1}{\sqrt{2}} ∑_{j=1}^{d}\{γ_{j}(\sqrt{−1}∂_{j} − (Bx)_{j}/2) + γ_{d+j}(\sqrt{K}x)_{j}\}.  (2.8)

For details of the Dirac matrices and the Dirac operators, we refer to [1].

Then we can prove the following lemmas. The proofs are easy and we omit them.

Lemma 2.4. Q^{2} = H_{(B,K)} + \frac{\sqrt{−1}}{2} ∑_{α<β} Ω̃_{α,β}γ_{α}γ_{β}.

Lemma 2.5. Setting ω_{β} = ∑_{α=1}^{2d} U_{α,β}γ_{α}, {ω_{β}}_{β=1}^{2d} is also a system of Dirac matrices and

∑_{α<β} Ω̃_{α,β}γ_{α}γ_{β} = ∑_{j=1}^{d} s_{j}ω_{j}ω_{d+j}.

Lemma 2.6. Q = \frac{1}{\sqrt{2}} ∑_{β=1}^{2d} R_{β}ω_{β}.

Now we prove Theorem 2.3.

Proof of Theorem 2.3. By Lemmas 2.4 and 2.5, we have

Q^{2} = H_{(B,K)} + \frac{\sqrt{−1}}{2} ∑_{j=1}^{d} s_{j}ω_{j}ω_{d+j}.  (2.9)

On the other hand, Lemma 2.6 implies

Q^{2} = \frac{1}{2} ∑_{α=1}^{2d} R_{α}^{2} + \frac{1}{2} ∑_{α<β}[R_{α}, R_{β}]ω_{α}ω_{β}.

Therefore, by the fundamental fact on the Dirac matrices that E_{2^{d}}, ω_{α}ω_{β}, α < β, are linearly independent, we obtain

H_{(B,K)} = \frac{1}{2} ∑_{α=1}^{2d} R_{α}^{2},  (2.10)
[R_{j}, R_{k}] = 0, [R_{d+j}, R_{d+k}] = 0  (2.11)

and

[R_{j}, R_{d+k}] = \sqrt{−1}δ_{j,k}s_{j}  (2.12)

for j, k = 1, 2, ..., d.

Note that R_{α}^{∗} = R_{α} by definition. Then we can prove the assertion (i) easily by (2.11) and (2.12). Moreover, by using (2.10) and (2.12), we get

∑_{j=1}^{d} A_{j}^{∗}A_{j} = \frac{1}{2} ∑_{α=1}^{2d} R_{α}^{2} + \frac{\sqrt{−1}}{2} ∑_{j=1}^{d}[R_{j}, R_{d+j}] = H_{(B,K)} − \frac{1}{2} ∑_{j=1}^{d} s_{j}. □

We end this section by giving a remark on the relations between the matrices Ω and Ω̃. When the symmetric matrix K is non-degenerate, we have

T Ω T^{−1} = Ω̃, T = (\begin{matrix}B/2 & E_{d} \\ −\sqrt{K} & 0 \end{matrix}) .

When K is degenerate, Ω and Ω̃ are not equivalent in general because rank(Ω) = d + rank(K), which follows from

(\begin{matrix}E_{d} & 0 \\ −B/2 & E_{d} \end{matrix}) Ω (\begin{matrix}E_{d} & 0 \\ −B/2 & E_{d} \end{matrix}) = (\begin{matrix}0 & E_{d} \\ −K & 0 \end{matrix}) .

This causes the various properties of the spectrum of H_{(B,K)}. For details, see Section 6 below.

3. The Eigenfunctions of H_{(B,K)}

We have obtained the eigenvalues of the quadratic operator H_{(B,K)} in the previous section. In this section we give the corresponding eigenfunctions and complete the proof of Theorem 2.2. We use the same notations as those in the previous section and assume that the symmetric matrix K is non-degenerate.

First of all we look for an eigenfunction corresponding to the least eigenvalue μ_{0} = ∑ s_{j}/2 = ∥Ω̃∥_{1}/4, where ∥Ω̃∥_{1} is the trace norm of Ω̃. For this purpose we let X = (X_{i,j}) be a complex symmetric (not Hermitian) matrix and set

ϕ_{0}(x) = exp(−⟨x, Xx⟩/2),

where ⟨ξ, η⟩ = ∑_{j=1}^{d} ξ_{j}η_{j} for ξ, η ∈ C^{d}.

Then, since X, \sqrt{K} are symmetric and B is skew-symmetric,

ϕ_{0}(x)^{−1}A_{j}ϕ_{0}(x) = 2^{−1/2} ∑_{i,k=1}^{d}\{(X_{i,k} + \sqrt{−1}B_{i,k}/2)(−\sqrt{−1}U_{k,j} + U_{k,d+j})

+ \sqrt{K}_{i,k}(U_{d+k,j} + \sqrt{−1}U_{d+k,d+j})\}x_{i}.

Now we set

U = (\begin{matrix}U_{(11)} & U_{(12)} \\ U_{(21)} & U_{(22)} \end{matrix}) , U_{1} = \sqrt{−1}U_{(11)} − U_{(12)}, U_{2} = U_{(21)} + \sqrt{−1}U_{(22)},

where U_{(ij)}’s are d × d matrices. Then A_{j}ϕ_{0} = 0 if

XU_{1} = \sqrt{K}U_{2} − \sqrt{−1}BU_{1}/2.  (3.1)

We show that (3.1) has a solution X which is symmetric and whose real part is strictly positive definite. For this purpose we show the following. Lemma 3.1. (i) \sqrt{K}U_{1}U_{2}^{∗} = U_{2}ΛU_{2}^{∗}, where U_{2}^{∗} = \overline{U_{2}^{t}} is the adjoint matrix of U_{2}. (ii) U_{1} and U_{2} are non-degenerate.

Proof. By the definition of U, we have

\begin{matrix}BU_{(11)} + \sqrt{K}U_{(21)} = −U_{(12)}Λ, \\ \sqrt{K}U_{(11)} = U_{(22)}Λ, \end{matrix}  \begin{matrix}BU_{(12)} + \sqrt{K}U_{(22)} = U_{(11)}Λ, \\ −\sqrt{K}U_{(12)} = U_{(21)}Λ. \end{matrix}  (3.2)

and

\sqrt{K}U_{1} = \sqrt{−1}\sqrt{K}(U_{(11)} + \sqrt{−1}U_{(12)}) = \sqrt{−1}(U_{(22)}Λ − \sqrt{−1}U_{(21)}Λ) = U_{2}Λ.

For the proof of (ii), it is sufficient to show that the Hermitian matrix U_{2}ΛU_{2}^{∗} is strictly positive definite. By (3.2) and the unitarity of U, we get

U_{2}ΛU_{2}^{∗} = U_{(21)}ΛU_{(21)}^{t} + U_{(22)}ΛU_{(22)}^{t} + \sqrt{−1}\sqrt{K}(U_{(11)}U_{(21)}^{t} + U_{(12)}U_{(22)}^{t})
= U_{(21)}ΛU_{(21)}^{t} + U_{(22)}ΛU_{(22)}^{t}.

Therefore U_{2}ΛU_{2}^{t} is real, symmetric and non-negative definite. Moreover, since the diagonal matrix Λ is strictly positive definite by the assumption and the 2d column vectors of U_{(21)} and U_{(22)} span R^{d}, U_{2}ΛU_{2}^{∗} is strictly positive definite. □

Now we give the ground state for the quadratic operator H_{(B,K)}. Theorem 3.2. Let X be the d × d matrix defined by

X = −\sqrt{−1}B/2 + \sqrt{K}U_{2}U_{1}^{−1}.

Then X is symmetric, Re(X) is strictly positive definite and ϕ_{0}(x) = exp(−⟨x, Xx⟩/2) is an eigenfunction of H_{(B,K)} corresponding to the least eigenvalue μ_{0}.

Proof. The symmetry of X can be proved from the definitions and we omit the details. To prove the strictly positive definiteness of Re(X), we note

U_{1}U_{2}^{∗} = U_{(11)}U_{(22)}^{t} − U_{(12)}U_{(21)}^{t} and Re(U_{2}U_{2}^{∗}) = E_{d},

which can be shown by the unitarity of U. Then we get Re(X)U_{1}U_{2}^{∗} = Re((X + \sqrt{−1}B/2)U_{1}U_{2}^{∗}) = Re(\sqrt{K}U_{2}U_{2}^{∗}) = \sqrt{K}.

Moreover, by using Lemma 3.1, we obtain (U_{1}U_{2}^{∗})^{∗} Re(X)U_{1}U_{2}^{∗} = U_{2}U_{1}^{∗}\sqrt{K} = U_{2}ΛU_{2}^{∗}

and

Re(X) = (U_{1}^{∗})^{−1}ΛU_{1}^{−1}.

Therefore Re(X) is strictly positive definite.

By the second assertion proved above, ϕ_{0} ∈ L^{2}(R^{d}). Moreover, since X solves (3.1), A_{j}ϕ_{0} = 0, j = 1, 2, ..., d.

The proof of the theorem is now completed. □

We proceed to the study on the general eigenfunctions of H_{(B,K)}. We set

ϕ_{n} = (A_{1}^{∗})^{n_{1}}(A_{2}^{∗})^{n_{2}} ⋯ (A_{d}^{∗})^{n_{d}}ϕ_{0} and μ_{n} = ∑_{j=1}^{d}(n_{j} + 1/2)s_{j}

for n = (n_{1}, n_{2}, ..., n_{d}) ∈ Z_{+}^{d}, where A_{j}^{∗}’s are the creation operators defined in the previous section.

Then the following is proved by the standard commutator calculus. Proposition 3.3. (i) ϕ_{n} is an eigenfunction of H_{(B,K)} corresponding to the eigenvalue μ_{n} for all n ∈ Z_{+}^{d} . (ii) A_{j}ϕ_{(n_{1},...,n_{j}+1,...,n_{d})} = (n_{j} + 1)s_{j}ϕ_{(n_{1},...,n_{j},...,n_{d})}. (iii) Denoting by (·, ·) and ∥ · ∥ the inner product and the norm in L^{2}(R^{d}),

(ϕ_{n}, ϕ_{n′}) = ∥ϕ_{0}∥^{2} ∏_{j=1}^{d} δ_{n_{j},n′_{j}}n_{j}!s_{j}^{n_{j}}.

