105

Spectral Analysis of Infinite‐dimensional
Dirac Operators on an Abstract
Boson‐Fermion Fock Space
Asao Arai

Department of Mathematics, Hokkaido University
Sapporo 060‐0810, JapanE‐mail: arai@math.sci.hokudai.ac.jp
Abstract

A review on spectral analysis of infinite dimensional Dirac type operatorson an abstract boson‐fermion Fock space is presented.

1

Introduction

For each pair (\mathscr{H}, \mathscr{K}) of complex Hilbert spaces, the tensor product Hilbert space\mathscr{F}(\mathscr{H}, \mathscr{K}):=\mathscr{F}_{b}(\mathscr{H})\otimes \mathscr{F}_{f}(\mathscr{K})of the boson Fock space\mathscr{F}_{b}(\mathscr{H}):=\bigoplus_{n=0}^{\infty}\bigotimes_{s}^{n} \mathscr{H}=\{ psi=\{ psi^{(n)}\ _{n=0}^{\infty}|\psi^{(n)}\in\bigotimes_{s}^{n} \mathscr{H},\sum_{n=0}^{\infty}\Vert\psi^{(n)}\Vert^{2}<\infty\}
over \mathscr{H} and the fermion Fock space\mathscr{F}_{f}(\mathscr{K}):=\bigoplus_{p=0}^{\infty}\wedge\mathscr{K}= p\{ phi=\{ phi^{(p)}\ _{p=0}^{\infty}|\phi^{(p)}\inp\wedge\mathscr{K},\sum_{p =0}^{\infty}\Vert\phi^{(p)}\Vert^{2}<\infty\}
over \mathscr{K} is defined, where \otimes_{s}^{n}\mathscr{H} denotes the n ‐fold symmetric tensor product of \mathscr{H}with \otimes_{s}^{0}\mathscr{H} :=\mathbb{C}, \wedge^{p}\mathscr{K} denotes the p‐fold anti‐symmetric tensor product of \mathscr{K} with\wedge^{0}\mathscr{K} :=\mathbb{C} and, for a vector \Psi in a Hilbert space, \Vert\Psi\Vert denotes the norm of \Psi . We callthe Hilbert space \mathscr{F}(\mathscr{H}, \mathscr{K}) the abstract boson‐fermion Fock space over (\mathscr{H}, \mathscr{K}) .
In a previous paper [2], the author introduced a general class of infinite‐dimensional

106
Dirac operators on \mathscr{F}(\mathscr{H}, \mathscr{K}) and clarified general mathematical structures behindsome supersymmetric quantum field models giving an abstract unification of them.

In particular, a path (functional) integral representation of analytical index of an
infinite dimensional Dirac operator was derived, which gives a kind of index theorem.
But spectral analysis of the infinite dimensional Dirac operators is still missing. Only

partial results are available [10]. In the present paper, we review some aspects ofspectral analysis of infinite dimensional Dirac operators.

2

Preliminaries

We first recall basic objects and facts associated with Fock spaces. See [11] for moredetails.

In general, for a linear operator A from a Hilbert space to a Hilbert space, wedenote its domain by D(A) .
For each vector f\in \mathscr{H} , there is a unique densely defined closed linear operatora(f) on \mathscr{F}_{b}(\mathscr{H}) such that its adjoint a(f)^{*} takes the following form:

D(a(f)^{*})= \{\psi\in \mathscr{F}_{b}(\mathscr{H})|\sum_{n={\imath} ^{\infty} \Vert\sqrt{n}S_{n}(f\otimes\psi^{(n-1)} \Vert^{2}<\infty\},(a(f)^{*}\psi)^{(0)}=0,(a(f)^{*}\psi)^{(n)}=\sqrt{n}S_{n}(f\otimes\psi^{(n-1)}), n\geq 1, \psi\in D(a(f)^{*}),

