Spectral Analysis of a Dirac Operator with a Meromorphic Potential

Asao Arai^{∗}and Kunimitsu Hayashi Department of Mathematics Hokkaido University Sapporo 060-0810, Japan

July 8, 2004

Abstract

We consider an operator Q(V) of Dirac type with a meromorphic potential given in terms of a function V of the form V (z) = λV_{1}(z) + μV_{2}(z), z ∈ C \ {0}, where V_{1} is a complex polynomial of 1/z, V_{2} is a polynomial of z, and λ and μ are non-zero complex parameters. The operator Q(V) acts in the Hilbert space L^{2}(IR^{2}; C^{4}) = ⊕^{4}L^{2}(IR^{2}). The main results we prove include: (i) the (essential) self-adjointness of Q(V); (ii) the pure discreteness of the spectrum of Q(V) ; (iii) if V_{1}(z) = z^{−p} and 4 ≤ deg V_{2} ≤ p + 2, then ker Q(V) ≠ {0} and dim ker Q(V) is independent of (λ, μ) and lower order terms of ∂V_{2}/∂z; (iv) a trace formula for dim ker Q(V).

1 Introduction and Main Results

In this paper we consider an operator of Dirac type with a meromorphic potential and analyze spectral properties of it. To define the operator and state the main results, we first need to introduce some basic objects.

As usual, we denote by σ_{1}, σ_{2}, σ_{3} the Pauli matrices:

σ_{1} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , σ_{2} = ( \begin{matrix}0 & −i \\ i & 0 \end{matrix} ) , σ_{3} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,  (1.1)

which give a self-adjoint representation of a Clifford algebra:

{σ_{j}, σ_{k}} = 2δ_{jk}, j, k = 1, 2, 3,  (1.2)

with {X, Y } := XY + Y X. We also define the following 2 × 2 matrices:

p_{+} := ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) = \frac{1}{2}(1 + σ_{3}), p_{−} := ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) = \frac{1}{2}(1 − σ_{3}).  (1.3)

∗Corresponding author: arai@math.sci.hokudai.ac.jp

We write z = x + iy ∈ C , x, y ∈ IR. We denote by D_{x} and D_{y} the generalized partial differential operators in the variables x and y respectively acting in L^{2}(IR^{2}) = L^{2}(C ), the Hilbert space of square integrable functions on IR^{2} with respect to the two-dimensional Lebesgue measure, and introduce

∂ := \frac{1}{2}(D_{x} − iD_{y}), ¯∂:= \frac{1}{2}(D_{x} + iD_{y}).  (1.4)

For a linear operator T on a Hilbert space, we denote its domain by D(T). For two linear operators S and T on a Hilbert space, D(T + S) := D(T) ∩ D(S) unless otherwise stated.

The meromorphic potential of the Dirac operator we consider is given in terms of a function of the following form:

V (z) = λV_{1}(z) + μV_{2}(z), z ∈ C^{×} := C \ {0},  (1.5)

where V_{1} is a complex polynomial of 1/z, V_{2} is a polynomial of z, and λ and μ are non-zero complex parameters, called the coupling constants.

In general, for a densely defined linear operator T on a Hilbert space, T^{∗} denotes the adjoint of T.

We introduce two Dirac-Weyl type operators acting in L^{2}(IR^{2}; C^{2}) = ⊕^{2}L^{2}(IR^{2}) by

Q_{−}(V) := −\frac{1}{2}σ_{1}D_{x} − \frac{1}{2}σ_{2}D_{y} + ip_{+}∂V − ip_{−}(∂V)^{∗}
= −\frac{1}{2}(σ_{1} + iσ_{2})∂ − \frac{1}{2}(σ_{1} − iσ_{2})¯∂ + ip_{+}(∂V) − ip_{−}(∂V)^{∗},  (1.6)
Q_{+}(V) := \frac{1}{2}(σ_{1} − iσ_{2})¯∂ + \frac{1}{2}(σ_{1} + iσ_{2})∂ − ip_{+}(∂V)^{∗} + ip_{−}(∂V).  (1.7)

Note that

D(Q_{−}(V)) = D(Q_{+}(V)) = D(D_{x}) ∩ D(D_{y}) ∩ D(∂V)  (1.8)

and hence

D(Q_{±}(V)) ⊃ C_{0}^{∞}(IR^{2} \ {0}; C^{2}),  (1.9)

the space of C^{2}-valued infinitely differentiable functions on IR^{2}\{0} with compact support in IR^{2} \ {0}. Therefore Q_{+}(V) and Q_{−}(V) are densely defined. Moreover it is easy to see that

Q_{−}(V) ⊂ Q_{+}(V)^{∗}, Q_{+}(V) ⊂ Q_{−}(V)^{∗}.  (1.10)

Hence Q_{±}(V) are closable.

The Dirac operator we consider in the present paper is defined by

Q(V) := ( \begin{matrix}0 & Q_{−}(V) \\ Q_{+}(V) & 0 \end{matrix} )  (1.11)
= iψ_{2} ¯∂ + iψ_{2}^{∗}∂ + iψ_{1}(∂V) − iψ_{1}^{∗}(∂V)^{∗},  (1.12)

acting in the Hilbert space

L^{2}(IR^{2}; C^{2}) ⊕ L^{2}(IR^{2}; C^{2}) = L^{2}(IR^{2}; C^{4}) = ⊕^{4}L^{2}(IR^{2}),  (1.13)

where

ψ_{1} := \frac{1}{2} ( \begin{matrix}0 & 1 + σ_{3} \\ 1 − σ_{3} & 0 \end{matrix} ) , ψ_{2} = \frac{1}{2} ( \begin{matrix}0 & iσ_{1} + σ_{2} \\ −iσ_{1} − σ_{2} & 0 \end{matrix} ) .  (1.14)

Note that

{ψ_{j}, ψ_{k}} = 0, j, k = 1, 2.  (1.15)

It follows from (1.9) and (1.10) that Q(V) is a symmetric operator with

D(Q(V)) ⊃ C_{0}^{∞}(IR^{2} \ {0}; C^{4}).  (1.16)

Remark 1.1 In practice, in analysis of Q_{±}(V) and Q(V), one works with only ∂V. Hence, from a purely mathematical point of view, it may be more suitable to define Q_{±}(V) and Q(V) with ∂V replaced by a meromorphic function W. But, in this respect, we follow the notations used in early works [6, 1, 2, 3].

In what follows, for a closable operator T, \overline{T} denotes the closure of T. For a polynomial P, we denote the degree of P by deg P. For a Fredholm operator T, we set

index(T) := dim ker T − dim ker T^{∗},  (1.17)

the index of F, where, for a linear operator A, ker A := {ψ ∈ D(A)|Aψ = 0}.

The Dirac operator Q(V) originally comes from supersymmetric quantum mechanics (SSQM) — it is called a supercharge of the N = 2 Wess-Zumino model — and the following facts are known:

(i) The case V_{1} = 0 and deg V_{2} ≥ 2 [6, 2]. In this case, Q(V) is essentially self-adjoint on C_{0}^{∞}(IR^{2}; C^{4}), \overline{Q_{±}(V)} are Fredholm and Q_{−}(V)^{∗} = \overline{Q_{+}(V)} with

ker \overline{Q_{−}(V)} = \{0\},
index(\overline{Q_{+}(V)}) = dim ker \overline{Q_{+}(V)} = deg V_{2} − 1.

Moreover the spectrum of \overline{Q(V)} is purely discrete.

(ii) The case V_{1}(z) = z^{−p} (p ∈ IN) and V_{2} = 0 [3]. In this case, Q(V) is essentially self-adjoint on on C_{0}^{∞}(IR^{2} \ \{0\}; C^{4}) and Q_{−}(V)^{∗} = \overline{Q_{+}(V)} with

ker \overline{Q_{−}(V)} = \{0\}, dim ker \overline{Q_{+}(V)} = p − 1.