In order to prove that the spectrum of H_{(B,K)} coincides with {μ_{n}}_{n∈Z_{+}^{d}} including the multiplicities, we show the completeness of the set {ϕ_{n}} in L^{2}(R^{d}). For this purpose we consider the generating function F (ξ, x), ξ ∈ C^{d} :

F (ξ, x) = ∑_{n∈Z_{+}^{d}} \frac{ξ^{n}}{n!} ϕ_{n}(x) = ∑_{n∈Z_{+}^{d}} (∏_{j=1}^{d} \frac{1}{n_{j}!}(ξ_{j}A_{j}^{∗})^{n_{j}} ) ϕ_{0}(x).

Lemma 3.4. F (ξ, x) = ϕ_{0}(x) exp(g(ξ, x)), where

g(ξ, x) = \sqrt{2}⟨(Re X)\overline{U_{1}}ξ, x⟩ − \frac{1}{2}⟨(Re X)\overline{U_{1}}ξ, \overline{U_{1}}ξ⟩.  (3.3)

Proof. We set f(t, x, ξ) = F (tξ, x). Then f is the unique solution of the following first order partial differential equation :

\frac{∂f}{∂t} − ∑_{j=1}^{d} ξ_{j}A_{j}^{∗}f = 0

or

\frac{∂f}{∂t} + ∑_{j=1}^{d} a_{j} \frac{∂f}{∂x_{j}} = b(x)f,  (3.4)

satisfying f|_{t=0} = ϕ_{0}, where a_{j} = (\overline{U_{1}}ξ)_{j}/\sqrt{2} and b(x) = ⟨\overline{XU_{1}}ξ, x⟩/\sqrt{2}.

(3.4) can be solved easily and we get f(t, ξ, x) = ϕ_{0}(x) exp(g(t, ξ, x)),

g(t, ξ, x) = \sqrt{2}⟨(Re X)\overline{U_{1}}ξ, x⟩t − ⟨(Re X)\overline{U_{1}}ξ, \overline{U_{1}}ξ⟩t^{2}/2.

Setting t = 1, we get (3.3). □

By this lemma and the non-degeneracy of the matrices Re X and U_{1}, we have the following theorem, which proves that the spectrum of H_{(B,K)} is given by {μ_{n}} including the multiplicities and completes the proof of Theorem 2.2. Theorem 3.5. {ϕ_{n}}_{n∈Z_{+}^{d}} forms a complete orthogonal basis of L^{2}(R^{d}) consisting of the eigenfunctions of H_{(B,K)}.

4. Generalized Hermite polynomials

We have given the explicit expressions of the eigenfunctions {ϕ_{n}}_{n∈Z_{+}^{d}} of the quadratic operator H_{(K,B)} defined by (2.1). By definition, ϕ_{n} is of the form G_{n}ϕ_{0} for some polynomial G_{n} in x = (x_{1}, x_{2}, ..., x_{d}) of order |n| = ∑ n_{j}.

Moreover, by Proposition 3.3 (iii) and Theorem 3.5, {G_{n}} forms a system of the complete orthogonal polynomials in L^{2}(R^{d}, |ϕ_{0}|^{2}dx).

In general the orthogonal polynomials satisfy some fundamental properties, e.g., some recurrence formulae. See, e.g., Szegö [26]. We will show some of these properties in this section. See also [18] for the case of d = 2.

At first we define an isometry T from L^{2}(R^{d}) = L^{2}(R^{d}, dx) onto L^{2}(R^{d}, |ϕ_{0}|^{2}dx) by

T φ = ϕ_{0}^{−1}φ, φ ∈ L^{2}(R^{d}),

and set

L = T H_{(B,K)}T^{−1} and A_{j}^{0} = T A_{j}T^{−1}.

L is so-called the ground state transformation of the Schrödinger operator H_{(B,K)}.

Then the following is easily verified. Proposition 4.1. (i) The system of the polynomials {G_{n}}_{n∈Z_{+}^{d}} forms a complete orthogonal basis of L^{2}(R^{d}, |ϕ_{0}|^{2}dx) satisfying LG_{n} = μ_{n}G_{n}. (ii) A_{j}^{0}G_{(n_{1},...,n_{j}+1,...,n_{d})} = (n_{j} + 1)s_{j}G_{(n_{1},...,n_{j},...,n_{d})}. (iii) (A_{j}^{0})^{∗}G_{(n_{1},...,n_{j}−1,...,n_{d})} = G_{(n_{1},...,n_{j},...,n_{d})}, where (A_{j}^{0})^{∗} is the formal adjoint of A_{j}^{0} with respect to the measure |ϕ_{0}|^{2}dx.

Secondly we show a recurrence formula for {G_{n}}. Proposition 4.2. Let U_{1} and X be the d × d matrices introduced in the previous section, then

G_{(n_{1},...,n_{j}+1,...,n_{d})}(x) = \sqrt{2}(U_{1}^{∗}(Re X)x)_{j}G_{(n_{1},......,n_{d})}(x)

− ∑_{k=1}^{d} n_{k}(U_{1}^{∗}(Re X)\overline{U_{1}})_{j,k}G_{(n_{1},...,n_{k}−1,...,n_{d})}(x).

Proof. By Lemma 3.4, we have

∑_{n∈Z_{+}^{d}} \frac{ξ^{n}}{n!} G_{n}(x) = exp(\sqrt{2}⟨(Re X)\overline{U_{1}}ξ, x⟩ − ⟨(Re X)\overline{U_{1}}ξ, \overline{U_{1}}ξ⟩/2).

By differentiating both hand sides with respect to ξ_{j} and by using the symmetry of X, we get the assertion of the proposition. □

5. An index of the Pauli Hamiltonian

In this section we calculate an index of the Pauli Hamiltonian P = Q^{2}, where Q is the Dirac operator defined by (2.8). We will see that the index is equal to one and will get an alternative proof of Theorem 2.2. We will use the same notations as those in the previous sections and assume that the symmetric matrix K is non-degenerate.

First of all we define an Hermitian matrix Γ by

Γ = ∏_{j=1}^{d} \sqrt{−1}γ_{d+j}γ_{j}.

Then the following is easily established by using the anti-commutativity (2.7) of the Dirac matrices {γ_{α}}_{α=1}^{2d}. Proposition 5.1. (P, Γ, Q) has the supersymmetry, that is, (i) P = Q^{2} ≧ 0, (ii) Γ^{2} is equal to the identity and (iii) {Q, Γ} = 0.

Since Γ is Hermitian and Γ^{2} is equal to the identity, it only has the eigenvalues ±1. We denote the corresponding eigenspaces by V_{±} and the restriction of P on L^{2}(R^{d}) ⨂ V_{±} by P_{±}. Then the supersymmetric index (the Witten index) of P is defined in the usual manner ([1]).

Definition. Ind(P) ≡ dim Ker(P_{+}) − dim Ker(P_{−}).

Before proceeding to the calculations of the index, we notice the following. The proof is easy and is omitted. Lemma 5.2. (Berezin’s formula) (i) \{\sqrt{−1}γ_{d+j}γ_{j}\}_{j=1}^{d} is a commutative system of Hermitian matrices and (\sqrt{−1}γ_{d+j}γ_{j})^{2} is equal to the identity, j = 1, 2, ..., d.

(ii) For any ℓ = 0, 1, ..., d − 1, and any j_{1}, j_{2}, ..., j_{ℓ} with 1 ≦ j_{1} < ⋯ < j_{ℓ} ≦ d,

Tr(Γ ∏_{k=1}^{ℓ} \sqrt{−1}γ_{d+j_{k}}γ_{j_{k}}) = 0.

(iii) Setting ω_{β} = ∑_{α=1}^{2d} U_{α,β}γ_{α} for an orthogonal matrix (U_{α,β}),

Tr(Γ ∏_{j=1}^{d} \sqrt{−1}ω_{d+j}ω_{j}) = 2^{d}.

We first prove : Proposition 5.3. Ind(P) = 1.

Proof. We note the following McKean-Singer identity ([1]) : for t > 0,

Ind(P) = ∫_{R^{d}} Tr(Γe^{−tP} (x, x))dx
= ∫_{R^{d}} Tr(Γe^{−tP} (t^{−1/2}x, t^{−1/2}x))t^{−d/2}dx,  (5.1)

where exp(−tP)(x, y), t > 0, x, y ∈ R^{d}, is the integral kernel of the semigroup generated by P.

For the integral kernel, we have the Feynman-Kac-Itô formula ([24]). We denote by E_{0}[·|A] the standard conditional expectation under the condition A with respect to the ddimensional Wiener measure on the path space starting from 0. Then we have, evaluating the integral kernel on the diagonal set,

\begin{aligned}e^{−tP} (x, x) = (2πt)^{−d/2}E_{0}[exp\{\sqrt{−1}S(t, x, w) & − F (t, x, w)\}|w_{t} = 0] \\ & × exp(−\sqrt{−1}t ∑_{α<β} Ω̃_{α,β}γ_{α}γ_{β}/2), \end{aligned} (5.2)

where S and F are the quadratic Wiener functionals defined by

S(t, x, w) = \frac{1}{2} ∫_{0}^{t}⟨B(x + w_{s}), ∘dw_{s}⟩

and

F (t, x, w) = \frac{1}{2} ∫_{0}^{t}⟨x + w_{s}, K(x + w_{s})⟩ds,

respectively. ⟨·, ·⟩ is the usual inner product in R^{d} and ∘dw_{s} is the Stratonovich stochastic integral. S is so-called the stochastic area of P. Lévy. We refer to [14] and [24] for the notions on the probability theory.