where S_{n} denotes the symmetrization operator (symmetrizer) on the n ‐fold tensor
product \otimes^{n}\mathscr{H} of \mathscr{H} . The operator a(f) (resp. a(f)^{*} ) is called the boson annihila‐tion (resp. creation) operator with test vector f.
There is a distinguished vector\Omega_{b}:=\{1,0,0, \cdots\}\in \mathscr{F}_{b}(\mathscr{H}),called the boson Fock vacuum in \mathscr{F}_{b}(\mathscr{H}) , which is vanished by the annihilation
operator:

a(f)\Omega_{b}=0, \forall f\in \mathscr{H}.
The set \{a(f), a(f)^{*}|f\in \mathscr{H}\} of boson annihilation operators and boson creation
operators obeys the canonical commutation relations (CCR) over \mathscr{H} :[a(f), a(g)^{*}]=\{f, g\rangle_{\mathscr{H}}, [a(f), a(g)]=0, f, g\in \mathscr{H}on the bosonic finite particle subspace\mathscr{F}_{b,0}(\mathscr{H}) :=\{\psi\in \mathscr{F}_{b}(\mathscr{H})|\exists n_{0}\in \mathbb{N}s.t. \psi^{(n)}=0, \forall n\geq n_{0}\},where [X, Y] :=XY-YX and \langle , \}_{\mathscr{H} denotes the inner product of \mathscr{H} (linear in
the second variable).

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In general, for a subset \mathscr{E} of a vector space, span ( \mathscr{E} ) or span \mathscr{E} denotes thesubspace generated by all the vectors of \mathscr{E}.
It is well known that, for each dense subspace \mathscr{D} of \mathscr{H} , the subspace\mathscr{F}_{b,fin}(\mathscr{D}) :=span\{\Omega_{b}, a(f_{1})^{*}\cdots a(f_{n} )^{*}\Omega_{b}|n\in \mathbb{N}, f_{j}\in \mathscr{D}, j=1, . n\}
is dense in \mathscr{F}_{b}(\mathscr{H}) . In fact, one has\mathscr{F}_{b,fin}(\mathscr{D})=\otimes_{s}^{n}\mathscr{D}\wedge,
the algebraic n ‐fold symmetric tensor product of \mathscr{D}.
We next move on to the fermion Fock space \mathscr{F}_{f}(\mathscr{K}) . For each u\in \mathscr{K} , there is aunique bounded linear operator b(u) on \mathscr{F}_{f}(\mathscr{K}) such that b(u)^{*} is given as follows:(b(u)^{*}\phi)^{(0)}=0, (b(u)^{*}\phi)^{(p)}=\sqrt{p}A_{p}(f\otimes\phi^{(p-1)} ), p\geq 1, \phi\in \mathscr{F}_{f}(\mathscr{K}),

where A_{p} is the anti‐symmetrization operator (anti‐symmetrizer) on\otimes^{p}\mathscr{K} . The op‐
erator b(u) (resp. b(u)^{*} ) is called the fermion annihilation (resp. creation) operator
with test vectoru.

The vector\Omega_{f}:=\{1,0,0, \}\in \mathscr{F}_{f}(\mathscr{K})is called the fermion Fock vacuum in \mathscr{F}_{f}(\mathscr{K}) , which is vanished by b(u) :b(u)\Omega_{f}=0, \forall u\in \mathscr{K}.

The set \{b(u), b(u)^{*}|u\in \mathscr{K}\} obeys the canonical anti‐commutation relations (CAR)
over\mathscr{K} :\{b(u), b(v)^{*}\}=\langle u, v\}_{\mathscr{K}}, \{b(u), b(v)\}=0, u, v\in \mathscr{K},
where\{X, Y\}:=XY+YX . It follows that

\Vert b(u)\Vert=\Vert u\Vert, \Vert b(u)^{*}\Vert=\Vert u\Vert, b(u)^{2}=0, (b(u)^{*})^{2}=0, \forall u\in \mathscr{K},
where, for a bounded linear operator T on a Hilbert space, \Vert T\Vert denotes the operator
norm of T.