But \overline{Q_{±}(V)} are not Fredholm.

As a next step of mathematical research, it is interesting to investigate the case where V_{1} ≠ 0 and V_{2} ≠ 0. This is one of the basic motivations of this work.

For a self-adjoint operator S, we denote by σ(S) and σ_{d}(S) the spectrum and the discrete spectrum of S respectively (σ_{d}(S) is the set of isolated eigenvalues of S with finite multiplicity). We say that the spectrum of S is purely discrete if σ(S) = σ_{d}(S).

For the function V_{1}, we define deg V_{1} by

deg V_{1} := p  (1.18)

if V_{1}(z) = ∑_{j=0}^{p} a_{j}z^{−j} with a_{p} ≠ 0 (a_{j} ∈ C , j = 0, ⋯, p).

We now state the main results. We work with the following hypothesis:

Hypothesis (A) deg V_{1} ≥ 1 and deg V_{2} ≥ 2.

Note that, under this hypothesis, the matrix valued potential iψ_{1}(∂V) − iψ_{1}^{∗}(∂V)^{∗} of the Dirac operator Q(V) has a pole at z = 0 with order deg V_{1} + 1 ≥ 2. This makes the mathematical analysis of Q(V) more difficult than the regular case V_{1} = 0.

Theorem 1.1 Assume (A). Then:

(i) The operator Q(V) is self-adjoint and C_{0}^{∞}(IR^{2} \ {0}; C^{4}) is a core of Q(V).

(ii) The operators Q_{±}(V) are closed and

Q_{+}(V)^{∗} = Q_{−}(V).  (1.19)

Theorem 1.2 Assume (A). Then:

(i) The spectrum of Q(V) is symmetric with respect to the origin, i.e., λ ∈ σ(Q(V)) ⟺ −λ ∈ σ(Q(V)).

(ii)

ker Q_{−}(V) = {0}.  (1.20)

(iii) The spectrum of Q(V) is purely discrete.

(iv) The operators Q_{±}(V) are Fredholm.

For the kernel of Q_{+}(V), we have the following theorem, which we regard as one of the most important results in the present paper.

Theorem 1.3 Let P (z) be a complex polynomial of z ∈ C with deg P ≤ q − 1 (q ∈ IN) and

U(z) := \frac{λ}{z^{p}} + μz^{q} + P (z), z ∈ C^{×}.  (1.21)

Let 4 ≤ q ≤ p + 2. Then ker Q(U) ≠ {0} and dim ker Q(U) is independent of (λ, μ) and P.

Remark 1.2 This theorem implies that dim ker Q(U) is determined by p and q only. But we have been unable to find an explicit formula for dim ker Q(U).

Remark 1.3 By Theorem 1.2-(ii), we have

ker Q(V) = ker Q_{+}(V) ⊕ {0}.  (1.22)

Hence

dim ker Q(V) = dim ker Q_{+}(V).  (1.23)

The basic methods we use to analyze the operator Q(V) is to consider its square Q(V)^{2} as was done in [6, 2, 3] (see also [10, Chapter 5]). This leads us to introduce the following operators:

H_{−}(V) := −\frac{1}{4}Δ + |∂V |^{2},  (1.24)
H_{+}(V) := H_{−}(V) + H_{I}(V),  (1.25)

both acting in L^{2}(IR^{2}; C^{2}), where

Δ := D_{x}^{2} + D_{y}^{2},  (1.26)

the two-dimensional generalized Laplacian, and

H_{I}(V) := ( \begin{matrix}0 & −i∂^{2}V \\ i(∂^{2}V)^{∗} & 0 \end{matrix} ) .  (1.27)

We remark that the definition of H_{+}(V) (resp. H_{−}(V)) here is different form that in [3], although it turns out that they coincide (see Theorem 2.5-(ii) and Theorem 2.6-(ii) in Section 2). We also note that the potential |∂V |^{2} (resp. H_{I}(V)) has a pole at z = 0 with order 2(deg V_{1} +1) ≥ 4 (resp. deg V_{1} +2). Hence H_{±}(V) are two-dimensional Schrödinger operators with strongly singular (matrix-valued) potentials.

We set

H(V) := H_{+}(V) ⊕ H_{−}(V) = ( \begin{matrix}H_{+}(V) & 0 \\ 0 & H_{−}(V) \end{matrix} ) .  (1.28)

Remark 1.4 In the context of SSQM, H(V) turns out to be the supersymmetric Hamiltonian of the N = 2 Wess-Zumino model (Theorem 1.4-(iii) below) and H_{+}(V) (resp. H_{−}(V)) is called the bosonic (resp. fermionic) part of H(V).

Theorem 1.4 Assume (A). Then:

(i) The operators H_{−}(V) and H_{+}(V) are nonnegative and self-adjoint. Moreover they are essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{2}).

(ii) The operator H(V) is nonnegative and self-adjoint. Moreover it is essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{4}).

(iii) The following operator equality holds:

Q(V)^{2} = H(V).  (1.29)

Theorem 1.5 Assume (A). Then:

(i) ker H_{−}(V) = {0}.

(ii) The operators H_{±}(V) have compact resolvent. In particular, the spectra of H_{±}(V) are purely discrete and

σ(H_{+}(V)) \ {0} = σ(H_{−}(V)) \ {0}.  (1.30)

Theorem 1.6 Let U be defined by (1.21) and 4 ≤ q ≤ p + 2. Then ker H_{+}(U) ≠ {0} and dim ker H_{+}(U) is independent of (λ, μ) and P.

In addition to the theorems stated above, we prove a trace formula for dim ker Q(V) in terms of H_{±}(V) (see Theorem 6.1 in Section 6).

The present paper is organized as follows. In Section 2 we discuss the (essential) self-adjointness problem of H_{±}(V) and Q(V). We prove Theorem 1.1 and Theorem 1.4. Section 3 is concerned with spectral properties of H_{±}(V) and Q(V). We prove Theorem 1.2 and Theorem 1.5. In Section 4 we prove the continuity of the ground state energy of H_{±}(V) (the infimum of the spectrum of H_{±}(V)) in the counpling constants λ and μ (Theorem 4.2). As a result, it is shown that dim ker H_{+}(V) (= dim ker Q_{+}(V)) is a constant independent of (λ, μ) (Theorem 4.3). These results also may be regarded as part of the main results of the present paper. Section 5 is devoted to proofs of Theorem 1.3 and Theorem 1.6. In the last section we establish the trace formula mentioned above.

2 Self-adjointness

In this section we prove Theorems 1.1 and 1.4.

Throughout this section, we assume (A) and write simply

Q = Q(V), Q_{±} = Q_{±}(V), H = H(V), H_{±} = H_{±}(V), H_{I} = H_{I}(V).

We denote by ⟨·, ·⟩ and ∥·∥ the inner product and the norm of L^{2}(IR^{2}; C^{r}) respectively (r ∈ IN):

⟨Ψ, Φ⟩ := ∫_{IR_{2}}⟨Ψ(r), Φ(r)⟩_{C}^{r}dr, (r = (x, y) ∈ IR^{2})  (2.1)
∥Ψ∥ := \sqrt{⟨Ψ, Ψ⟩}, Ψ, Φ ∈ L^{2}(IR^{2}; C^{r}),  (2.2)

where ⟨·, ·⟩_{C}^{r} is the standard inner product of C^{r}.