Let us consider the asymptotics of the integrand of (5.1). Then, by (5.2) and by using the self-similarity of the Wiener measure, we get

t^{−d/2} Tr(Γe^{−tP} (t^{−1/2}x, t^{−1/2}x))

= (2πt^{2})^{−d/2}e^{−⟨x,Kx⟩/2} Tr(Γ exp(t ∑_{j=1}^{d} s_{j}\sqrt{−1}ω_{d+j}ω_{j}/2)) · (1 + o(1))
= (2π)^{−d/2}e^{−⟨x,Kx⟩/2} ∏_{j=1}^{d} s_{j} · (1 + o(1))

as t ↓ 0 for each x ∈ R^{d}. We have used Lemmas 2.5 and 5.2 at the second and the third lines, respectively.

The estimate above is pointwise and depends on x. However, since the magnetic field is constant and the scalar potential grows quadratically at infinity, the magnetic field does not contribute to the leading asymptotics of the integrals over R^{d} as t ↓ 0 (cf. [16]). That is, we can show

Ind(P) = lim_{t↓0} ∫_{R^{d}} Tr(Γe^{−tP} (x, x))dx

= ∫_{R^{d}} lim_{t↓0} t^{−d/2} Tr(Γe^{−tP} (t^{−1/2}x, t^{−1/2}x))dx

= ∫_{R^{d}}(2π)^{−d/2}e^{−⟨x,Kx⟩/2}dx ∏_{j=1}^{d} s_{j}

= (det K)^{−1/2} ∏_{j=1}^{d} s_{j}.

By (2.4), we have proved that the index is equal to one and completed the proof of the proposition. □

Secondly we show the following proposition, which proves Theorem 3.5 if it is combined with Proposition 5.3.

Proposition 5.4. Ind(P) = dim Ker(H_{(B,K)} − μ_{0}).

Proof. Let {ω_{β}}_{β=1}^{2d} be the system of the Dirac matrices introduced in the statement of Lemma 2.5. Since \sqrt{−1}ω_{d+j}ω_{j}, j = 1, 2, ..., d, are Hermitian and (\sqrt{−1}ω_{d+j}ω_{j})^{2} is equal to the identity, their eigenvalues are only ±1. Letting ε_{j} be 1 or −1, we put

V (ε_{1}, ε_{2}, ..., ε_{d}) = \{v ∈ C^{2^{d}}; \sqrt{−1}ω_{d+j}ω_{j}v = ε_{j}v, j = 1, 2, ..., d\}.

Then, since \{\sqrt{−1}ω_{d+j}ω_{j}\} are commutative, the eigenspace V_{±} of Γ are decomposed by

V_{±} = ⨁_{ε_{1}ε_{2}···ε_{d}=±1} V (ε_{1}, ε_{2}, ..., ε_{d}) (direct sum).

Therefore, letting P (ε_{1}, ε_{2}, ..., ε_{d}) be the restriction of P on L^{2}(R^{d}) ⨂ V (ε_{1}, ε_{2}, ..., ε_{d}), we get

Ker(P_{±}) = ⨁_{ε_{1}ε_{2}···ε_{d}=±1} Ker(P (ε_{1}, ε_{2}, ..., ε_{d})).

On the other hand, we have

P = H_{(B,K)} − ∑_{j=1}^{d} s_{j}\sqrt{−1}ω_{d+j}ω_{j}/2 ≧ H_{(B,K)} − μ_{0}

by (2.9) because ω_{d+j}ω_{j} = −ω_{j}ω_{d+j}. Since μ_{0} is the least eigenvalue of H_{(B,K)}, we get

Ker(P (ε_{1}, ε_{2}, ..., ε_{d})) = \begin{cases}Ker(H_{(B,K)} − μ_{0}) & if ε_{1} = ε_{2} = ⋯ = ε_{d} = 1, \\ \{0\} & otherwise. \end{cases}

Therefore we obtain dim Ker(P_{+}) = dim Ker(P (1, 1, ..., 1)) = dim Ker(H_{(B,K)} − μ_{0})

and

dim Ker (P_{−}) = 0.

The proof is completed. □

6. Quadratic Hamiltonians and harmonic oscillators

In this section, we determine the spectrum of the quadratic operator defined by (2.1) without assuming that the matrix K is strictly positive. For this we first show that the corresponding classical mechanics is transformed to that of a harmonic oscillator by a symplectic transform and then we use the theory of the metaplectic representation (cf.[5]).

Let Sp(d; R) be the group of linear symplectic transforms in R^{2d} and 𝔰𝔭(d; R) be its Lie algebra. The classical mechanics corresponding to (2.1) is given by (2.2) and (2.3). Note that Ω ∈ 𝔰𝔭(d; R).

We set r = rank(K) and ℓ = rank(Ω̃)/2. Then, it is easy to see

d + r ≧ 2ℓ.

Taking Lemma 2.1 into account, we denote the non-zero eigenvalues of Ω by ±\sqrt{−1}s_{j}, s_{j} > 0, j = 1, 2, ..., ℓ, including the multiplicities. For ℓ ≦ m ≦ d, we define T_{m,ℓ} ∈ 𝔰𝔭(d; R) by

T_{m.ℓ} = (\begin{matrix}0 & E′_{m} \\ −(Λ′_{ℓ})^{2} & 0 \end{matrix}) ,

where

E′_{m} = (\begin{matrix}E_{m} & 0 \\ 0 & 0 \end{matrix}) , Λ′_{ℓ} = (\begin{matrix}Λ_{ℓ} & 0 \\ 0 & 0 \end{matrix}) ,

E_{m} is the m × m unit matrix and Λ_{ℓ} is the ℓ × ℓ diagonal matrix whose nonzero elements are s_{1}, ..., s_{ℓ}. It is easy to show that T_{m,ℓ} is diagonalizable if and only if m = ℓ.

Then we first show : Theorem 6.1. There exists an S ∈ Sp(d; R) such that

S^{−1} Ω S = T_{m,ℓ}, m = d + r − ℓ. (6.1)

Proof. We only give the matrix S. Let us first consider the case of d + r = 2ℓ. We let U_{i} = (\begin{matrix}a_{i} \\ b_{i} \end{matrix}) , a_{i}, b_{i} ∈ C^{d}, i = 1, 2, ..., ℓ, be the eigenvectors of Ω corresponding the the eigenvalue \sqrt{−1}s_{i} so that

((K + s_{i}^{2})a_{i}, a_{i}) = 2s_{i}^{2}

and, if s_{i} = s_{j} for i ≠ j,

((K + s_{i}^{2})a_{i}, a_{j}) = 0

and we set X_{i} = Re(U_{i}) and Y_{i} = Im(U_{i}/s_{i}), where (·, ·) is the usual inner product in C^{d}, (a, b) = ∑_{i=1}^{d} \overline{a_{i}}b_{i} for a, b ∈ C^{d}. Moreover we let Z_{i} = (\begin{matrix}c_{i} \\ d_{i} \end{matrix}) , c_{i}, d_{i} ∈ R^{d}, i = 1, 2, ..., 2n, n = d−ℓ, be the eigenvectors of Ω corresponding to the zero eigenvalue so that

(c_{i}, Bc_{j}) = (c_{n+i}, Bc_{n+j}) = 0 and (c_{i}, Bc_{n+j}) = −δ_{i,j}

for i, j = 1, ..., n. Note that Kc_{i} = 0. Then we can show that the matrix

S = (X_{1} ⋯ X_{l}Z_{1} ⋯ Z_{n}Y_{1} ⋯ Y_{l}Z_{n+1} ⋯ Z_{2n})

is symplectic and satisfies (6.1).

We next consider the case of d + r > 2ℓ. We change the coordinate so that

K = (\begin{matrix}K^{(1)} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) , B = (\begin{matrix}B^{(1)} & B^{(2)} & B^{(3)} \\ −(B^{(2)})^{t} & 0 & 0 \\ −(B^{(3)})^{t} & 0 & B^{(0)} \end{matrix}) ,

where K^{(1)} is the r × r nondegenerate diagonal matrix, B^{(1)} is an r × r matrix, and

B^{(0)} = (\begin{matrix}B_{1}^{(0)} \\ & . & . & . \\ & & & & B_{s}^{(0)} \end{matrix}) , B_{i}^{(0)} = (\begin{matrix}0 & −β_{i} \\ β_{i} & 0 \end{matrix}) ,

β_{i} > 0, s = ℓ − r. We set h = m − ℓ. Then the size of B^{(2)} is r × h.

We take X_{i}, Y_{i}, i = 1, 2, ..., ℓ as in the last case. We take the eigenvectors Z_{i} = (\begin{matrix}c_{i} \\ d_{i} \end{matrix}) , i = 1, 2, ..., 2s, of Ω corresponding to the eigenvalue zero as follows: for i = 1, 2, ..., s,

c_{i} = (0 ⋯ 1^{(r+h+2i−1)−th} ⋯ 0)^{t}/\sqrt{β_{i}}, c_{s+i} = (0 ⋯ 1^{(r+h+2i)−th} ⋯ 0)^{t}/\sqrt{β_{i}}.

For the h × h positive definite symmetric matrix

A = I + (B^{(2)})^{t}(K^{(1)})^{−1}B^{(2)},

let λ_{1}, ..., λ_{h} be its eigenvalues and {n_{1}, ..., n_{h}} be the corresponding orthonormal eigenvectors. For i = 1, 2, ..., h, put q_{i}^{(1)} = 0 ∈ R^{r}, q_{i}^{(2)} = n_{i}/\sqrt{λ_{i}} ∈ R^{h}, q_{i}^{(3)} = 0 ∈

R^{2s}, q_{i} = (\begin{matrix}q_{i}^{(1)} \\ q_{i}^{(2)} \\ q_{i}^{(3)} \end{matrix}) ∈ R^{d}, r_{i} = −Bq_{i}/2 ∈ R^{d} and W_{i} = (\begin{matrix}q_{i} \\ r_{i} \end{matrix}) ∈ R^{2d}. We see that

{Z_{1}, ..., Z_{2s}, W_{1}, ..., W_{h}} is a basis of the eigenspace of the eigenvalue zero.