For each dense subspace \mathscr{D} of \mathscr{K} , the subspace\mathscr{F}_{f,fin}(\mathscr{D}) :=span\{\Omega_{f}, b(u_{1})^{*}\cdots b(u_{p} )^{*}\Omega_{f}p\in \mathbb{N}, u_{k}\in \mathscr{D}, k=1, , p\},is dense in \mathscr{F}_{f}(\mathscr{K}) .

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3

Exterior Differential Operators on the Boson‐
Fermion Fock SpaceFor a linear operator L on a Hilbert space, we set

C^{\infty}(L):= \bigcap_{n=1}^{\infty}D(L^{n}),the C^{\infty} ‐domain of L . If L is self‐adjoint, then C^{\infty}(L) is dense.
Let A be a densely defined closed linear operator from \mathscr{H} to \mathscr{K} . Then, by von
Neumann’s theorem, A^{*}A and AA^{*} are non‐negative self‐adjoint operators on \mathscr{H}and \mathscr{K} respectively and hence C^{\infty}(A^{*}A) and C^{\infty}(AA^{*}) are dense in \mathscr{H} and \mathscr{K}respectively. Therefore the algebraic tensor product\mathscr{D}_{A}^{\infty} :=\mathscr{F}_{b,fin}(C^{\infty}(A^{*}A))\otimes \mathscr{F}_{f,fin}(C^{\infty}(AA^{*}))\wedge
is dense in the boson‐fermion Fock space \mathscr{F}(\mathscr{H}, \mathscr{K}) .

Proposition 3.1 There exists a unique den\mathcal{S}ely defined closed linear operator d_{A} on\mathscr{F}(\mathscr{H}, \mathcal{K}) such that the following (i) and (ii) hold:(i) \mathscr{D}_{A}^{\infty}\subset D(d_{A}) and \mathscr{D}_{A}^{\infty} is a core of d_{A}.

(ii) For each vector \Psi\in \mathscr{D}_{A}^{\infty} of the form

\Psi=a(f_{1})^{*}\cdots a(f_{n})^{*}\Omega_{b}\otimes b(u_{1})^{*}\cdots b(u_{p})^{*}\Omega_{f}, n, p\geq 0,
where a(f_{1})^{*}\cdots a(f_{n})^{*}\Omega_{b} (resp. b(u_{1})^{*}\cdots b(u_{p})^{*}\Omega_{f}) with n=0 (resp. p=0 )should read \Omega_{b} (resp. \Omega_{f} ), d_{A} acts as
d_{A}\Psi=0

for n=0,d_{A} \Psi=\sum_{j=1}^{n}a(f_{1})^{*}\cdots\overline{a(f_{j})^{*} \cdots a(f_{n})^{*}\Omega_{b}\otimes b(Af_{j})^{*}b(u_{1})^{*}\cdots b(u_{p})^{*} \Omega_{f}for n\geq 1 , where \overline{a(f_{j})^{*} indicates the omission of a(f_{j})^{*}
leaves \mathscr{D}_{A}^{\infty} invariant.

In particular_{f}d_{A}Moreover, the following (iii) -(v) hold:(iii) \mathscr{D}_{A}^{\infty}\subset D(d_{A}^{*}) and d_{A}^{*}\Psi=0 for p=0,

d_{A}^{*} \Psi=\sum_{k=1}^{p}(-1)^{k-1}a(A^{*}u_{k})^{*}a(f_{1})^{*}

.

.

.

a(f_{n})^{*}\Omega_{b}\otimes b(u_{1})^{*}

for p\geq 1 . In particular, d_{A}^{*} leaves \mathscr{D}_{A}^{\infty} invariant.

.

.

.

\overline{b(u_{k})'}

.

.

.

b(u_{p})^{*}\Omega_{f}109(iv) D(d_{A}^{2})=D(d_{A}) and, for all \Psi\in D(d_{A}), d_{A}^{2}\Psi=0.
(v) Let B be a bounded linear operator from \mathscr{H} to \mathcal{K} with D(B)=\mathscr{H} . Then,for all \Psi\in \mathscr{D}_{A}^{\infty} and \alpha, \beta\in \mathbb{C},\alpha d_{A}\Psi+\beta d_{B}\Psi=d_{\alpha A+\beta B}\Psi.
We call the operator d_{A} the exterior differential operator on \mathscr{F}(\mathscr{H}, \mathscr{K}) associ‐ated with A.