Lemma 2.1 For all Ψ ∈ C_{0}^{∞}(IR^{2} \ {0}; C^{2}),

\frac{1}{4}∥ΔΨ∥^{2} + ∥(∂V)^{2}Ψ∥^{2} ≤ ∥H_{−}Ψ∥^{2} + ∥(∂^{2}V)Ψ∥^{2}.  (2.3)

Proof. This fact is stated in [3, Lemma 2.7]. For the reader’s convenience, we give a proof (cf. the proof of [1, Lemma 2.7]). It is obvious that

C_{0}^{∞}(IR^{2} \ {0}; C^{2}) ⊂ D(H_{±}).  (2.4)

We have

∂ ¯∂ ⊂ \frac{1}{4}Δ, ¯∂∂ ⊂ \frac{1}{4}Δ

and

∂ ¯∂Ψ = ¯∂∂Ψ = \frac{1}{4}ΔΨ, Ψ ∈ C_{0}^{∞}(IR^{2} \ \{0\}; C^{2}).

Let Ψ ∈ C_{0}^{∞}(IR^{2} \ {0}; C^{2}). Then

∥H_{−}Ψ∥^{2} = \frac{1}{4}∥ΔΨ∥^{2} + ∥(∂V)^{2}Ψ∥^{2} − C_{V}

with

C_{V} := ℜ⟨¯∂∂Ψ, (∂V)^{∗}(∂V)Ψ⟩ + ℜ⟨∂ ¯∂Ψ, (∂V)^{∗}(∂V)Ψ⟩.

By integration by parts, we see that

C_{V} = ∥(∂^{2}V)Ψ∥^{2} − ∥(∂V)∂Ψ∥^{2} − ∥(∂V)¯∂Ψ∥^{2}.

Hence

∥H_{−}Ψ∥^{2} = \frac{1}{4}∥ΔΨ∥^{2} + ∥(∂V)^{2}Ψ∥^{2} − ∥(∂^{2}V)Ψ∥^{2} + ∥(∂V)∂Ψ∥^{2} + ∥(∂V)¯∂Ψ∥^{2}. (2.5)

Hence (2.3) follows.

Lemma 2.2 For every ε > 0, there exists a constant b_{ε} > 0 such that

|∂^{2}V (z)|^{2} ≤ ε|∂V (z)|^{4} + b_{ε}, z ∈ C \ {0}.  (2.6)

Proof. See [3, Lemma 2.8].

Lemma 2.3 Let A be a densely defined closable linear operator from a Hilbert space H to a Hilbert space K. Suppose that there exists a dense subspace D of H such that D ⊂ D(A^{∗}A) and A^{∗}A is essentially self-adjoint on D. Then

\overline{A^{∗}A} = Ā^{∗}Ā.  (2.7)

Proof. Let S = \overline{A^{∗}A}. For all ψ ∈ D(S), there exists a sequence \{ψ_{n}\}_{n} with ψ_{n} ∈ D (n ∈ IN) such that ψ_{n} → ψ, Sψ_{n} → Sψ (n → ∞). Since Sψ_{n} = A^{∗}Aψ_{n}, it follows that {Aψ_{n}}_{n} is a Cauchy sequence in K. Hence ψ ∈ D(Ā) and Aψ_{n} → Āψ (n → ∞). Since A^{∗} is closed, it follows that Āψ ∈ D(A^{∗}) and A^{∗}Āψ = Sψ. This means that S ⊂ A^{∗}Ā = Ā^{∗}Ā. By the von Neumann theorem ([8, Theorem X.25]), Ā^{∗}Ā is self-adjoint. Thus (2.7) follows.

Lemma 2.4 Let A and B be closed linear operators from a Hilbert space H to a Hilbert space K such that A + B is closable. Suppose that there exist a core D of \overline{A + B} and closed linear operators A′, B′ from H to K such that D ⊂ D(A) ∩ D(B), D(A) = D(A′), D(B) = D(B′) and

∥A′ψ∥ + ∥B′ψ∥ ≤ C(∥(A + B)ψ∥ + ∥ψ∥), ψ ∈ D,  (2.8)

where C > 0 is a constant. Then A + B is closed and

∥A′ψ∥ + ∥B′ψ∥ ≤ C(∥(A + B)ψ∥ + ∥ψ∥), ψ ∈ D(A + B).  (2.9)

Proof. Let ψ ∈ D(\overline{A + B}). Then there exists a sequence \{ψ_{n}\}_{n} with ψ_{n} ∈ D such that ψ_{n} → ψ and (A + B)ψ_{n} → (\overline{A + B})ψ (n → ∞). By (2.8), \{A′ψ_{n}\}_{n} and \{B′ψ_{n}\}_{n} are Cauchy sequences. Since A′ and B′ are closed, it follows that ψ ∈ D(A′) ∩ D(B′) = D(A)∩D(B) = D(A+B) and A′ψ_{n} → A′ψ, B′ψ_{n} → B′ψ (n → ∞). Hence D(\overline{A + B}) ⊂ D(A + B) and (2.9) holds. In particular, \overline{A + B} = A + B. Thus A + B is closed.

Theorem 2.5

(i) The operator H_{−} is essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{2}) and H_{−} is nonnegative, self-adjoint with

(1 − ε)∥(∂V)^{2}Ψ∥^{2} + ∥\frac{1}{4}ΔΨ∥^{2} ≤ ∥H_{−}Ψ∥^{2} + b_{ε}∥Ψ∥^{2}, Ψ ∈ D(H_{−}),  (2.10)

where ε ∈ (0, 1) is an arbitrary constant and b_{ε} > 0 is a constant depending on ε.

(ii) Operator equality

H_{−} = \overline{Q}_{−}^{∗} \overline{Q}_{−}  (2.11)

holds.

(iii) \overline{Q}_{−} = Q_{−}, i.e., Q_{−} is closed.

(iv) For all Ψ ∈ D(Q_{−}),

∥Q_{−}Ψ∥^{2} = \frac{1}{4}∥(−Δ)^{1/2}Ψ∥^{2} + ∥(∂V)Ψ∥^{2}.  (2.12)

Proof. (i) The essential self-adjointness of H_{−} on C_{0}^{∞}(IR^{2} \ {0}; C^{2}) is already proved in [3, Theorem 2.12]. The nonnegativity of \overline{H}_{−} is obvious. Thus we need only to show the self-adjointness of H_{−} with D(H_{−}) = D(Δ) ∩ D(|∂V |^{2}) (this was not proven in [3]). By (2.3) and (2.6), we see that (2.10) holds for all Ψ ∈ C_{0}^{∞}(IR^{2} \ {0}; C^{2}). Hence, by applying Lemma 2.4 with A = −Δ/4 and B = |∂V |^{2}, we conclude that H_{−} is closed and (2.10) holds. Hence H_{−} is self-adjoint.

(ii) By direct computations, we have

H_{−}Ψ = Q_{−}^{∗}Q_{−}Ψ, Ψ ∈ C_{0}^{∞}(IR^{2} \ {0}; C^{2}).  (2.13)

Hence, applying Lemma 2.3, we obtain (2.11).

(iii) Since we have established the self-adjointness and the nonnegativity of H_{−}, the operator H_{−}^{1/2} exists. By definition (1.24) of H_{−} and a limiting argument, one can show that D(H_{−}^{1/2}) ⊂ D((−Δ)^{1/2}) ∩ D(∂V) and

∥H_{−}^{1/2}Ψ∥^{2} = \frac{1}{4}∥(−Δ)^{1/2}Ψ∥^{2} + ∥(∂V)Ψ∥^{2}, Ψ ∈ D(H_{−}^{1/2}).  (2.14)

On the other hand, (2.11) implies that D(H_{−}^{1/2}) = D(\overline{Q}_{−}) and

∥H_{−}^{1/2}Ψ∥^{2} = ∥\overline{Q}_{−}Ψ∥^{2}, Ψ ∈ D(H_{−}^{1/2}).  (2.15)

Hence, in particular, it follows that D(\overline{Q}_{−}) ⊂ D((−Δ)^{1/2}) ∩ D(∂V). Note that the right hand side is equal to D(Q_{−}). Thus D(\overline{Q}_{−}) ⊂ D(Q_{−}), which means that \overline{Q}_{−} = Q_{−}.