For i = 1, 2, ..., h, set v_{i}^{(1)} = (K^{(1)})^{−1}B^{(2)}q_{i}^{(2)} ∈ R^{r}, v_{i}^{(3)} = (B^{(0)})^{−1}(B^{(3)})^{t}v_{i}^{(1)} ∈ R^{2s} and v_{i}^{(2)} ∈ R^{h} as follows :

(v_{1}^{(2)} ⋯ v_{h}^{(2)}) = −BC^{−1}D^{t}ED

where B = (n_{1} ⋯ n_{h}), C is the h × h non-degenerate diagonal matrix whose nonzero element is \sqrt{λ_{1}}, ..., \sqrt{λ_{h}}, D is the (r + 2s) × h matrix defined by

D = (\begin{matrix}v_{1}^{(1)} & ⋯ & v_{h}^{(1)} \\ v_{1}^{(3)} & ⋯ & v_{h}^{(3)} \end{matrix}) ,

and E is the (r + 2s) × (r + 2s) matrix defined by

E = \frac{1}{2} (\begin{matrix}B^{(1)} & B^{(3)} \\ −(B^{(3)})^{t} & B^{(0)} \end{matrix}) .

Moreover, we set v_{i} = (\begin{matrix}v_{i}^{(1)} \\ v_{i}^{(2)} \\ v_{i}^{(3)} \end{matrix}) ∈ R^{d}, w_{i} = q_{i} − Bv_{i}/2 ∈ R^{d} and V_{i} = (\begin{matrix}v_{i} \\ w_{i} \end{matrix}) ∈ R^{2d} for

i = 1, 2, ..., h. It is easy to see ΩV_{i} = W_{i}.

Then we can show that the matrix

S = (X_{1} ⋯ X_{ℓ}W_{1} ⋯ W_{h}Z_{1} ⋯ Z_{s}Y_{1} ⋯ Y_{ℓ}V_{1} ⋯ V_{h}Z_{s+1} ⋯ Z_{2s})

is symplectic and satisfies (6.1). □

By (6.1), we get the following. Define the classical Hamiltonian H_{cl}^{m,ℓ} by

H_{cl}^{m,ℓ}(x, p) = \frac{1}{2} ∑_{j=1}^{m} p_{j}^{2} + \frac{1}{2} ∑_{j=1}^{ℓ} s_{j}^{2}x_{j}^{2}

and set (\begin{matrix}x′ \\ p′ \end{matrix}) = S^{−1} (\begin{matrix}x \\ p \end{matrix}) . Then the following identity holds :

H_{cl}^{m,ℓ}(x′, p′) = H_{cl}(x, p),  (6.2)

where H_{cl} is the classical Hamiltonian defined by (2.2). This identity holds also when we put p = −\sqrt{−1}∇.

Now, by using the metaplectic representation (cf.[5]), we get the following: Theorem 6.2. H_{(B,K)} is unitary equivalent to the (possibly degenerate) harmonic oscillator

H_{osc}^{m,ℓ} = \frac{1}{2} ∑_{j=1}^{m} ∂_{j}^{2} + \frac{1}{2} ∑_{j=1}^{ℓ} s_{j}^{2}x_{j}^{2}, m = d + r − ℓ,

defined on L^{2}(R^{d}).

Proof. Let G be the subgroup of the unitary transforms on L^{2}(R^{d}) defined in pp.71 of [5] and 𝔊 be its Lie algebra. In [5], a two to one homomorphism ρ from G onto Sp(d; R) is constructed and it is shown that its differentiation dρ is an isomorphism from 𝔊 to 𝔰𝔭(d; R) such that

\sqrt{−1}H_{(B,K)} ∈ 𝔊, dρ(\sqrt{−1}H_{(B,K)}) = Ω.

Therefore, for the matrix S in Theorem 6.1, there is a unitary transform U ∈ G such that ρ(U) = S and it satisfies U^{−1}H_{(B,K)}U = H_{osc}^{m,ℓ}. □

Theorem 6.2 allows us to determine the spectra of H_{(B,K)} :

Theorem 6.3. (i) If K is non-degenerate, then σ(H_{(B,K)}) consists only of the eigenvalues with finite multiplicities and is given by (2.5) including the multiplicities. (ii) If K is degenerate and d + rank(K) = rank(Ω̃) = 2ℓ, then σ(H_{(B,K)}) consists only of the eigenvalues with infinite multiplicities and is given by

{∑_{j=1}^{ℓ}(n_{j} + 1/2)s_{j}; n_{j} ∈ Z_{+}, j = 1, 2, ..., ℓ}.

(iii) If d + rank(K) > rank(Ω̃) = 2ℓ, then σ(H_{(B,K)}) is equal to [μ_{0}, ∞) and is absolutely continuous, where μ_{0} = ∑_{j=1}^{ℓ} s_{j}/2.

Remark. When d = 2, this result has been proved in [18] by a different method.

It should be noted that d+rank(K) = rank(Ω̃) if and only if the matrix Ω in the classical Hamilton equation is diagonalizable and that, in this case, the spectrum of H_{(B,K)} is pure point.

7. Semiclassical Asymptotics of Eigenvalues

In the following sections, we discuss the asymptotics of the spectrum of H(ℏ) defined by (1.2) as ℏ ↓ 0. We are interested in the leading asymptotics of the low lying eigenvalues.

Setting λ = 1/ℏ, we consider the Schrödinger operator H(λ) defined by the following instead of H(ℏ) :

H(λ) = H_{(A,V,v)}(λ) = \frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}∂_{j} − λA_{j})^{2} + λ^{2}V + λv, ∂_{j} = ∂/∂x_{j}. (7.1)

We use the same notation because there is no fear of confusion.

Throughout the following sections, we assume the following.

(A.1) (i) V is a non-negative C^{∞} function and the set of the zeros (sometimes called the wells) of V is not empty ; (ii) v is a real C^{∞} function bounded from below ; (iii) A = (A_{1}, A_{2}, ..., A_{d}) is a R^{d}-valued C^{∞} function.

Under this assumption, H(λ) is essentially self-adjoint on C_{0}^{∞}(R^{d}) ([22]). We will denote the unique self-adjoint realization on L^{2}(R^{d}) by the same notation H(λ).

Moreover we identify the vector potential A with the differential 1-form ∑ A_{j}dx_{j} as usual and we denote it also by A. Then the differential 2-form dA represents the corresponding magnetic field.

In this section, we assume the non-degeneracy of the Hessian matrices of V at all of the wells. The main theorem in this section is typical and fundamental in the study of the semiclassical approximation of the eigenvalues. We will discuss a more general case in the following sections.

First of all we give a fundamental proposition. We assume also the following.

(A.2)  lim inf_{|x|→∞} V (x) > 0.

Proposition 7.1. Assume (A. 1) and (A. 2). Then, for any m ∈ N, there exists a constant λ_{m} such that H(λ) has more than m discrete spectrum {μ_{n}(λ)}, μ_{1}(λ) ≦ μ_{2}(λ) ≦ ⋯, below the essential spectrum for any vector potential A and for any λ > λ_{m}.

The proof is omitted here because, when we give an upper bound for μ_{n}(λ) by using the Rayleigh-Ritz technique (cf. [25]), this proposition will be proved simultaneously. For details, see Section 9.

To proceed further, we make one more assumption.

(A.3) V^{−1}(0) is a finite subset {p^{a}}_{a=1}^{N} of R^{d} and the Hessian matrix K^{a} = (∂_{i}∂_{j}V (p^{a})) at each p^{a} is non-degenerate.

Setting B_{i,j}(x) = ∂_{j}A_{i}(x) − ∂_{j}A_{i}(x), i, j = 1, 2, ..., d, B(x) = (B_{i,j}(x)) and B^{a} = B(p^{a}), a = 1, 2, ..., N, we consider the quadratic operator H^{a} = H_{(B^{a},K^{a})} defined by (2.1) and let {μ_{n^{a}}^{a}}_{n^{a}=1}^{∞} be the increasing sequence of the eigenvalues of H^{a}, counting the multiplicities.

Moreover we define the increasing sequence {μ_{n}}_{n=1}^{∞} so that

{μ_{n}}_{n=1}^{∞} = ⋃_{a=1}^{N}{v(p^{a}) + μ_{n^{a}}^{a}}_{n^{a}=1}^{∞}

holds including the multiplicities.

Then the main theorem in this section is the following. Theorem 7.2. ([17]) Under the assumptions (A. 1)– (A. 3),

lim_{λ→∞} \frac{1}{λ} μ_{n}(λ) = μ_{n} for any n ∈ N. (7.2)

We end this section by giving some remarks on Theorem 7.2.

Essentially we have only to consider H(λ) in the neighborhoods of the wells of V and we can approximate it by the quadratic operator H_{(λB^{a},λ^{2}K^{a})} + λv(p^{a}) near each well p^{a}.

In order to illustrate this folklore, it might be useful to refer to the cases when we can consider the trace Tr(exp(−tH(λ))), t > 0, of the semigroup generated by H(λ). For example, if we assume moreover that there exists a constant δ > 0 such that V (x) ≧ |x|^{δ} holds for sufficiently large |x|, then we can prove

lim_{λ→∞} Tr(exp(−tH(λ))) = ∑_{a=1}^{N} Tr(exp(−tH^{a})) = ∑_{n=1}^{∞} exp(−tμ_{n})

and (7.2) by virtue of the continuity theorem of the Laplace transformation.

If we consider the problem on a compact Riemannian manifold, the additional assumptions on V are not necessary and we can prove (7.2) under the assumptions corresponding to (A.1) and (A.3). See [17] and [29].

In [17], it is assumed that the wells of V are also stationary points of the magnetic field (B_{i,j}(x)) by a technical reason. We found that such an assumption is not necessary to prove (7.2). For details, see Section 10.

8. Degenerate Wells

In this section, continued from the previous section, we discuss the asymptotic behavior of the eigenvalues of the Schrödinger operator H(λ) defined by (7.1). We admit that the Hessian matrices of V at the wells are degenerate and that the set of the wells contains a compact submanifold of R^{d}. For the related topics when A ≡ 0, see the works of Helffer-Sjöstrand [9] and [10].