4

Infinite Dimensional Dirac Operators

The Dirac operator on \mathscr{F}(\mathscr{H}, \mathscr{K}) associated with A is defined by
QA :=d_{A}+d_{A}^{*}.

Theorem 4.1 The operator QA i\mathcal{S} self‐adjoint and unbounded from above and below.
The Laplace‐Beltrami‐de Rham operator on \mathscr{F}(\mathscr{H}, \mathscr{K}) associated withdefined byAis\triangle_{A}:=d_{A}^{*}d_{A}+d_{A}d_{A}^{*}.
Theorem 4.2

5\triangle_{A}=Q_{A}^{2}.

Supersymmetric StructureLet\mathscr{F}_{+}:=\mathscr{F}_{b}(\mathscr{H})\otimes(\oplus_{p=0}^{\infty} \wedge^{2p}\mathscr{K}) (even forms),:=\mathscr{F}_{b}(\mathscr{H})\otimes(\oplus_{p=0}^{\infty}\wedge^{2p+1} \mathscr{K}) (odd forms).

\mathscr{F}_{-}

Then we have the orthogonal decomposition\mathscr{F}(\mathscr{H}, \mathscr{K})=\mathscr{F}_{+}\oplus \mathscr{F}_{-}.
Let P\pm:\mathscr{F}(\mathscr{H}, \mathscr{K})arrow \mathscr{F}_{\pm} be the orthogonal projections. Then the operator\Gamma:=P_{+}-P_{-}.

is unitary, self‐adjoint and the grading operator for the above orthogonal decompo‐
sition.

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Proposition 5.1 (anti‐commutativity) Operator equality Q_{A}\Gamma=-\Gamma Q_{A} holds.
Corollary 5.2 (spectral symmetry) The spectrum \sigma(Q_{A}) of QA is reflection sym‐
metric with respect to the origin of \mathbb{R}:\sigma(Q_{A})=\sigma(-Q_{A}) .
The quadruple SQFT_{A} :=(\mathscr{F}(\mathscr{H}, \mathscr{K}), Q_{A}, \triangle_{A}, \Gamma) is a supersymmetric quantum
theory in the abstract sense [1], where QA is a self‐adjoint supercharge, \triangle_{A} is thesupersymmetric Hamiltonian and \Gamma is the state‐sign operator. We remark that
SQFT_{A} gives a unification of some supersymmetric free quantum field models [2,3, 4, 5, 6].

6

Relations with Second Quantization Operators

For each self‐adjoint operator S on \mathscr{H} , one can define the bosonic second quantiza‐tion of S by

d\Gamma_{b}(S):=\oplus_{n=0}^{\infty}d\Gamma_{b}^{(n)}(S)withd \Gamma_{b}^{(0)}(S) :=0, d\Gamma_{b}^{(n)}(S) :=\sum_{j={\imath} ^{n}I\otimes \otimes I\otimes jth\smile S\otimes I\otimes \otimes I, n\geq 1,
where, for a closable operator T on a Hilbert space, \overline{T} denotes the closure of T . It
follows that d\Gamma_{b}(S) is self‐adjoint. If S\geq 0 , then d\Gamma_{b}(S)\geq 0 . Moreover,0\in\sigma_{p}(d\Gamma_{b}(S)) , \Omega_{b}\in ker(d\Gamma_{b}(S)).