(iv) This follows from (2.14), (2.15) and part (iii).

Theorem 2.6

(i) The operator H_{+} is self-adjoint and essentially self-adjoint on every core of H_{−}. In particuler, H_{+} is essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{2}).

(ii) Operator equality

H_{+} = \overline{Q}_{+}^{∗}\overline{Q}_{+}  (2.16)

holds. In particular, H_{+} is nonnegative.

(iii) \overline{Q}_{+} = Q_{+}, i.e., Q_{+} is closed.

(iv) For all Ψ = (Ψ_{1}, Ψ_{2}) ∈ D(Q_{+}),

∥Q_{+}Ψ∥^{2} = \frac{1}{4}∥(−Δ)^{1/2}Ψ∥^{2} + ∥(∂V)Ψ∥^{2} + q_{H_{I}}(Ψ),  (2.17)

where

q_{H_{I}}(Ψ) := ∫_{IR_{2}} {Ψ_{1}(r)^{∗}(−i)(∂^{2}V)(z)Ψ_{2}(r) + Ψ_{2}(r)^{∗}i(∂^{2}V)(z)^{∗}Ψ_{1}(r)} dr. (2.18)

Proof. (i) Let H_{I} be defined by (1.27). Then we have by (2.6)

∥H_{I}Ψ∥^{2} ≤ ε∥(∂V)^{2}Ψ∥^{2} + b_{ε}∥Ψ∥^{2}, Ψ ∈ D((∂V)^{2}),

where ε ∈ (0, 1) is arbitrary and b_{ε} > 0 is a constant. This inequality and (2.10) imply that

∥H_{I}Ψ∥^{2} ≤ \frac{ε}{1 − ε}∥H_{−}Ψ∥^{2} + c_{ε}∥Ψ∥^{2}, Ψ ∈ D(H_{−}),  (2.19)

where c_{ε} > 0 is a constant. Hence H_{I} is infinitesimally small with respect to H_{−}. Therefore, by the Kato-Rellich theorem [8, Theorem X.12], H_{+} = H_{−} + H_{I} is self-adjoint with D(H_{+}) = D(H_{−}) and bounded from below. Moreover it is essentially self-adjoint on every core of H_{−}.

(ii) It is straightforward to see that

H_{+}Ψ = Q_{+}^{∗}Q_{+}Ψ, Ψ ∈ C_{0}^{∞}(IR^{2} \ {0}; C^{2}).

Hence a simple application of Lemma 2.3 yields (2.16).

(iii) By a limiting argument and definition (1.25) of H_{+}, we can show that D(H_{+}^{1/2}) = D(H_{−}^{1/2}) and

∥H_{+}^{1/2}Ψ∥^{2} = ∥H_{−}^{1/2}Ψ∥^{2} + q_{H_{I}}(Ψ), Ψ ∈ D(H_{+}^{1/2}) = D(H_{−}^{1/2}).  (2.20)

By (2.16), we have D(H_{+}^{1/2}) = D(\overline{Q}_{+}). Since D(H_{−}^{1/2}) = D((−Δ)^{1/2}) ∩ D(∂V) = D(Q_{+}), it follows that D(\overline{Q}_{+}) = D(Q_{+}). This means that \overline{Q}_{+} = Q_{+}.

(iv) This follows from (2.14), (2.20) and part (iii).

Corollary 2.7 The operator H defined by (1.28) is a nonnegative self-adjoint operator. It is essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{4}).

Lemma 2.8 Let S be a symmeric operator on a Hilbert space K and D be a dense subspace of K such that D ⊂ D(S^{2}) and S^{2} is essentially self-adjoint on D. Then S is essentially self-adjoint on D and its closure is essentially self-adjoint on all other cores of \overline{S^{2}}.

Proof. Let N := \overline{S^{2}} + 1. Then N ≥ 1 and ∥Sψ∥ ≤ ∥Nψ∥, ψ ∈ D. Obviously |⟨Sψ, Nψ⟩−⟨Nψ, Sψ⟩| = 0, ψ ∈ D. Hence, by a simple application of Glimm-Jaffe-Nelson type commutator theorem [8, Theorem X.37], we see that S is essentially self-adjoint on D ans its closure is essentially self-adjoint on all other cores for N.

Theorem 2.9

(i) The operator Q is essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{4}) and every core of H is a core of Q̄.

(ii) The operator Q is self-adjoint.

(iii) Operator equalities (1.19) and (1.29) hold.

Proof. (i) It is obvious that C_{0}^{∞}(IR^{2} \ {0}; C^{4}) ⊂ D(Q^{2}). By direct computations, we have

HΨ = Q^{2}Ψ, Ψ ∈ C_{0}^{∞}(IR^{2} \ {0}; C^{4}).  (2.21)

Hence, by Lemma 2.8, the assertion of part (i) follows.

(ii) By the closedness of Q_{±} (Theorem 2.5-(iii) and Theorem 2.6-(iii)), Q is closed. This fact and part (i) imply that Q is self-adjoint.

(iii) Operator equality (1.19) follows from the self-adjointness of Q. Since we have established the self-adjointness of Q, a simple application of Lemma 2.3 gives (1.29).

In summary we have proved Theorem 1.1 and Theorem 1.4.

3 Spectral Properties

Throughout this section, we write simply Q = Q(V), Q_{±} = Q_{±}(V), H_{±} = H_{±}(V).

3.1 Proof of Theorem 1.2-(i)

We have the following orthogonal decomposition:

L^{2}(IR^{2}; C^{4}) = H_{+} ⊕ H_{−}  (3.1)

with

H_{+} := \{( \begin{matrix}f \\ g \\ 0 \\ 0 \end{matrix} ) |f, g ∈ L^{2}(IR^{2})\} ≅ L^{2}(IR^{2}; C^{2}),  (3.2)

H_{−} := \{( \begin{matrix}0 \\ 0 \\ f \\ g \end{matrix} ) |f, g ∈ L^{2}(IR^{2})\} ≅ L^{2}(IR^{2}; C^{2}).  (3.3)

We denote the orthogonal projections onto H_{±} by Γ_{±} respectively. Let

Γ := Γ_{+} − Γ_{−}.  (3.4)

Then Γ is self-adjoint and unitary:

Γ^{∗} = Γ, Γ^{2} = I,  (3.5)

where I denotes identity.

Lemma 3.1 For all Ψ ∈ D(Q), ΓΨ ∈ D(Q) and

QΓΨ + ΓQΨ = 0.  (3.6)

Proof. For all Ψ = (Ψ_{+}, Ψ_{−}) ∈ H_{+} ⊕ H_{−}, we have ΓΨ = (Ψ_{+}, −Ψ_{−}). Hence, if Ψ ∈ D(Q) = D(Q_{+}) ⊕ D(Q_{−}), then ΓΨ ∈ D(Q). By direct computations, we have QΓΨ = −ΓQΨ, Ψ ∈ D(Q).

By Lemma 3.1, we have operator equality ΓQΓ^{−1} = −Q. It follows from the unitary invariance of spectrum that σ(Q) = σ(−Q), which implies Theorem 1.2-(i).

3.2 Proofs of Theorem 1.2-(ii) and Theorem 1.5-(i)

Let Ψ ∈ ker H_{−}. Then H_{−}Ψ = 0. Then, by (2.10), Ψ ∈ ker Δ = {0}. Hence Ψ = 0. Thus Theorem 1.5-(i) follows. This fact and (2.11) imply that ker Q_{−} = {0}. Hence Theorem 1.2-(ii) is proved.