Besides the assumptions (A.1) and (A.2), we assume the following :

(A.4)(i) V^{−1}(0) is decomposed into the disjoint union as

V^{−1}(0) = M_{1} ⋃ M_{2} ⋃ M,

where M_{1} = {p^{a}}_{a∈Λ_{1}}, M_{2} = {q^{b}}_{b∈Λ_{2}} are finite subsets of R^{d} and M is a compact submanifold of R^{d} ; (ii) The Hessian matrix K^{a} = K(p^{a}) of V at p^{a}, a ∈ Λ_{1}, is non-degenerate ; (iii) The Hessian matrix K(q^{b}) of V at q^{b}, b ∈ Λ_{2}, is degenerate ; (iv) The rank of the Hessian matrix of V at q ∈ M is equal to the codimension of M.

Denoting the spectrum of a self-adjoint operator L on L^{2}(R^{d}) by σ(L), we define the increasing sequence {μ_{n}}_{n=1}^{∞} in the same way as in the previous section such that

{μ_{n}}_{n=1}^{∞} = ⋃_{a∈Λ_{1}}{σ(H^{a}) + v(p^{a})}

holds including the multiplicities, where H^{a} = H_{(B^{a},K^{a})} is the quadratic operator defined by (2.1).

For q ∈ M_{2} ∪ M also, we consider the quadratic operator H^{q} = H_{(B(q),K(q))}. Note that, since K(q) is degenerate for q ∈ M_{2} ∪ M, σ(H^{q}) consists only of the eigenvalues with infinite multiplicities or it is absolutely continuous by Theorem 6.3.

We set

ν = min{inf σ(H^{q}); q ∈ M_{2} ⋃ M}.  (8.1)

When V has only the finite non-degenerate wells, we define ν = ∞.

Then we can mention the main result in this section. Theorem 8.1. Assume (A. 1), (A. 2) and (A. 4). Then, for every eigenvalue μ_{n}(λ) of H(λ) below the essential spectrum, it holds that

lim_{λ→∞} \frac{1}{λ} μ_{n}(λ) = min\{μ_{n}, ν\}.  (8.2)

The proof of the theorem will be given in the following Sections 9 and 10.

The degenerate convergence in (8.2) is intuitively clear. If we admit the one-to-one correspondence of the eigenvalues {μ_{n}(λ)} of H(λ) and those of the direct sum of the quadratic operators, which we have exactly in the previous section, ν in the essential spectrum can correspond to any number of μ_{n}(λ)’s.

9. Upper Bounds

The purpose of this section is to prove the following : Proposition 9.1. Under the assumptions of Theorem 8.1, H(λ) has any number of the discrete spectrum μ_{n}(λ) below the essential spectrum if λ is sufficiently large and

lim sup_{λ→∞} \frac{1}{λ} μ_{n}(λ) ≦ min\{μ_{n}, ν\}.  (9.1)

The method of the proof, essentially borrowed from Simon [25], is based on the Rayleigh-Ritz principle. The eigenfunctions for the quadratic operators are important in order to construct the test functions. See also [17].

Before proceeding to the proof, we prepare some lemmas. For p ∈ V^{−1}(0), we consider the following quadratic operator :

H_{p}^{0}(λ) = \frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}∂_{j} − λA_{0,j}^{p})^{2} + \frac{λ^{2}}{2} ∑_{i,j=1}^{d} K_{i,j}^{p}x_{i}x_{j} + λv(p)

and

H_{p}^{ε}(λ) = H_{p}^{0}(λ) + ε^{2}λ^{2}|x|^{2},

where ε > 0 and A_{0,j}^{p}(x) = ∑_{k=1}^{d} ∂_{k}A_{j}(p)x_{k}. Moreover we set H_{p}^{ε} = H_{p}^{ε}(1). Then we have the following. Lemma 9.2. For each p ∈ V^{−1}(0), there is an increasing sequence {μ_{n}^{p}(ε)}_{n=1}^{∞} of eigenvalues and the complete orthonormal basis {ϕ_{n}^{p,ε}}_{n=1}^{∞} of L^{2}(R^{d}) consisting of the corresponding eigenfunctions of H_{p}^{ε}. Moreover ϕ_{n}^{p,ε} decays exponentially at infinity and the order of the decay depends on ε > 0.

Proof. Let θ_{0}^{p}(x) = B(p)x/2, B(p)_{i,j} = ∂_{j}A_{i}(p) − ∂_{i}A_{j}(p). Since d(A_{0}^{p} − θ_{0}^{p}) = 0 if we identify the vector potentials with the differential 1-forms, there exists a homogeneous quadratic polynomial g^{p} such that A_{0}^{p} − θ_{0}^{p} = dg^{p}. Then the gauge invariance says

H_{p}^{ε}(λ) = (G_{λ}^{p})^{∗} H_{p}^{s,ε}(λ) G_{λ}^{p},

where G_{λ}^{p} is the multiplication operator defined by exp(\sqrt{−1}λg^{p}) and

H_{p}^{s,ε}(λ) = \frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}∂_{j} − λθ_{0,j}^{p})^{2} + \frac{λ^{2}}{2} ∑_{i,j=1}^{d} K_{i,j}^{p}x_{i}x_{j} + ε^{2}λ^{2}|x|^{2} + λv(p).

For the operator H_{p}^{s,ε}(λ), the corresponding eigenvalues and eigenfunctions are given by Theorem 2.2 and Proposition 3.3. Combining these results, we can complete the proof. □

We consider also the following quadratic operators : for p ∈ V^{−1}(0),

P_{p}^{0}(λ) = \frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}∂_{j} − λA_{j}^{p})^{2} + \frac{λ^{2}}{2} ∑_{i,j=1}^{d} K_{i,j}^{p}(x_{i} − p_{i})(x_{j} − p_{j}) + λv(p)  (9.2)

and

P_{p}^{ε}(λ) = P_{p}^{0}(λ) + ε^{2}λ^{2}|x − p|^{2},

where A_{j}^{p}(x) = ∑_{k=1}^{d} ∂_{k}A_{j}(p)(x_{k} − p_{k}).

Then, defining a unitary operator U_{λ}^{p} in L^{2}(R^{d}) by

(U_{λ}^{p}f)(x) = λ^{d/4}f(λ^{1/2}(x − p)), f ∈ L^{2}(R^{d}),

it is easy to verify the following. Lemma 9.3. P_{p}^{0}(λ) = λU_{λ}^{p}H_{p}^{0}(U_{λ}^{p})^{−1}.

We let h be a C^{∞} function such that 0 ≦ h ≦ 1, h(x) = 1 if |x| ≦ 1 and supp(h) ⊆ {|x| ≦ 2} and set

h_{λ}^{p}(x) = h(λ^{2/5}(x − p)).

Moreover, setting

f^{p}(x) = ∑_{j=1}^{d} A_{j}(p)(x_{j} − p_{j}) and F_{λ}^{p}(x) = exp(\sqrt{−1}λf^{p}(x)),

we define a function φ_{p,n}^{ε,λ} by

φ_{p,n}^{ε,λ}(x) = \overline{F_{λ}^{p}(x)}h_{λ}^{p}(x)(U_{λ}^{p}ϕ_{n}^{p,ε})(x).

Lemma 9.4. There exists a constant C_{ε}, dependent on ε and independent of λ, such that

(φ_{p,k}^{ε,λ}, φ_{p,ℓ}^{ε,λ}) = δ_{k,ℓ} + O(e^{−C_{ε}λ^{1/5}})  (9.3)

as λ ↑ ∞, where (·, ·) is the inner product in L^{2}(R^{d}) and the large O depends on k and ℓ.

Proof. By definitions, we have

(φ_{p,k}^{ε,λ}, φ_{p,ℓ}^{ε,λ}) = δ_{k,ℓ} + ∫_{R^{d}}(h(λ^{−1/10}y)^{2} − 1)\overline{ϕ_{k}^{p,ε}(y)}ϕ_{ℓ}^{p,ε}(y)dy

and

|ϕ_{n}^{p,ε}(y)| ≦ |G_{n}^{p,ε}(y)| exp(−C_{ε}|x|^{2})  (9.4)

for some polynomial G_{n}^{p,ε} and some positive constant C_{ε}, both of which depends on ε in general.

The rest of the proof is easy. □

Moreover we show the following. Lemma 9.5. It holds that

(φ_{p,k}^{ε,λ}, H(λ)φ_{p,ℓ}^{ε,λ}) = λ(ϕ_{k}^{p,ε}, H_{p}^{0}ϕ_{ℓ}^{p,ε}) + O(λ^{4/5})

as λ ↑ ∞, where the large O depends on k, ℓ and ε.

Proof. We set H̃^{p}(λ) = F_{λ}^{p}H(λ)(F_{λ}^{p})^{∗}. Then we have

(φ_{p,k}^{ε,λ}, H(λ)φ_{p,ℓ}^{ε,λ}) = (h_{λ}^{p}U_{λ}^{p}ϕ_{k}^{p,ε}, H̃^{p}(λ)h_{λ}^{p}U_{λ}^{p}ϕ_{ℓ}^{ε,λ}).

Setting

I_{1} = (h_{λ}^{p}U_{λ}^{p}ϕ_{k}^{p,ε}, P_{p}^{0}(λ)h_{λ}^{p}U_{λ}^{p}ϕ_{ℓ}^{p,ε})

and

I_{2} = (h_{λ}^{p}U_{λ}^{p}ϕ_{k}^{p,ε}, (H̃^{p}(λ) − P_{p}^{0}(λ))h_{λ}^{p}U_{λ}^{p}ϕ_{ℓ}^{p,ε}),

we show

I_{1} = λ(ϕ_{k}^{p,ε}, H_{p}^{0}ϕ_{ℓ}^{p,ε}) + O(e^{−C_{ε}λ^{1/5}}) and I_{2} = O(λ^{4/5}).

