Konstantin Pankrashkin
email: konstantin.pankrashkin@math.u-psud.fr

Serge Richard
email: richard@math.univ-lyon1.fr

One-dimensional Dirac operators with zero-range interactions: Spectral, scattering, and topological results

Keywords: 2010 Mathematics Subject Classification: 81U15, 35Q41 Dirac operators, zero-range interaction, wave operators, index theorem

The spectral and scattering theory for 1-dimensional Dirac operators with mass m and with zero-range interactions are fully investigated. Explicit expressions for the wave operators and for the scattering operator are provided. These new formulae take place in a representation which links, in a suitable way, the energies -∞ and +∞, and which emphasizes the role of ±m. Finally, a topological version of Levinson's theorem is deduced, with the threshold effects at ±m automatically taken into account.

Introduction

In a series of recent works on scattering theory, it has been shown that rather explicit formulae for the wave operators do exist and that these operators share structural properties amongst several models. Such new formulae were then at the root of a topological approach of Levinson's theorem. More precisely, it has been shown that this famous theorem, which allows one to compute the number of bound states of a physical system in terms of the scattering part of that system, is in fact an index theorem. Let us stress that an index theorem automatically means a strong robustness of the mentioned link between spectral and scattering properties under perturbations. We refer to [START_REF] Bellissard | Scattering theory for lattice operators in dimension d ≥ 3[END_REF][START_REF] Isozaki | On the wave operators for the Friedrichs-Faddeev model[END_REF][START_REF] Kellendonk | Levinson's theorem and higher degree traces for Aharonov-Bohm operators[END_REF][START_REF] Kellendonk | Levinson's theorem for Schrödinger operators with point interaction: a topological approach[END_REF][START_REF] Kellendonk | On the structure of the wave operators in one dimensional potential scattering[END_REF][START_REF] Kellendonk | On the wave operators and Levinson's theorem for potential scattering in R 3[END_REF][START_REF] Pankrashkin | Spectral and scattering theory for the Aharonov-Bohm operators[END_REF][START_REF] Schulz-Baldes | The density of surface states as the total time delay[END_REF][START_REF] Richard | New formulae for the wave operators for a rank one interaction[END_REF][START_REF] Richard | New expressions for the wave operators of Schrödinger operators in R 3[END_REF][START_REF] Richard | Explicit formulas for the Schrödinger wave operators in R 2[END_REF] for such explicit formulae in the context of Schrödinger operators, Aharonov-Bohm operators or for the Friedrichs-Faddeev model, and for their applications.

However, despite the variety of models already investigated with this approach, all these models share one common property: the operators describing them have a continuous spectrum made of one single connected part. This feature, which may seem harmless, has been in fact very convenient for the construction of the C * -algebraic framework surrounding the topological approach. Therefore, one of the motivations for looking at Dirac operators was that its continuous spectrum is made of two unbounded disjoint parts. In addition, even if some Levinson's type results exist for this model (see for example [START_REF] Clemence | Low-energy scattering and Levinson's theorem for a one-dimensional Dirac equation[END_REF][START_REF] Klaus | On the Levinson theorem for Dirac operators[END_REF][START_REF] Lin | Levinson theorem for Dirac particles in one dimension[END_REF][START_REF] Ma | Levinson theorem for Dirac particles[END_REF]), it has never been argued that this relation is of topological nature1 . Thus, this work is a first attempt to derive explicit formulae for the wave operators in the context of relativistic operators, and to deduce some topological consequences of such formulae. Now, as any first attempt, the model under consideration is rather simple (see [START_REF] Kellendonk | Levinson's theorem for Schrödinger operators with point interaction: a topological approach[END_REF] for its counterpart in the Schrödinger case). In fact, we investigate 1-dimensional massive Dirac operators with the simplest interactions, namely the so called "zero-range" or "point" interactions. More precisely, we consider the scattering theory for a pair of self-adjoint operators (H 0 , H CD ), where H 0 is the usual free Dirac operator with mass m > 0 and H CD is any of the (four parameters family of) self-adjoint extensions which can be constructed from the restriction of H 0 to functions which vanish at 0. Note that such extensions have already been studied in several papers, see for example [START_REF] Alonso | Delta-type Dirac point interactions and their nonrelativistic limits[END_REF][START_REF] Behrndt | Scattering matrices and Weyl functions[END_REF][START_REF] Benvegnù | Relativistic point interaction[END_REF][START_REF] Carlone | On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set[END_REF], but our aim and results are different.