3.3 Proof of Theorem 1.5-(ii)

Lemma 3.2 Let W be a Lebesgue measurable function on IR^{n} (n ∈ IN). Suppose that W is bounded from below with W (x) → ∞ as |x| → ∞ (x ∈ IR^{n}) and D(|W |^{1/2})∩D((−Δ)^{1/2}) is dense in L^{2}(IR^{n}) (Δ is the n-dimensional generalized Laplacian). Then H_{W} := −Δ+W defined as a sum of quadratic forms is a self-adjoint operator with compact resolvent. In particular, H_{W} has purely discrete spectrum and a complete set of eigenfunctions.

Proof. Similar to the proof of [9, Theorem XIII] (In this theorem, it is assumed that W ∈ L_{loc}^{1}(IR^{n}). But this condition is not necessary. In this respect, it is sufficient to assume the denseness of D(|W |^{1/2}) ∩ D((−Δ)^{1/2}) which ensures the existence of the self-adjoint operator H_{W} defined as a sum of quadratic forms).

We write

H_{−} = L ⊕ L  (3.7)

with

L = −\frac{1}{4}Δ + |∂V |^{2}  (3.8)

acting in L^{2}(IR^{2}).

Lemma 3.3 The operator L has compact resolvent. In particular, L has purely discrete spectrum and a complete set of eigenfunctions.

Proof. It is easy to see that |∂V |^{2} → ∞ as |z| → ∞. Hence we can apply Lemma 3.2 to obtain the desired result.

By Lemma 3.3 and (3.7), H_{−} has compact resolvent. In particular it has purely discrete spectrum. Since H_{I} is infinitesimally small with respect to H_{−} as already shown, it follows from a general theorem [8, Theorem XIII.68] that H_{+} has compact resolovent. Thus the pure discreteness of spectra of H_{±} is proven.

The following fact (spectral supersymmetry) is well known ([4] and [10, Corollary 5.6]):

Lemma 3.4 For every densely defined closed linear operator C from a Hilbert space to a Hilbert space,

σ(C^{∗}C) \ {0} = σ(CC^{∗}) \ {0},  (3.9)
σ_{p}(C^{∗}C) \ {0} = σ_{p}(CC^{∗}) \ {0},  (3.10)

where σ_{p}(T) denotes the point spectrum of an operator T (the set of all eigenvalues of T) and the multiplicity of λ ∈ σ_{p}(C^{∗}C) \ {0} is equal to that of λ ∈ σ_{p}(CC^{∗}) \ {0}.

The spectral property (1.30) follows from an application of Lemma 3.4 with (1.19), (2.11) and (2.16).

3.4 Proof of Theorem 1.2-(iii)

This follows from Therem 1.4-(iii) and the functional calculus.

3.5 Proof of Theorem 1.2-(iv)

We have already seen that Q_{±} are densely defined and closed. From the definition of Q, we have

ker Q = ker Q_{+} ⊕ ker Q_{−}.

By Theorem 1.2-(iii), dim ker Q < ∞. Hence dim ker Q_{−}^{∗} = dim ker Q_{+} < ∞ and dim ker Q_{+}^{∗} = dim ker Q_{−} < ∞. The pure discreteness of the spectrum of Q implies also that Ran(Q_{±}) (the range of Q_{±}) are closed. Thus Q_{±} are Fredholm.

4 Continuity of Ground State Energies and Constancy of dim ker H_{+}(V) in Coupling Constants

For a self-adjoint operator S on a Hilbert space, we definne

E_{0}(S) := inf σ(S),  (4.1)

provided that S is bounded from below. By abuse of nomenclature, we call E_{0}(S) the ground state energy of S.

In this section, as a preliminary to the next section, we prove the continuiuty of the ground state energies of H_{±}(V) on the coupling constants λ and μ.

To make explicit the dependence of H_{±}(V) on λ and μ, we write

H_{±}(λ, μ) := H_{±}(V).  (4.2)

For a self-adjoint operator S on a Hilbert space, we denote by ρ(S) the resolvent set of S: ρ(S) = C \ σ(S).

Lemma 4.1 Let (λ_{0}, μ_{0}) ∈ (C^{×})^{2} be arbitrarily fixed. Let z ∈ ρ(H_{±}(λ_{0}, μ_{0})). Then z ∈ ρ(H_{±}(λ, μ)) for all (λ, μ) ∈ (C^{×})^{2} in a neighborhood N_{0} ⊂ (C^{×})^{2} of (λ_{0}, μ_{0}) and

∥(H_{±}(λ, μ) − z)^{−1} − (H_{±}(λ_{0}, μ_{0}) − z)^{−1}∥ ≤ C(|λ − λ_{0}| + |μ − μ_{0}|)  (4.3)

for all (λ, μ) ∈ N_{0}, where C > 0 is a constant independent of (λ, μ) ∈ N_{0}.

Proof. It is sufficient to consider the case where |λ−λ_{0}|+|μ−μ_{0}| ≤ δ_{1} with a constant δ_{1} > 0. Throughout the proof, C_{j}’s denote constants independent of λ and μ. Let

D := D(−Δ) ∩ D(|∂V_{1}|^{2}) ∩ D(|∂V_{2}|^{2}).  (4.4)

Then

D(H_{±}(λ, μ)) = D ⊕ D, (λ, μ) ∈ (C^{×})^{2}.  (4.5)

Let Ψ ∈ D ⊕ D. Then

[H_{+}(λ, μ) − H_{+}(λ_{0}, μ_{0})]Ψ = (|λ|^{2} − |λ_{0}|^{2})|∂V_{1}|^{2}Ψ + (|μ|^{2} − |μ_{0}|^{2})|∂V_{2}|^{2}Ψ
+2ℜ{(λ^{∗}μ − λ_{0}^{∗}μ_{0})(∂V_{1})^{∗}(∂V_{2})}Ψ
+(λ − λ_{0})H_{I}(V_{1})Ψ + (μ − μ_{0})H_{I}(V_{2})Ψ.

Hence

∥[H_{+}(λ, μ) − H_{+}(λ_{0}, μ_{0})]Ψ∥
≤ C_{1}(|λ − λ_{0}| + |μ − μ_{0}|)
×(∥(λ_{0}∂V_{1})^{2}Ψ∥ + ∥H_{I}(λ_{0}V_{1})Ψ∥ + ∥(μ_{0}∂V_{2})^{2}Ψ∥ + ∥H_{I}(μ_{0}V_{2})Ψ∥).

By (2.10) and (2.19), we have

∥[H_{+}(λ, μ) − H_{+}(λ_{0}, μ_{0})]Ψ∥
≤ C_{2}(|λ − λ_{0}| + |μ − μ_{0}|)(∥H_{−}(λ_{0}∂V_{1})Ψ∥ + ∥H_{−}(μ_{0}V_{2})Ψ∥ + ∥Ψ∥).

Note that

H_{+}(λ_{0}, μ_{0}) = H_{−}(λ_{0}V_{1}) + 2ℜ[(λ_{0}∂V_{1})^{∗}(μ_{0}∂V_{2})] + |μ_{0}∂V_{2}|^{2} + H_{I}(λ_{0}V_{1} + μ_{0}V_{2}),
= H_{−}(μ_{0}V_{2}) + 2ℜ[(λ_{0}∂V_{1})^{∗}(μ_{0}∂V_{2})] + |λ_{0}∂V_{1}|^{2} + H_{I}(λ_{0}V_{1} + μ_{0}V_{2})

By this fact and the closedness of H_{+}(λ_{0}, μ_{0}), there exists a constant C_{3} > 0 such that

∥H_{−}(λ_{0}V_{1})Ψ∥ ≤ C_{3}(∥H_{+}(λ_{0}, μ_{0})Ψ∥ + ∥Ψ∥),
∥H_{−}(μ_{0}V_{2})Ψ∥ ≤ C_{3}(∥H_{+}(λ_{0}, μ_{0})Ψ∥ + ∥Ψ∥).