For this purpose, we write

I_{1} = I_{11} + I_{12} + I_{13} + I_{14},

where

\begin{matrix}I_{11} = (h_{λ}^{p}U_{λ}^{p}ϕ_{k}^{p,ε}, h_{λ}^{p}P_{p}^{0}(λ)U_{λ}^{p}ϕ_{ℓ}^{p,ε}), \\ I_{13} = −(h_{λ}^{p}U_{λ}^{p}ϕ_{k}^{p,ε}, ⟨∇h_{λ}^{p}, ∇U_{λ}^{p}⟩ϕ_{ℓ}^{p,ε}), \end{matrix} \begin{matrix}I_{12} = −(h_{λ}^{p}U_{λ}^{p}ϕ_{k}^{p,ε}, (Δh_{λ}^{p})U_{λ}^{p}ϕ_{ℓ}^{p,ε})/2, \\ I_{14} = −λ(h_{λ}^{p}U_{λ}^{p}ϕ_{k}^{p,ε}, \sqrt{−1}⟨A^{p}, ∇h_{λ}^{p}⟩U_{λ}^{p}ϕ_{ℓ}^{p,ε}). \end{matrix}

By the same way as in the proof of Lemma 9.4, we get I_{11} = (U_{λ}^{p}ϕ_{k}^{p,ε}, P_{p}^{0}(λ)U_{λ}^{p}ϕ_{ℓ}^{p,ε}) + O(e^{−C_{ε}λ^{1/5}}).

Therefore Lemma 9.3 allows us to conclude

I_{11} = λ(ϕ_{k}^{p,ε}, H_{p}^{0}ϕ_{ℓ}^{p,ε}) + O(e^{−C_{ε}λ^{1/5}}).

Since the supports of Δh_{λ}^{p} and ∇h_{ϕ}^{p} are included in the set {x; λ^{−2/5} ≦ |x−p| ≦ 2λ^{−2/5}}, we have only to consider the integral over this set for I_{12}, I_{13}, I_{14} and it is easy to get

|I_{1i}| = O(e^{−C_{ε}λ^{1/5}}), i = 2, 3, 4.

By expanding H̃(λ) − P_{p}^{0}(λ), we can prove I_{2} = O(λ^{4/5}) by the same way as in [17] and [25]. The details are omitted. □

Now we are in the position to give a proof of Proposition 9.1.

Proof of Proposition 9.1. By the assumptions (A.1) and (A.2), there exists C > 0 such that

inf σ_{ess}(H(λ)) ≧ Cλ^{2}

for sufficiently large λ, where σ_{ess}(H(λ)) is the essential spectrum of H(λ). Therefore the first assertion follows from Lemmas 9.4, 9.5 and the Rayleigh-Ritz principle.

We set Ψ_{k,ℓ}^{p,ε} = (ϕ_{k}^{p,ε}, H_{p}^{0}ϕ_{ℓ}^{p,ε}) and let ν_{m}^{p}(ε) be the maximum eigenvalue of the m × m symmetric matrix (Ψ_{k,ℓ}^{p,ε})_{k,ℓ=1,2,...,m}. Moreover we set

ϕ_{ξ}^{p,ε} = ∑_{k=1}^{m} ξ_{k}ϕ_{k}^{p,ε} for ξ ∈ C^{m}.

Then we have

ν_{m}^{p}(ε) = sup{(ϕ_{ξ}^{p,ε}, H_{p}^{0}ϕ_{ξ}^{p,ε}); |ξ| = 1} ≦ sup{(ϕ_{ξ}^{p,ε}, H_{p}^{ε}ϕ_{ξ}^{p,ε}); |ξ| = 1} = μ_{m}^{p}(ε). Now we define n_{0} ∈ N by

n_{0} = sup{n; μ_{n} ≦ ν}.

For n = 1, 2, ..., n_{0}, there exists p_{n} ∈ M_{1} and m(n) ∈ N such that

lim_{ε↓0} ν_{m(n)}^{p_{n}}(ε) = lim_{ε↓0} μ_{m(n)}^{p_{n}}(ε) = μ_{n}.

Therefore we may assume that {ν_{m(k)}^{p_{k}}(ε)}_{k=1}^{n_{0}} is an increasing sequence.

Then, by the Rayleigh-Ritz principle, we have

lim sup_{λ↑∞} \frac{1}{λ} μ_{n}(λ) ≦ ν_{m(n)}^{p_{n}}(ε)

and, letting ε tend to 0, we get

lim sup_{λ↑∞} \frac{1}{λ} μ_{n}(λ) ≦ μ_{n}

for n = 1, 2, ..., n_{0}.

Next we choose any q ∈ M_{2} ∪ M such that

ν = inf σ(P_{q}^{0}(1)),

where P_{q}^{0}(λ) is the quadratic operator defined by (9.2). Then we get, by Lemmas 9.4, 9.5 and the Rayleigh-Ritz principle again,

lim sup_{λ↑∞} \frac{1}{λ} μ_{n}(λ) ≦ ν_{n−n_{0}}^{q}(ε)

for every n > n_{0}. Since ν_{k}^{q}(ε) tends to ν as ε ↓ 0 for every k = 1, 2, ..., we get

lim sup_{λ↑∞} \frac{1}{λ} μ_{n}(λ) ≦ ν for every n > n_{0}.

We have completed the proof of Proposition 9.1. □

10. Lower Bounds

The purpose of this section is to prove

lim inf_{λ→∞} \frac{1}{λ} μ_{n}(λ) ≧ min\{μ_{n}, ν\}  (10.1)

and complete the proof of Theorem 8.1.

We begin with a general proposition.

Proposition 10.1. Letting W_{1}, W_{2}, ..., W_{d} be C^{∞} functions and θ be a differential 1-form on R^{d}, we set

σ(x) = (σ_{α,β}(x)) = (\begin{matrix}−dθ(x) & ∇W (x) \\ −(∇W (x))^{t} & 0 \end{matrix}) ,

where dθ is identified with the d×d skew-symmetric matrix (∂_{i}θ_{j}−∂_{j}θ_{i}) and ∇W = (∂_{i}W_{j}). Then, for any ξ ∈ ⋀ T^{∗}(R^{d}) ⊗ C with (ξ, ξ) = 1 and for any system of Dirac matrices {γ_{α}}_{α=1}^{2d}, the operator P_{ξ} defined by

P_{ξ} = \frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}∂_{j} − θ_{j})^{2} + \frac{1}{2} ∑_{j=1}^{d} W_{j}^{2} + \frac{1}{4}(ξ, ∑_{α,β=1}^{2d} σ_{α,β}\sqrt{−1}γ_{α}γ_{β}ξ)

is a positive operator, where (·, ·) is the natural inner product in ⋀ T^{∗}(R^{d}) ⊗ C.

Proof. Consider the Dirac operator

Q = \frac{1}{\sqrt{2}} ∑_{j=1}^{d}\{γ_{j}(\sqrt{−1}∂_{j} − θ_{j}) + γ_{d+j}W_{j}\}.

Then, noting that we have adopted the inner product which is complex linear in the second variable, we get

(φ, P_{ξ}φ)_{L^{2}} = ∥Q(φξ)∥_{L^{2}}^{2}

for any φ ∈ C_{0}^{∞}(R^{d}), where (·, ·)_{L^{2}} is the inner product in L^{2}(R^{d}) and the norm in the right hand side is that of L^{2}(⋀ T^{∗}(R^{d}) ⊗ C). □

Remark. By Proposition 10.1, we can show the following extension of the results in [27]: if the matrix valued function σ(x) is uniformly continuous, then

lim inf_{λ→∞} \frac{1}{λ} inf σ(L(λ)) ≧ inf_{x∈R^{d}} \frac{∥σ(x)∥_{1}}{4} ,

where L(λ) is the Friedrichs extension of the symmetric operator

\frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}∂_{j} − λθ_{j})^{2} + \frac{1}{2} ∑_{j=1}^{d}(λW_{j})^{2}

on C_{0}^{∞}(R^{d}) and ∥ · ∥_{1} is the trace norm. For the related results, see also [8] and [19].

Before proceeding to the proof of (10.1), we show the following.

Lemma 10.2. There exists a differential 1-form α = ∑_{i=1}^{d} α_{i}dx_{i} such that (i) dα = dA, (ii) α_{k}(p^{a}) = 0 and (iii) ∂_{j}α_{k}(p^{a}) = ∂_{j}A_{k}(p^{a}) for every j, k = 1, 2, ..., d and every a ∈ Λ_{1}.

Proof. Let j_{a}, a ∈ Λ_{1}, be C^{∞} functions with compact disjoint supports such that j_{a} is equal to one near p^{a} and the supports do not intersect with M_{2} ∪ M. Moreover we set

g_{a}(x) = ∑_{i=1}^{d} A_{i}(p^{a})(x_{i} − p_{i}^{a})

Then, setting

f(x) = ∑_{a∈Λ_{1}} j_{a}(x)g_{a}(x) and α_{k}(x) = A_{k}(x) − ∂_{k}f(x),

we can easily show that α satisfies the assertion of the lemma. □

Note that, denoting by F_{λ} the multiplication operator defined by exp(\sqrt{−1}λf), we have

H̃(λ) ≡ H_{(α,V,v)}(λ) = F_{λ}H_{(A,V,v)}(λ)F_{λ}^{∗}.  (10.2)

For the proof of (10.1), we introduce the IMS localization formula ([1] and [25]) in a convenient form for our purpose.

Fix arbitrary ε > 0. Then, since M is assumed to be compact, there exists a finite subset {r^{c}}_{c∈Λ_{3}} of M such that

⋃_{c∈Λ_{3}} B(r^{c}, ε) ⊃ M,

where B(r^{c}, ε) is the ε-ball with center r^{c}. We set Λ = Λ_{1} ∪ Λ_{2} ∪ Λ_{3} and ε_{c} = λ^{−2/5} for c ∈ Λ_{1} and ε_{c} = ε for c ∈ Λ_{2} ∪ Λ_{3}, respectively. Thus, denoting p^{c} and q^{c} also by r^{c} for c ∈ Λ_{1} ∪ Λ_{2}, we get a finite covering {B(r^{c}, ε_{c})}_{c∈Λ} of V^{−1}(0).