Similarly, for each self‐adjoint operator T on \mathscr{K} , one can define the fermionicsecond quantization of T by

d\Gamma_{f}(T):=\oplus_{p=0}^{\infty}d\Gamma_{f}^{(p)}(T)withd \Gamma_{f}^{(0)}(T) :=0, d\Gamma_{f}^{(p)}(T) :=\sum_{j=1}^{p} I\otimes\cdots\otimes I\otimes jthT\smile\otimes I\otimes\cdots\otimes I, p\geq 1.
It follows that d\Gamma_{f}(T) is self‐adjoint. If T\geq 0 , then d\Gamma_{f}(T)\geq 0 . Moreover,0\in\sigma_{p}(d\Gamma_{f}(T)) , \Omega_{f}\in ker(d\Gamma_{f}(T))

.

111
111
As we have already mentioned, the operator A yields the non‐negative self‐adjoint

operators A^{*}A and AA^{*} . Therefore A^{*}A (resp. AA^{*} ) may be a one‐particle Hamilto‐
nian for a boson (resp. fermion). Then the Hamiltonian of a non‐interacting system
consisting of such bosons and fermions is given by

H(A) :=d\Gamma_{b}(A^{*}A)\otimes I+I\otimes d\Gamma_{f}(AA^{*}).

It follows that H(A) is a non‐negative self‐adjoint operator acting in \mathscr{F}(\mathscr{H}, \mathscr{K}) and

0\in\sigma_{p}(H(A)) , \Omega_{b}\otimes\Omega_{f}\in kerH(A).

Theorem 6.1 H(A)=\triangle_{A} . In particular, H(A) is a supersymmetric Hamiltonian.

7

Spectra of H(A) and Q_{A}

In what follows, we assume that \mathscr{H} and \mathscr{K} are separable. For a linear operator Tfrom a Hilbert space to a Hilbert space, we set
nul T:=\dim kerT\in\{0\}\cup \mathbb{N}\cup\{+\infty\}.

Theorem 7.1\sigma(H(A) = \{0\}\cup\overline{(\bigcup_{n=1}^{\infty}\{\sum_{j=1}^{n} \lambda_{j}\lambda_{j}\in\sigma(A^{*}A)\backslash \{0\},j=1,\cdots,n\}) ,\{0\}\cup(\bigcup_{n=1}^{\infty}\{\sum_{j=1}^{n}\lambda_{j}\lambda_{j} \in\sigma_{p}(A^{*}A)\backslash \{0\},j=1, \cdots , n\})\sigma_{p}(H(A))=.

Theorem 7.2 The spectrum \sigma(Q_{A}) and the point spectrum \sigma_{p}(Q_{A}) of Q_{A} are sym‐
metric with respect to the origin and\sigma(Q_{A})\sigma_{p}(Q_{A})==\{0\} cup(\overline{\bigcup_{n=1}^{\infty}\{ pm\sqrt{\sum_{j-1}^{n}\lambda_{j} }\lambda_{j}\in\sigma(A^{*}A)\backslash\{0\},j=1,\cdots,n\})\{0\} cup(\bigcup_{n=1}^{\infty}\{ pm\sqrt{\sum_{j- 1}^{n}\lambda_{j} \lambda_ {j}\in\sigma_{p}(A^{*}A)\backslash \{0\},j=1, \cdots, n\})withnul (Q_{A}-\lambda)= nul (Q_{A}+\lambda) ,\lambda\in\sigma_{p}(Q_{A}) .

,

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8

A Simple Perturbation

In this section, we consider a simple perturbation of QA via a perturbation of d_{A}.
Let

g\in D(A)\backslash \{0\}, v\in D(A^{*})\backslash \{0\}andd(\alpha):=d_{A}+\alpha a(g)\otimes b(v)^{*}
with a constant \alpha\in \mathbb{C} being a perturbation parameter. It is easy to see that d(\alpha)is densely defined with D(d(\alpha))\supset \mathscr{D}_{A}^{\infty} andd(\alpha)^{2}=0on\mathscr{D}_{A}^{\infty}.

Moreover, d(\alpha)^{*} is densely defined with \mathscr{D}_{A}^{\infty}\subset D(d(\alpha)^{*}) andd(\alpha)^{*}=d_{A}^{*}+\alpha^{*}a(g)^{*}\otimes b(v)on\mathscr{D}_{A}^{\infty}.