Hence

∥[H_{+}(λ, μ) − H_{+}(λ_{0}, μ_{0})]Ψ∥ ≤ C_{4}(|λ − λ_{0}| + |μ − μ_{0}|)(∥H_{+}(λ_{0}, μ_{0})Ψ∥ + ∥Ψ∥). (4.6)

Let z ∈ ρ(H_{+}(λ_{0}, μ_{0})). Then we have

H_{+}(λ, μ) − z = {1 + [H_{+}(λ, μ) − H_{+}(λ_{0}, μ_{0})](H_{+}(λ_{0}, μ_{0}) − z)^{−1}}
×(H_{+}(λ_{0}, μ_{0}) − z).

This formula and (4.6) imply the assertion for H_{+}(λ, μ) to be proved. Similarly we can prove the properties of H_{−}(λ, μ) as stated in Lemma 4.1.

Let

E_{±}(λ, μ) := E_{0}(H_{±}(λ, μ)).  (4.7)

Theorem 4.2 The functions E_{±} : (λ, μ) ↦ E_{±}(λ, μ) are continuous on (C^{×})^{2}.

Proof. By the nonnegativity of H_{±}(λ, μ) and the functional calculus, we have

\frac{1}{E_{±}(λ, μ) + 1} = ∥(H_{±}(λ, μ) + 1)^{−1}∥.

By Lemma 4.1, the right hand side is continuous in (λ, μ).

Theorem 4.3 Suppose that, for some (λ_{0}, μ_{0}) ∈ (C^{×})^{2}, E_{+}(λ_{0}, μ_{0}) = 0. Then:

(i) For all (λ, μ) ∈ (C^{×})^{2}, E_{+}(λ, μ) = 0 and hence ker H_{+}(λ, μ) ≠ {0}.

(ii) The quantity dim ker H_{+}(λ, μ) is a constant independent of (λ, μ) ∈ (C^{×})^{2}.

Proof. (i) By Theorem 1.5-(i), we have

E_{−}(λ, μ) > 0, ∀(λ, μ) ∈ (C^{×})^{2}.  (4.8)

In particular ε_{0} := E_{−}(λ_{0}, μ_{0}) > 0. By the continuity of E_{±}(λ, μ) in (λ, μ) (Theorem 4.2), for every ε ∈ (0, ε_{0}/2), there exists a constant r_{ε} > 0 such that, if |λ − λ_{0}| + |μ − μ_{0}| < r_{ε}, then

0 ≤ E_{+}(λ, μ) < ε, |E_{−}(λ, μ) − ε_{0}| < ε.

Hence, if |λ − λ_{0}| + |μ − μ_{0}| < r_{ε}, then

E_{−}(λ, μ) > ε_{0} − ε > ε > E_{+}(λ, μ).

Therefore E_{−}(λ, μ) ≠ E_{+}(λ, μ) for |λ − λ_{0}| + |μ − μ_{0}| < r_{ε}, which, togehter with (1.30), implies that E_{+}(λ, μ) = 0 for |λ − λ_{0}| + |μ − μ_{0}| < r_{ε}. By (4.8), we can repeat this process and see that E_{+}(λ, μ) = 0 for all (λ, μ) ∈ (C^{×})^{2}. Thus the facts of part (i) follow.

(ii) We denote by P (λ, μ) the orthogonal projection onto ker H_{+}(λ, μ). By the assumption, 0 ∈ σ(H_{+}(λ_{0}, μ_{0})) and a := inf σ(H_{+}(λ_{0}, μ_{0})) \ {0} > 0. Let 0 < ε < a and C_{ε} := {z ∈ C | |z| = ε}. Then C_{ε} ⊂ ρ(H_{+}(λ_{0}, μ_{0})). By Theorem 4.2, there exists a constant δ > 0 such that, if |λ − λ_{0}| + |μ − μ_{0}| < δ, then E_{0}(H_{+}(λ, μ)) = 0 (hence ker H_{+}(λ, μ) ≠ {0}) and C_{ε} ⊂ ρ(H_{+}(λ, μ)). Therefore, for such (λ, μ), we have

P (λ, μ) = −\frac{1}{2πi} ∫_{C_{ε}}(H_{+}(λ, μ) − z)^{−1}dz.

Using Lemma 4.1, one can show that P (λ, μ) is continuous in (λ, μ) in operator norm. Hence, by a general theorem (e.g., [9, p.14, Lemma]), dim Ran(P (λ, μ)) is a constant on the region {(λ, μ) ∈ (C^{×})^{2}||λ − λ_{0}| + |μ − μ_{0}| < δ}. Repeating this process, we see that dim Ran(P (λ, μ)) is a constant independent of (λ, μ) ∈ (C^{×})^{2}. On the other hand, dim Ran(P (λ, μ)) = dim ker H_{+}(λ, μ). Thus the assertion of part (ii) follows.

5 Kernels of H_{+}(V) and Q_{+}(V)

In this section we prove Theorem 1.3 and Theorem 1.6.

5.1 A special case

We first prove Theorem 1.6 and Theorem 1.3 for the function U with P = 0:

Theorem 5.1 Let λ, μ ∈ C^{×} and

W_{λ,μ}(z) := \frac{λ}{z^{p}} + μz^{q}, z ∈ C^{×}.  (5.1)

Let 4 ≤ q ≤ p + 2. Then

ker H_{+}(W_{λ,μ}) ≠ {0}.  (5.2)

Moreover, dim ker H_{+}(W_{λ,μ}) is independent of λ and μ.

Corollary 5.2 Let 4 ≤ q ≤ p + 2. Then

ker Q_{+}(W_{λ,μ}) ≠ {0}.  (5.3)

Moreover, dim ker Q_{+}(W_{λ,μ}) is independent of λ and μ.

Proof. By Theorem 2.6-(ii) and (iii), we have

ker Q_{+}(W_{λ,μ}) = ker H_{+}(W_{λ,μ}).  (5.4)

By this fact and Theorem 5.1, we obtain the desired result.

To prove Theorem 5.1, we need some preliminaries. We first recall a known fact:

Lemma 5.3 [6, 2] Suppose that deg V_{2} ≥ 2. Then:

(i) The operators H_{±}(V_{2}) are nonnegative and self-adjoint with D(H_{±}(V_{2})) = D(Δ)∩ D(|∂V_{2}|^{2}). Moreover they are essentially self-adjoint on C_{0}^{∞}(IR^{2}; C^{2}).

(ii) The spectra of H_{±}(V_{2}) are purely discrete and

dim ker H_{+}(V_{2}) = deg V_{2} − 1,  (5.5)
dim ker H_{−}(V_{2}) = 0.  (5.6)

.

The following fact may be interesting to keep in mind, although it is not necessary in the arguments below.

Proposition 5.4 For all complex polynomials Y of z ∈ C , H_{±}(Y) are not essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{2}).

Proof. For a constant μ ∈ C \ {0} and q ∈ IN, we define

M_{q}(z) := μz^{q}, z ∈ C .  (5.7)

Applying [8, Theorem X.7, Theorem X.10, Theorem X.11], one can prove that H_{−}(M_{q}) is not essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{2}).

For a general polynomial Y with deg Y = q, we write Y (z) = M_{q}(z) + P_{1}(z) with deg P_{1} ≤ q − 1. Then |∂P_{1}|^{2} + 2ℜ{(∂M_{q})^{∗}(∂P_{1})} is infinitesimally small with respect to |∂M_{q}|^{2}. Using this fact and the Kato-Rellich theorem, one can show that H_{−}(Y) is not essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{2}).