We consider a partition of unity {h_{c}}_{c∈Λ′}, Λ′ = Λ ∪ {0} subordinate to {B(r^{c}, ε_{c})}_{c∈Λ}, that is, {h_{c}} is a system of non-negative functions such that

\begin{matrix}supp(h_{c}) ⊂ B(r^{c}, 2ε_{c}), \\ ∑_{c∈Λ′} h_{c}(x)^{2} ≡ 1, \end{matrix}  \begin{matrix}supp(h_{0}) ⊂ \overline{⋂_{c∈Λ}\{R^{d} \ B(r^{c}, ε_{c})\}} ≡ B(λ, ε), \\ ∑_{c∈Λ′} |∇h_{c}(x)|^{2} < ∞. \end{matrix}

Then we have the following identity, called the IMS localization formula :

H(λ) = ∑_{c∈Λ′} h_{c}H(λ)h_{c} − \frac{1}{2} ∑_{c∈Λ′} |∇h_{c}|^{2}.  (10.3)

This formula is well known when A = 0 and it is easy to show that it holds without any change in the presence of magnetic fields. We may assume that

∑_{c∈Λ′} |∇h_{c}(x)|^{2} ≦ C_{1}(ε^{−2} + λ^{4/5})  (10.4)

for some positive constant C_{1}. We will denote the constants independent of λ and ε by C_{i}’s.

Next let H_{c}(λ, ε) be the operator H(λ) defined on L^{2}(B(r^{c}, 2ε_{c})) and L^{2}(B(λ, ε)) for c ∈ Λ and c = 0, respectively, with the Dirichlet boundary conditions. Then, by (10.3) and (10.4), we have

inf σ(H(λ)) ≧ inf_{c∈Λ′}{inf σ(H_{c}(λ, ε))} − C_{1}(ε^{2} + λ^{4/5}).  (10.5)

As the first step to the proof of (10.1), we give a lower bound for σ(H_{c}(λ, ε)).

Since V (x) ≧ min{C_{ε}, C_{2}λ^{−4/5}}, x ∈ B(λ, ε), for some C_{ε}, C_{2} > 0, we have

inf σ(H_{0}(λ, ε)) ≧ min{C_{ε}λ^{2}, C_{2}λ^{6/5}}.  (10.6)

For c ∈ Λ, we consider the quadratic operator H_{c} = H_{(B^{c},K^{c})}, where B_{i,j}^{c} = ∂_{i}A_{j}(r^{c}) − ∂_{j}A_{i}(r^{c}) and K_{i,j}^{c} = ∂^{2}V (r^{c})/∂x_{i}∂x_{j}. We set

n(r^{c}) = inf σ(H_{c}) = ∑_{j=1}^{d} s_{j}^{c}/2,

where s_{j}^{c} ≧ 0 and ±\sqrt{−1}s_{j}^{c}, j = 1, 2, ..., d, are the eigenvalues of the 2d × 2d matrix Ω_{(B^{c},K^{c})} defined by (2.3). Note that at least one of s_{j}^{c}’s is zero for c ∈ Λ_{2} ∪ Λ_{3} since K^{c} is degenerate.

Then we show : Lemma 10.3. Under the assumptions (A. 1), (A. 2) and (A. 4), there exists a positive constant C_{3}, independent of λ and ε, such that

inf σ(H_{c}(λ, ε)) ≧ λ{n(r^{c}) + v(r^{c}) − C_{3}ε}  (10.7)

holds for every c ∈ Λ and any sufficiently large λ.

Proof. We give the proof for c ∈ Λ_{3}. The proof of the case c ∈ Λ_{1} ∪ Λ_{2} is similar and easier.

Fix c ∈ Λ_{3} and choose a coordinate system (y_{1}, y_{2}, ..., y_{d}) around r^{c} such that y̌ = (y_{ℓ+1}, ..., y_{d}) is a coordinate in V^{−1}(0) ∩ B(r^{c}, 2ε), where d − ℓ = dim M. Moreover we set ŷ = (y_{1}, ..., y_{ℓ}) and

ˆ∇^{2}V (y) = (∂^{2}V/∂y_{i}∂y_{j})_{i,j=1,...,ℓ}.

Then, by the assumption (A.4), there exist C_{4} > 0 and C_{5} > 0 such that

V (y) ≧ \frac{1}{2}(1 − C_{4}ε)⟨ŷ, ˆ∇^{2}V (r^{c})ŷ⟩_{ℓ}

and

v(y) ≧ v(r^{c}) − C_{5}ε

hold for any y ∈ B(r^{c}, 2ε), where ⟨·, ·⟩_{ℓ} is the usual inner product in R^{ℓ}.

Now we apply Proposition 10.1 by setting W_{j}(y) = λ(\sqrt{ˆ∇^{2}V (r^{c})}ŷ)_{j}, j = 1, 2, ..., ℓ,

W_{ℓ+1} = ⋯ = W_{d} = 0,

where \sqrt{ˆ∇^{2}V} is the positive symmetric root of ˆ∇^{2}V. Then we get

inf σ(H_{c}(λ, ε)) ≧ λ(L_{ξ}^{c}/2 + v(r^{c}) − C_{5}ε),  (10.8)

where

L_{ξ}^{c} = −(ξ, ∑_{α<β} σ_{α,β}^{c}\sqrt{−1}γ_{α}γ_{β}ξ),

σ^{c} = (σ_{α,β}^{c}) = (\begin{matrix}−dA(r^{c}) & \sqrt{∇_{y}^{2}V (r^{c})} \\ −\sqrt{∇_{y}^{2}V (r^{c})} & 0 \end{matrix}) .

As in Section 2, we choose a 2d × 2d orthogonal matrix U = (U_{α,β}) such that

U^{t} σ^{c} U = (\begin{matrix}0 & Λ^{c} \\ −Λ^{c} & 0 \end{matrix}) ,

Λ^{c} is a diagonal matrix with diagonal elements s_{j}^{c}, and define ω_{β}, β = 1, 2, ..., 2d, by

ω_{β} = ∑_{α=1}^{2d} U_{α,β}γ_{α}.

Then {ω_{β}}_{β=1}^{2d} is also a system of Dirac matrices and we get

L_{ξ}^{c} = (ξ, ∑_{j=1}^{d} s_{j}^{c}\sqrt{−1}ω_{d+j}ω_{j}ξ).

Letting ξ be an element in V (1, 1, ..., 1) defined in the proof of Proposition 5.4, we obtain

L_{ξ}^{c} = ∑_{j=1}^{d} s_{j}^{c}.

Combining this with (10.8), we get (10.7). □

Now we are in the position to complete the proof of (10.1). The method in the first part is essentially borrowed from [25]. We refer also to [17].

Proof of (10.1). We set n_{0} = sup{n; μ_{n} ≦ ν}.

For n = 1, (10.5) and (10.7) imply

lim inf_{λ→∞} \frac{1}{λ} μ_{1}(λ) = lim inf_{λ→∞} \frac{1}{λ} inf σ(H(λ))

≧ min_{c∈Λ}{n(r^{c}) + v(r^{c}) − C_{3}ε}.

Since μ_{1} = min{n(r^{c}) + v(r^{c}); c ∈ Λ} and ε is arbitrary, we get

lim inf_{λ→∞} \frac{1}{λ} μ_{1}(λ) ≧ μ_{1}.

If μ_{1} has the multiplicity n_{1} ≧ 2,

lim inf_{λ→∞} \frac{1}{λ} μ_{n_{1}}(λ) ≧ lim inf_{λ→∞} \frac{1}{λ} μ_{1}(λ) = μ_{1} = μ_{n_{1}}.

Next we let n be such that μ_{n} < μ_{n+1} ≦ μ_{n_{0}} and choose μ with μ_{n} < μ < μ_{n+1}.

We use the IMS localization formula. At first we note that

(ϕ, h_{c}H(λ)h_{c}ϕ) ≧ νλ∥h_{c}ϕ∥^{2}  (10.9)

holds for every c ∈ Λ_{2} ∪ Λ_{3} and every ϕ ∈ C_{0}^{∞}(R^{d}), where ∥ · ∥ and (·, ·) are the norm and the inner product in L^{2}(R^{d}), respectively. Moreover, by (10.6), we have

(ϕ, h_{0}H(λ)h_{0}ϕ) ≧ min{C_{ε}λ^{2}, C_{2}λ^{6/5}}∥h_{0}ϕ∥^{2} ≧ μλ∥h_{0}ϕ∥^{2}  (10.10)

if ε > 0 is fixed and λ is sufficiently large.

For a ∈ Λ_{1}, using the operators P_{p}^{0}(λ) and H̃(λ) defined by (9.2) and (10.2), we write

(ϕ, h_{a}H(λ)h_{a}ϕ) = I_{1,a}^{λ}(ϕ) + I_{2,a}^{λ}(ϕ),

where

I_{1,a}^{λ}(ϕ) = (ϕ, h_{a}F_{λ}^{∗}P_{p^{a}}^{0}(λ)F_{λ}h_{a}ϕ)

and

I_{2,a}^{λ}(ϕ) = (ϕ, h_{a}F_{λ}^{∗}(H̃(λ) − P_{p^{a}}^{0}(λ))F_{λ}h_{a}ϕ).

Now let L_{λ}^{a}, a ∈ Λ_{1}, be the orthogonal projections onto the eigenspace of P_{p^{a}}^{0}(λ) corresponding to the eigenvalues less than λμ.

Then we have

I_{1,a}^{λ}(ϕ) = (F_{λ}h_{a}ϕ, P_{p^{a}}^{0}(λ)(1 − L_{λ}^{a})F_{λ}h_{a}ϕ) + (F_{λ}h_{a}ϕ, P_{p^{a}}^{0}(λ)L_{λ}^{a}F_{λ}h_{a}ϕ)
= ((1 − L_{λ}^{a})F_{λ}h_{a}ϕ, P_{p^{a}}^{0}(λ)(1 − L_{λ}^{a})F_{λ}h_{a}ϕ) + (F_{λ}h_{a}ϕ, P_{p^{a}}^{0}(λ)L_{λ}^{a}F_{λ}h_{a}ϕ)
≧ λμ∥F_{λ}h_{a}ϕ∥^{2} + (ϕ, h_{a}F_{λ}^{∗}L_{λ}^{a}P_{p^{a}}^{0}(λ)F_{λ}h_{a}ϕ).  (10.11)

For the estimate for I_{2,a}^{λ}(ϕ), we write

H̃(λ) − P_{p^{a}}^{0}(λ) = J_{1}^{a}(λ) + J_{2}^{a}(λ),

where J_{1}^{a}(λ) is the first order differential operator defined by

J_{1}^{a}(λ) = −\sqrt{−1}λ ∑_{j=1}^{d}\{α_{j}(x) − ∑_{k=1}^{d} ∂_{k}α_{j}(p^{a})(x_{k} − p_{k}^{a})\}\frac{∂}{∂x_{j}}

and J_{2}^{a}(λ) is the multiplication operator defined by the function

\frac{−\sqrt{−1}λ}{2}(div α(x) − div α(p^{a})) + \frac{λ^{2}}{2} \{|α(x)|^{2} − ∑_{j=1}^{d}(∑_{k=1}^{d} ∂_{k}α_{j}(p^{a})(x_{k} − p_{k}^{a}))^{2}\}

+ λ^{2}(V (x) − \frac{1}{2} ∑_{i,j=1}^{d} K_{i,j}^{a}(x_{i} − p_{i}^{a})(x_{j} − p_{j}^{a})) + λ(v(x) − v(p^{a})).