Hence d(\alpha) is closable. We denote the closure of d(\alpha)[\mathscr{D}_{A}^{\infty} by \overline{d}(\alpha) .
Lemma 8.1 For all \Psi\in D(\overline{d}(\alpha)),\overline{d}(\alpha)\Psi is in D(\overline{d}(\alpha)) and\overline{d}(\alpha)^{2}\Psi=0.
Using the operator \overline{d}(\alpha) , one can define a perturbed Dirac operator:

Q(\alpha):=\overline{d}(\alpha)+\overline{d}(\alpha)^{*}
We note that

Q(\alpha)=Q_{A}+V_{g,v}(\alpha)on\mathscr{D}_{A}^{\infty}

with

V_{g,v}(\alpha) :=\alpha a(g)\otimes b(v)^{*}+\alpha^{*}a(g)^{*}\otimes b(v)8.1.

Self‐adjointness of Q(\alpha)

Let T_{g,v}:\mathscr{H}arrow \mathscr{K} be defined by

T_{g,v}f:=\{g, f\rangle v, f\in \mathscr{H}.
It is obvious that T_{g,v} is a bounded linear operator (a one‐rank operator). Hence

A(\alpha):=A+\alpha T_{g,v}
is a densely defined closed linear operator with D(A(\alpha))=D(A) .

113Remark 8.2 Perturbations of a linear operator by one‐rank or two‐rank operators

have been studied in various contexts. See, e.g. [12, 13] and references therein.

Lemma 8.3 (a key lemma) For all \alpha\in \mathbb{C} , the following operator equality hold_{\mathcal{S}} :\overline{d}(\alpha)=d_{A(\alpha)}.
Theorem 8.4(i) For all \alpha\in \mathbb{C}, Q(\alpha)i\mathcal{S} self‐adjoint andQ(\alpha)=Q_{A(\alpha)}.
(ii) For all \alpha\in \mathbb{C}, Q(\alpha) is essentially self‐adjoint on \mathscr{D}_{A}^{\infty}.

(iii) For all \alpha\in \mathbb{C},Q(\alpha)=\overline{Q_{A}+V_{g,v}(\alpha)}.

(iv) The operator \Gamma leaves D(Q(\alpha)) invariant and\Gamma Q(\alpha)+Q(\alpha)\Gamma=0on

D(Q(\alpha)) .

(v) For all \Psi\in \mathscr{D}_{A}^{\infty} , the vector‐valued function:\alpha\mapsto Q(\alpha)\Psi is strongly contin‐uous on \mathb {C} . Moreover, for all z\in \mathbb{C}\backslash \mathbb{R}, (Q(\alpha)-z)^{-1} is strongly continuous
in \alpha\in \mathbb{C}.

8.2

Spectra of Q(\alpha)

Theorem 8.5 For all \alpha\in \mathbb{C}, \sigma(Q(\alpha)) and \sigma_{p}(Q(\alpha)) are symmetric with respect tothe origin and\sigma(Q \alpha) =\{0\} cup(\overline{\bigcup_{n={\imath}^{\infty} \{ pm\sqrt{\sum_{j-1}^{n}\lambda_{j}\lambda_{j}\in\sigma(A \alpha)^{*} A(\alpha) \backslash\{0\},j=1,\cdots,n\}),\sigma_{p}(Q(\alpha) =\{0\}\cup(\bigcup_{n=1}^{\infty}\{\pm\sqrt{\sum_{j- 1} ^{n}\lambda_{j} \lambda_{j}\in\sigma_{p}(A(\alpha)^{*}A(\alpha) \backslash \{0\} , j=1, \cdots, n\})withnul (Q(\alpha)-\lambda)= nul (Q(\alpha)+\lambda) ,\lambda\in\sigma_{p}(Q(\alpha)) .

This theorem shows that the spectrum and the point spectrum of Q(\alpha) arecompletely determined from those of A(\alpha)^{*}A(\alpha)\backslash \{0\}.