By (2.10), H_{I}(Y) is infinitesimally small with respect to H_{−}(Y). Hence, by a perturbation argument again, we can show that H_{+}(Y) is not essentially self-adjoint on C_{0}^{∞}(IR^{2} \ {0}; C^{2}).

Because of the fact stated in Proposition 5.4, the following lemma, which is a key to the proof of Theorem 5.1, is non-trivial.

Lemma 5.5 Let M_{q} be defined by (5.7) and q ≥ 4. Then

inf_{Ψ∈C_{0}^{∞}(IR^{2}\{0};C^{2}),∥Ψ∥=1}⟨Ψ, H_{+}(M_{q})Ψ⟩ = 0.  (5.8)

Proof. Let K_{ν}(z) be the modified Bessel function of the third kind (e.g., [5, §7.2.2]) and

Φ_{m} = μr^{q−1}K_{m/q}(2μr^{q})(2π)^{−1/2}e^{i(q−1−m)θ},
(r = |z|, θ = arg z, m = 1,…, q − 1).

Put

Ω_{m} = ( \begin{matrix}Φ_{m} \\ −iΦ_{q−m}^{∗} \end{matrix} ) .

Then it was shown in [2, Proposition 3.2] that, for m = 1, ⋯, q−1, Ω_{m} ∈ ker H_{+}(M_{q})\{0}:

H_{+}(M_{q})Ω_{m} = 0.

Let q ≥ 4. Then, by the regularity of Ω_{m} (cf. [2, p.2975, Remark (1)]) and (2.16), we see that Ω_{m} ∈ D(Q_{+}(M_{q})) and

Q_{+}(M_{q})Ω_{m} = 0.  (5.9)

Let f be a C_{0}^{∞}-function on IR^{2} such that f(r) = 1 for |r| ≤ 1, f(r) = 0 for |r| ≥ 2 and |f(r)| ≤ 1 for 1 ≤ |r| ≤ 2 and g ∈ C_{0}^{∞}(IR^{2}) be such that g(0) = 1. For n ∈ IN, we define

h_{n}(r) = g (\frac{r}{n}) [1 − f(nr)].

Then h_{n} ∈ C_{0}^{∞}(IR^{2} \ {0}) and

lim_{n→∞} h_{n}(r) = 1, ∀r ∈ IR^{2} \ {0}.

Let

Ψ_{n}^{(m)}(r) = h_{n}(r)Ω_{m}(r).

Then Ψ_{n}^{(m)} ∈ C_{0}^{∞}(IR^{2} \ {0}; C^{2}) and

lim_{n→∞} ∥Ψ_{n}^{(m)}∥ = ∥Ω_{m}∥ ≠ 0,

which implies that ∥Ψ_{n}^{(m)}∥ ≠ 0 for all n ≥ n_{0} with some n_{0} ∈ IN. Hence we can define a sequence {Ψ̃_{n}^{(m)}}_{n≥n_{0}} of unit vectors in C_{0}^{∞}(IR^{2} \ {0}; C^{2}) by

Ψ̃_{n}^{(m)} := \frac{Ψ_{n}^{(m)}}{∥Ψ_{n}^{(m)}∥}.

Using (5.9), we see that

Q_{+}(M_{q})Ψ_{n}^{(m)} = \frac{1}{2}\{(σ_{1} − iσ_{2})(¯∂h_{n}) + (σ_{1} + iσ_{2})(∂h_{n})\}Ω_{m}.

By the fact that ∥σ_{j}∥ = 1 (j = 1, 2), we have

∥Q_{+}(M_{q})Ψ_{n}^{(m)}∥ ≤ ∥(D_{x}h_{n})Ω_{m}∥ + ∥(D_{y}h_{n})Ω_{m}∥.

By direct computations, we have

(D_{#}h_{n})(r) = \frac{1}{n}(D_{#}g) (\frac{r}{n}) [1 − f(nr)] − ng (\frac{r}{n}) (D_{#}f)(nr),

where # = x, y. Hence

∥(D_{#}h_{n})Ω_{m}∥ ≤ I_{n}^{1/2} + J_{n}^{1/2}

with

I_{n} := \frac{1}{n^{2}} ∫_{IR_{2}} |(D_{#}g) (\frac{r}{n}) [1 − f(nr)]|^{2} ∥Ω_{m}(r)∥^{2}dr,

J_{n} := n^{2} ∫_{IR_{2}} |g (\frac{r}{n}) (D_{#}f)(nr)|^{2} ∥Ω_{m}(r)∥^{2}dr.

We have

0 ≤ I_{n} ≤ \frac{1}{n^{2}} ∥D_{#}g∥_{∞}^{2}∥Ω_{m}∥^{2},

where, for u : IR^{2} → C , we define ∥u∥_{∞} := sup_{r∈IR^{2}} |u(r)|. Hence

lim_{n→∞} I_{n} = 0.

On the other hand, by change of variables, we have

0 ≤ J_{n} ≤ ∥g∥_{∞}^{2} ∫_{IR_{2}} |(D_{#}f)(r)|^{2} ∥Ω_{m} (\frac{r}{n})∥^{2} dr.

Under the present assumption q ≥ 4, we can take m ≥ 2. Then it follows from the asymptotic properties of the Bessel function K_{ν}

K_{ν}(r) ∼ \sqrt{π/(2r)}e^{−r} (r → ∞),

K_{ν}(r) ∼ \frac{const.}{r^{ν}} (r → 0) (ν ≥ 1)

that Ω_{m} is bounded on IR^{2} and

lim_{r→0} Ω_{m}(r) = 0, r ∈ IR^{2}.

Hence, by the Lebesgue dominated convergence theorem, we obtain

lim_{n→∞} J_{n} = 0.

Thus we have for m = 2, ⋯, q − 1

lim_{n→∞} ∥Q_{+}(M_{q})Ψ_{n}^{(m)}∥ = 0.

This implies that

lim_{n→∞}⟨Ψ̃_{n}^{(m)}, H_{+}(M_{q})Ψ̃_{n}^{(m)}⟩ = 0 (m = 2, ⋯, q − 1).

Thus (5.8) is proved.

Proof of Theorem 5.1

By Theorem 4.3, we need only to prove (5.2) for sufficiently small |λ|. Throughout the proof, we write W_{λ} = W_{λ,μ}. Let f(z) = z^{−p}. Then we have

H_{+}(W_{λ}) = H_{+}(M_{q}) + F_{λ} + λH_{I}(f),

where

F_{λ}(z) := |λ|^{2}|(∂f)(z)|^{2} + 2ℜ{(λ∂f)^{∗}∂M_{q}(z)}.

By the variational principle, we have

E_{0}(H_{+}(W_{λ})) ≤ ⟨Ψ, H_{+}(M_{q})Ψ⟩ + ⟨Ψ, F_{λ}Ψ⟩ + λ⟨Ψ, H_{I}(f)Ψ⟩

for all Ψ ∈ D(H_{+}(W_{λ})) with ∥Ψ∥ = 1. Hence

lim sup_{λ→0} E_{0}(H_{+}(W_{λ})) ≤ ⟨Ψ, H_{+}(M_{q})Ψ⟩.

We have

inf_{Ψ∈D(H_{+}(W_{λ})),∥Ψ∥=1}⟨Ψ, H_{+}(M_{q})Ψ⟩ ≤ inf_{Ψ∈C_{0}^{∞}(IR^{2}\{0};C^{2}),∥Ψ∥=1}⟨Ψ, H_{+}(M_{q})Ψ⟩.

By Lemma 5.5, the right hand side is equal to zero. Hence

lim_{λ→0} E_{0}(H_{+}(W_{λ})) = 0.