For the term related J_{1}^{a}(λ), we estimate as follows:

|(ϕ, h_{a}F_{λ}^{∗}J_{1}^{a}(λ)F_{λ}h_{a}ϕ)| ≦ (λ^{4/5}∥h_{a}ϕ∥^{2} + λ^{−4/5}∥J_{1}^{a}(λ)F_{λ}h_{a}ϕ∥^{2})/2.

Since the coefficients of J_{1}^{a}(λ) is of O(|x − p^{a}|^{2}) as x → p^{a}, we have

∥J_{1}^{a}(λ)F_{λ}h_{a}ϕ∥^{2} ≦ C_{6}λ^{2/5}∥∇(F_{λ}h_{a}ϕ)∥^{2}.

Moreover, since α is of O(|x − p^{a}|) as x → p^{a}, we have

∥∇(F_{λ}h_{a}ϕ)∥^{2} ≦ 2(∥(\sqrt{−1}∇ − λα)(F_{λ}h_{a}ϕ)∥^{2} + ∥λαh_{a}ϕ∥^{2})

≦ 2∥(\sqrt{−1}∇ − λα)(F_{λ}h_{a}ϕ)∥^{2} + C_{7}λ^{6/5}∥h_{a}ϕ∥^{2}.

Combining these, we obtain

∥(ϕ, h_{a}F_{λ}^{∗}J_{1}^{a}(λ)F_{λ}h_{a}ϕ)| ≦ C_{8}λ^{−2/5}(h_{a}ϕ, H(λ)h_{a}ϕ) + C_{9}λ^{4/5}∥h_{a}ϕ∥^{2}. (10.12)

For the term related to J_{2}^{a}(λ), it is easy to prove (cf. [17])

|(ϕ, h_{a}F_{λ}^{∗}J_{2}^{a}(λ)F_{λ}h_{a}ϕ)| ≦ C_{10}λ^{4/5}∥h_{a}ϕ∥^{2}.  (10.13)

By (10.11), (10.12) and (10.13), we get

\begin{aligned}∑_{a∈Λ_{1}}(ϕ, h_{a}H(λ)h_{a}ϕ) ≧ λμ ∑_{a∈Λ_{1}} & ∥h_{a}ϕ∥^{2} + (ϕ, L_{n}(λ)ϕ) \\ & − C_{11}λ^{4/5}∥ϕ∥^{2} − C_{12}λ^{−2/5}(ϕ, H(λ)ϕ), \end{aligned} (10.14)

where L_{n}(λ) is the operator of rank at most n given by

L_{n}(λ) = ∑_{a∈Λ_{1}} h_{a}F_{λ}^{∗}L_{λ}^{a}P_{p^{a}}^{0}(λ)F_{λ}h_{a}.

Combining this with (10.9) and (10.10), we obtain

(1 + C_{12}λ^{−2/5})(ϕ, H(λ)ϕ) ≧ λμ∥ϕ∥^{2} + (ϕ, L_{n}(λ)ϕ) − C_{11}λ^{4/5}∥ϕ∥^{2}

and

lim inf_{λ→∞} \frac{1}{λ} μ_{n+1}(λ) ≧ μ, μ_{n} < μ < μ_{n+1}.

Since μ is arbitrary, we have proved

lim inf_{λ→∞} \frac{1}{λ} μ_{n+1}(λ) ≧ μ_{n+1}.

Finally we consider n > n_{0}. We have ν < μ_{n}. Then, choosing μ = ν, we can prove (10.10) and (10.14) by the same way. Therefore, combining these with (10.9) and summing up, we get

lim inf_{λ→∞} \frac{1}{λ} μ_{n}(λ) ≧ ν.

The proof of (10.1) and Theorem 8.1 is now completed. □

11. Magnetic Well

In this section, we consider the Schrödinger operator H_{A}(λ) defined by

H_{A}(λ) = \frac{1}{2} ∑_{j=1}^{d}(\sqrt{−1}∂_{j} − λA_{j})^{2}

on L^{2}(R^{d}), where A_{j}, j = 1, 2, ..., d are real smooth functions. We set as before B_{i,j}(x) = ∂_{j}A_{i}(x) − ∂_{i}A_{j}(x) and B(x) = (B_{i,j}(x)).

Denoting by s(x) the trace norm ∥B(x)∥_{1} of B(x), we assume the following :

(A.5) (i) lim inf_{|x|→∞} s(x) ≡ s_{∞}^{−} > s_{0} ≡ inf{s(x); x ∈ R^{d}} ;

(ii) sup_{x∈R^{d}} \frac{|(∇B_{j,k})(x)|}{1 + s(x)} < ∞, for any j, k = 1, 2, ..., d.

Following [7], we call each connected component of the set {x ∈ R^{d}; s(x) = s_{0}} a magnetic well.

The purpose of this section is to show the following : Theorem 11.1. Under the assumption (A. 5), H_{A}(λ) has any number of the discrete spectrum {μ_{n}(λ)} below its essential spectrum if λ is sufficiently large and it holds that

lim_{λ↑∞} \frac{1}{λ} μ_{n}(λ) = s_{0}/4

for every n ∈ N.

For the related results, see [3], [15], [20] and [27].

Proof. Modifying the proof of the Persson’s theorem (cf. [1] and [20]), we can prove

inf σ_{ess}(H_{A}(λ)) ≧ s_{∞}^{−}λ/4 − Cλ^{3/4}  (11.1)

where σ_{ess}(H_{A}(λ)) is the essential spectrum of H_{A}(λ) and C is a constant. It is a consequence of the inequality

(H_{A}(λ)ϕ, ϕ) ≧ ∥ϕ∥^{2} inf{(λs(x)/4 − Cλ^{3/4}); x ∈ supp ϕ}  (11.2)

for ϕ ∈ C_{0}^{∞}(R^{d}), which dues to Theorem 3.1 of [7].

Next we choose any p ∈ R^{d} such that s(p) = s_{0}. Changing the coordinate if necessary, we may assume that B(p) is of the form

B(p) = (\begin{matrix}B′(p) & 0 \\ 0 & 0 \end{matrix}) ,

where B′(p) is a skew-symmetric non-degenerate matrix.

Now we consider the quadratic operator

H_{p}(λ) = \frac{1}{2} ∑_{j=1}^{r}(\sqrt{−1}∂_{j} − λ(B(p)x)_{j}/2)^{2} − \frac{1}{2} ∑_{j=r+1}^{d} ∂_{j}^{2},

H_{p}^{r}(λ) = \frac{1}{2} ∑_{j=1}^{r}(\sqrt{−1}∂_{j} − λ(B′(p)x̂)_{j}/2)^{2},

where r is the half of the rank of the matrix B(p) and x̂ = (x_{1}, ..., x_{2r}). The spectrum of H_{p}^{r}(λ) on L^{2}(R^{r}) consists only of the eigenvalues with infinite multiplicities and the least eigenvalue is λs_{0}/4. We denote by {ϕ_{k}^{λ}(x̂)}_{k=1}^{∞} an orthonormal basis of the eigenspace of H_{p}^{r}(λ) corresponding to the least eigenvalue λs_{0}/4.

Moreover we set

φ_{k}^{λ,ε}(x) = ϕ_{k}^{λ}(x̂) ∏_{j=r+1}^{d} \frac{1}{ε\sqrt{2π}} exp(−x_{j}^{2}/2ε^{2})

for ε > 0. Then, by the same way as in Section 9, we can prove

(φ_{k}^{λ,ε}, H_{A}(λ)φ_{ℓ}^{λ,ε}) = λδ_{k,ℓ}s_{0}/4 + O(λ^{4/5})

as λ ↑ ∞, where the large O depends on ε. Therefore, by (11.1), the Rayleigh-Ritz principle enables us to prove that H_{A}(λ) has any number of the discrete spectrum {μ_{n}(λ)} below σ_{ess}(H_{A}(λ)) and that

lim sup_{λ↑∞} \frac{1}{λ} μ_{n}(λ) ≦ s_{0}/4 for every n.

On the other hand, (11.2) implies inf σ(H_{A}(λ)) ≧ λs_{0}/4 − Cλ^{3/4} and

lim inf_{λ↑∞} \frac{1}{λ} μ_{n}(λ) ≧ s_{0}/4.

We have completed the proof of Theorem 11.1. □

Remark. For the operator H_{A}(λ), recently Helffer-Mohamed [7] and Montgomery [20] have discussed the asymptotic behavior of the eigenvalues {μ_{n}(λ)}, in particular, of the least eigenvalue in detail when s_{0} = 0. Moreover the results of Hempel-Herbst [12] on the convergence of the resolvents show the zeros of A and dA are important for this problem. See also Nakamura [21].

Acknowledgement. The authors are most grateful to Professor David Elworthy for the invitation to Mathematics Research Center, University of Warwick, where some of this work was done, and his warm hospitality. They would like to thank Professor B. Driver for fruitful discussions on the quadratic Hamiltonians. The identity (6.2) is essentially due to him. They would like to thank also Professors B. Helffer, N. Ikeda and A. Truman for valuable discussions and comments.

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