114
Identification of the domain of Q(\alpha)

8.3

Recall that |A| :=(A^{*}A)^{1/2} acting in \mathscr{H} . It follows that A is injective if and only if|A| is injective.

Theorem 8.6 Suppose that A is injective and g\in D(|A|^{-1}) . Then_{f} for all |\alpha|<
1/(\Vert v\Vert\Vert|A|^{-1}g\Vert), Q(\alpha) is self‐adjoint with D(Q(\alpha))=D(Q_{A}) andQ(\alpha)=Q_{A}+V_{g,v}(\alpha).

Moreover, Q(\alpha) is essentially self‐adjoint on any core for Q_{A}.
Proof. The essential part of the proof is to show that V_{g,v}(\alpha) is Q_{A} ‐boundedwith a relative upper bound |\alpha||v|\Vert|A|^{-1}g\Vert . Then one needs only to apply the

Kato‐Rellich theorem. For more details, see the proof of [10, Theorem 17].

\bul etKernel of Q(\alpha)9

We now investigate the kernel of Q(\alpha) . We need a classification for conditions on\{A, g, v\} :(C.1)

A is injective,v\in D(A^{-1}) and \{g, A^{-1}v\}\neq 0 . In this case we introduce a

constant\alpha_{0}:=-\frac{1}{\langle g,A^{-1}v\rangle} .
(C.2)

A^{*} is injective,
constant(9.1)

g\in D(A^{*-1}) and \langle v, A^{*-1}g\rangle\neq 0 . In this case we introduce a\beta_{0}:=-\frac{1}{\langle A^{*^{-1} g,v\rangle}.
(C.3) (a) A is injective and v\not\in D(A^{-1}) or (b) A is injective and v\in D(A^{-1}) with\{g, A^{-1}v\rangle=0.
(C.4) (a) A^{*} is injective and g\not\in D(A^{*-1}) or (b) A^{*} is injective g\in D(A^{*-1}) with\langle v, A^{*-1}g\rangle=0.

We first consider the kernel of A(\alpha) and A(\alpha)^{*}.
Lemma 9.1

(i) Suppose that (C. 1) holds. ThenkerA(\alpha) = \{0\}, \alpha\neq\alpha_{0},kerA(\alpha_{0}) = \{cA^{-1}v|c\in \mathbb{C}\}.

115(ii) Suppose that (C.2) holds. ThenkerA(\alpha)^{*} = \{0\}, \alpha\neq\beta_{0},kerA(\beta_{0})^{*} = \{cA^{*-1}g|c\in \mathbb{C}\}.
(iii) Suppose that (C.3) holds. Then, for all \alpha\in \mathbb{C},kerA(\alpha) = \{0\}.
(iv) Suppose that (C.4) holds. Then, for all \alpha\in \mathbb{C},kerA(\alpha)^{*} = \{0\}.
Theorem 9.2(i) Assume (C.1). ThenkerQ(\alpha_{0})=\oplus_{n,p=0}^{\infty}[(\otimes^{n}\{zA^{-1}v|z\in \mathbb{C} \})\otimes\wedge^{p}(kerA(\alpha_{0})^{*})].
and hence nul Q(\alpha_{0})=\infty.

Moreover, for all \alpha\neq\alpha_{0},kerQ(\alpha)=\oplus_{p=0}^{\infty}\mathbb{C}\otimes\wedge^{p}(kerA(\alpha)^{*})(ii)

A_{S\mathcal{S}}ume (C.2).

.

Then

kerQ(\beta_{0})=\oplus_{n=0}^{\infty}\{[\otimes_{s}^{n}ker(A(\beta_{0}))] \otimes[\mathbb{C}\oplus span(\{A^{*-}g\})]\},
kerQ(\alpha)=\oplus_{n=0}^{\infty}[\otimes_{s}^{n}kerA(\alpha)\otimes \mathbb{C}], \alpha\neq\beta_{0}.
(iii) Assume (C.3)

Then, for all \alpha\in \mathbb{C},kerQ(\alpha)=\oplus_{p=0}^{\infty}[\mathbb{C}\otimes\wedge^{p}(ker(A(\alpha) ^{*})].
(iv)

A_{\mathcal{S}}sume (C.4).