Therefore, for every ε > 0, there exists a constant δ_{ε} > 0 such that

0 ≤ E_{0}(H_{+}(W_{λ})) < ε, ∀|λ| < δ_{ε}.  (5.10)

On the other hand, we have

H_{−}(W_{λ}) = H_{−}(M_{q}) + F_{λ}.

Under the condition that p + 2 ≥ q, we can show that

F_{λ}(z) ≥ −C_{p,q}|λ|^{2(q−1)/(p+q)}, ∀z ∈ C^{×},

with a constant C_{p,q} > 0. Hence, by the variational principle again, we have

E_{0}(H_{−}(W_{λ})) ≥ E_{0}(H_{−}(M_{q})) − C_{p,q}|λ|^{2(q−1)/(p+q)}.

By Lemma 5.3-(ii), we have E_{0}(H_{−}(M_{q})) > 0. Hence one can take a constant η > 0 such that E_{0}(H_{−}(M_{q})) − η > 0. Then, for all λ ∈ C^{×} satisfying

C_{p,q}|λ|^{2(q−1)/(p+q)} < E_{0}(H_{−}(M_{q})) − η,

we have E_{0}(H_{−}(W_{λ})) > η > 0. By (1.30), we have

σ(H_{−}(W_{λ})) \ {0} = σ(H_{+}(W_{λ})) \ {0}.  (5.11)

It follows from this property and (5.10) that, for all sufficiently small |λ|, E_{0}(H_{+}(W_{λ})) = 0. Thus (5.2) follows for all sufficiently small |λ|.

5.2 A more general case

Let U be given by (1.21). and

R := ( \begin{matrix}i∂P & 0 \\ 0 & −i(∂P)^{∗} \end{matrix} ) .

Then we have

Q_{−}(U) = Q_{−}(W_{λ,μ}) + R.  (5.12)

Lemma 5.6 The operator R is relatively compact with respect to Q_{−}(W_{λ,μ}).

Proof. Let {Ψ_{n}}_{n} be a sequence with Ψ_{n} ∈ D(Q_{−}(W_{λ,μ})) such that

∥Ψ_{n}∥ + ∥Q_{−}(W_{λ,μ})Ψ_{n}∥ ≤ 1.

Then ∥Ψ_{n}∥ ≤ 1 and, by (2.12),

∥(−Δ)^{1/2}Ψ_{n}∥ ≤ 2, ∥(∂W_{λ,μ})Ψ_{n}∥ ≤ 1.

Let B_{ℓ} := {r ∈ IR^{2}| |r| ≤ ℓ} (ℓ > 0) and χ_{ℓ} be the characteristic function of B_{ℓ}. It is well known (or easy to see ) that, for all ℓ > 0, χ_{ℓ}(−Δ + 1)^{−1/2} is a compact operator on L^{2}(IR^{2}). Hence there is a subsequence {Ξ_{m}}_{m} of {Ψ_{n}} such that, for all ℓ ∈ IN, {χ_{ℓ}Ξ_{m}}_{m} is strongly convergent in L^{2}(IR^{2}; C^{2}). Let a ∈ IN. Then

∥(1 − χ_{a})RΞ_{m}∥ = ∥(1 − χ_{a})R(∂W_{λ,μ})^{−1}(∂W_{λ,μ})Ξ_{m}∥
≤ ∥(1 − χ_{a})R(∂W_{λ,μ})^{−1}∥_{∞}.

It is easy to see that, for all sufficiently large a ∈ IN

∥(1 − χ_{a})R(∂W_{λ,μ})^{−1}∥_{∞} ≤ \frac{C}{a}

with a constant C > 0. Hence, for every ε > 0, there is a constant a such that

∥(1 − χ_{a})R(Ξ_{m} − Ξ_{n})∥ < ε, m, n ∈ IN.

We fix such an a. Since |∂P | is bounded on every bounded set of IR^{2}, it follows from the strong convergence of {χ_{ℓ}Ξ_{m}}_{m} that there is a constant n_{0} ∈ IN such that

∥χ_{a}R(Ξ_{m} − Ξ_{n})∥ < ε, m, n ≥ n_{0}.

Hence we obtain

∥R(Ξ_{m} − Ξ_{n})∥ < 2ε, m, n ≥ n_{0}.

This means that {RΞ_{m}}_{m} is a strongly convergent sequence in L^{2}(IR^{2}; C^{2}). Thus R is relatively compact with respect to Q_{−}(W_{λ,μ}).

Lemma 5.7 Suppose that 4 ≤ q ≤ p + 2. Then

dim ker Q_{+}(U) = dim ker Q_{+}(W_{λ,μ}).  (5.13)

Proof. By Lemma 5.6 and the Fredholmness of Q_{−}(W_{λ,μ}), we can apply a stability theorem ([7, p.238, Theorem 5.26]) for Fredholm index to obtain

index(Q_{−}(W_{λ,μ})) = index(Q_{−}(U)).

Using Theorem 1.1-(ii) and Theorem 1.2-(ii), we obtain (5.13).

Theorem 1.3 now follows from Corollary 5.2, Lemma 5.7 and (1.23). This fact together with Theorem 2.6-(ii) and (iii) yields Theorem 1.6.

6 A Trace Formula

Finally, we present a formula for dim ker Q_{+}(V) in terms of H_{±}(V).

Theorem 6.1 Assume (A). Then, for all t > 0, e^{−tH_{±}(V)} are trace class and

dim ker Q(V) = dim ker Q_{+}(V) = tr e^{−tH_{+}(V)} − tr e^{−tH_{−}(V)}  (6.1)

independently of t > 0, where tr means trace.

Proof. It is easy to see that

|∂V (z)|^{2} ≥ ω^{2}|z|^{2} − c,

where ω > 0 and c are positive constants depending on λ and μ. Hence

H_{−}(V) ≥ H_{os} ⊕ H_{os} − c,

where

H_{os} := −\frac{1}{4}Δ + ω^{2}|z|^{2}

acting in L^{2}(IR^{2}). It is well known that H_{os} is the Hamiltonian of a two-dimensional quantum harmonic oscillator and

σ(H_{os}) = σ_{d}(H_{os}) = {ω(n + m + 1)|n, m ∈ {0} ∪ IN} ,

where each eigenvalue ω(n + m + 1) is simple. In particular, for all t > 0, e^{−tH_{os}} is trace class on L^{2}(IR^{2}). Therefore, via the min-max principle (Theorem XIII.1 and Problem 1 (p.364) in [9]), one can show that, for all t > 0, e^{−tH_{−}(V)} is trace class and

tr e^{−tH_{−}(V)} ≤ e^{ct}tr e^{−tH_{os}}.

It follows from (2.19) and a general theorem [8, Theorem X.18] that H_{I}(V) is infinitesimally form-bounded with respect to H_{−}(V). Hence we have

H_{+}(V) ≥ (1 − ε)H_{−}(V) − b_{ε},

where ε ∈ (0, 1) is arbitrary and b_{ε} > 0 is a constant. Therefore, by the min-max principle again, e^{−tH_{+}(V)} is trace class for all t > 0. It follows from these facts and Theorem 1.4-(ii) that e^{−tQ(V)^{2}} is trace class for all t > 0. Hence, by applying a well known formula on Fredholm index (e.g., [10, Theorem 5.19]), we have

index(Q_{+}(V)) = tr [Γe^{−tQ(V)^{2}}] ,

where Γ is defined by (3.4). By Theorem 1.1-(ii), Theorem 1.2-(ii) and (1.23), the left hand side is equal to dim ker Q_{+}(V) = dim ker Q(V), while the right hand side is equal to

tr e^{−tH_{+}(V)} − tr e^{−tH_{−}(V)}

by Theorem 1.4-(iii). Thus (6.1) follows.

Remark 6.1 The right hand side of (6.1) has a functional integral representation. But here we do not go into the details. It will be discussed in a separate paper.

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