Then, for all \alpha\in \mathbb{C},kerQ(\alpha)=\oplus_{n=0}^{\infty}[\otimes_{s}^{n}kerA(\alpha)\otimes \mathbb{C}].
Corollary 9.3(i) Assume (C.1) and (C.2). ThenkerQ(\alpha_{0})= span(\{a(A^{-1}v)^{*}n\Omega_{b}\otimes b(A^{*^{-1} g)^{*}j\Omega_{f}|n\geq 0, j=0,1\})

kerQ(\alpha)=\{c\Omega_{b}\otimes\Omega_{f}|c\in \mathbb{C}\}, \alpha\neq\alpha_{0}.

,116(ii) Assume (C.1) and (C.4). ThenkerQ(\alpha_{0})=\overline{span(\{a(A^{-1}v)^{*}n\Omega_{b}\otimes\Omega_{f} |n\geq 0\})},
kerQ(\alpha)=\{c\Omega_{b}\otimes\Omega_{f}|c\in \mathbb{C}\}, \alpha\neq\alpha_{0}.
(iii) Assume (C.2) and (C.3). ThenkerQ(\beta_{0})=span(\{\Omega_{b}\otimes b(A^{*-1}g)^{*}\Omega_{f}|jj=0,1\}).

kerQ(\alpha)=\{c\Omega_{b}\otimes\Omega_{f}|c\in \mathbb{C}\}, \alpha\neq\beta_ {0}.
(iv) Assume (C.3) and (C.4). Then, for all \alpha\in \mathbb{C},kerQ(\alpha)=\{c\Omega_{b}\otimes\Omega_{f}|c\in \mathbb{C}\}.

10

Non‐zero Eigenvalues of Q(\alpha)Hypothesis (A)(i) \mathscr{H}=\mathcal{K} ;(ii) A is an injective and nonnegative self‐adjoint operator;(iii) g=v\in D(A^{-1}) .

Under Hypothesis (A), the constant \alpha_{0} defined by (9.1) takes the form

\alpha_{0}=-\frac{1}{\{v,A^{-1}v\rangle}<0.
Theorem 10.1 Let Hypothesis (A) be \mathcal{S} atisfied and \alpha<\alpha_{0}(<0) . Then, there
exists a unique constant x_{0}(\alpha)<0 such that \alpha\langle v, (x_{0}(\alpha)-A)^{-1}v } =1 and, for alln\in\{0\}\cup \mathbb{N},\pm\sqrt{n}x_{0}(\alpha)\in\sigma_{p}(Q(\alpha)).

with eigenvectors[Q(\alpha)\pm\sqrt{n}x_{0}(\alpha)]\{a(\phi.)^{*n-p}\Omega_{b}\otimes b(\phi_{\alpha})^{*p}\Omega_{f}\}\in ker(Q(\alpha)\mp\sqrt{n}x_{0}(\alpha))(n\geq p\geq 0)

,

where

\phi_{\alpha}:=(x_{0}(\alpha)-A)^{-1}v.
Moreover, x_{0}(\alpha) , as a function of \alpha<\alpha_{0} , is strictly monotone increasing on( ‐00, \alpha_{0}) with \lim_{\alphaarrow-\infty}x_{0}(\alpha)=-\infty and \lim_{\alphaarrow\alpha_{0}}x_{0}(\alpha)=0.
Note that Theorem 10.1 holds even if Q_{A} has no non‐zero eigenvalues. This is
an interesting phenomenon. Since the condition \alpha<\alpha_{0}<0 implies that |\alpha|>|\alpha_{0}|,the phenomenon may be regarded as a strong coupling effect.

117
Acknowledgement
This work is supported by KAKENHI

15K04888 from JSPS.

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378‐393.

[2] A. Arai, A general class of infinite‐dimensional Dirac operators and path inte‐gral representation of their index, J. Funct. Anal. 105 (1992), pp.342‐408.
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