More concretely, after reviewing some properties of free Dirac operator H 0 , we provide in Section 2 a parametrization of all the mentioned self-adjoint extensions using the machinery of boundary triples, and we obtain an explicit resolvent formula in terms of the parameters (C, D). Based on these formulae, the spectral properties of the Dirac operator with a zero-range interaction are then described in Proposition 2.3.

In Section 3, we develop the scattering theory for our model. We first recall the main definition of the wave operators in the time-dependent framework of scattering theory as well as in its stationary approach. The spectral representation of the free operator H 0 is then provided. Based on the stationary expressions for the wave operators, some rather explicit formulae could be derived for them in the spectral representation of H 0 , but the results would not be very convenient (some unbounded operators would still be present).

Our main surprise, and one of the asset of this work, is that the wave operators can be computed very explicitly in another unitarily equivalent representation which we have called the upside-down representation. The reason for this name comes from the fact that in this representation the thresholds values ±m are sent to ±∞ while any neighbourhood of the points ±∞ is then located near the point 0. In this representation, which takes place in the Hilbert space L 2 (R; C 2 ), we first show that the wave operators for the pair (H 0 , H CD ) exist and that the stationary approach leads to the same operators than the time dependent approach. In addition, restricting our attention to one of the wave operators only, we provide an explicit expression for this operator in terms of a product of a continuous function of the position operator X and a continuous function of its conjugate operator D = -i d dx , see formula (3.10). Note that the X-factor is tightly dependent of the parameters (C, D), while the factor containing D does not depend on them at all. We also note that these factors admit a suitable asymptotic behavior, which allows one to develop the algebraic framework.

In the last section of this paper, we deduce the topological consequences of the explicit formula derived in Section 3. In particular, we derive a topological version of Levinson's theorem which relates the number of bound states of the operator H CD to the winding number of a certain function which involves the scattering operator but also other operators related to threshold effects. The main result of this section in contained in Theorem 4.1. Let us stress that in our approach, the threshold effects are automatically taken into account, namely we don't have to calculate separately the contributions due to the possible half-bound states located at ±m. Note also that the discrepancy of the contributions of the scattering operator for positive or negative energies appears naturally in our framework.

As a conclusion, let us emphasize that the spectral and the scattering theory for Dirac operators with zero-range interactions are fully developed in this work, and that new and explicit formulae for the wave operators are also provided. We expect that such formulae as well as the topological approach of Levinson's theorem will still hold for Dirac operators perturbed by more general potentials.

Acknowledgments. The work was partially supported by ANR NOSEVOL and GDR DYN-QUA.

2 Framework and spectral results

Dirac operators with boundary conditions at the origin

Let H be the Hilbert space L 2 (R; C 2 ) with scalar product and norm denoted by ⟨•, •⟩ and ∥ • ∥.

Its elements are written

f = ( f 1 f 2 )
. For m > 0, we consider the free Dirac operator H 0 defined by

H 0 = ( 0 -1 1 0 ) d dx + ( m 0 0 -m ) (2.1) on the domain dom(H 0 ) = H 1 (R; C 2 ).
Here, H 1 (R; C 2 ) denotes the Sobolev space on R of order 1 and with values in C 2 . A standard computation shows that H 0 is a self-adjoint operator in H, and its spectrum equals (∞, -m] ∪ [m, +∞) and is absolutely continuous. In what follows we need an explicit expression for the Green function of H 0 , that is, for the integral kernel G 0 of its resolvent. For that purpose, for z

∈ C \ σ(H 0 ) we set k = k(z) = √ z 2 -m 2
, where the branch of the square root is fixed by the condition ℑ √ λ > 0 for λ < 0. As a consequence it follows that ℑk(z) > 0 for all z / ∈ σ(H 0 ). Then, let us note that for any z ∈ C the equality (H 0z)(H 0 + z) = -∆z 2 + m 2 holds on H 2 (R; C 2 ); here ∆ denotes the usual Laplacian in L 2 (R; C 2 ). Therefore, for any z / ∈ σ(H 0 ) we have

(H 0 -z) -1 = (H 0 + z) ( -∆ -(z 2 -m 2 ) ) -1 ,
and we infer from this equality that

G 0 (x, y; z) = i 2k ( z + m -d/dx d/dx z -m ) e ik|x-y| = ( i m+z k sgn(x -y) -sgn(x -y) i k m+z ) e ik|x-y| 2 . (2.2)
Now, let us denote by H the restriction of H 0 to the domain

dom(H) = { f ∈ H 1 (R; C 2 ) | f (0) = 0 } .
This operator is not self-adjoint any more, but symmetric with deficiency indices (2, 2), see for example [START_REF] Behrndt | Scattering matrices and Weyl functions[END_REF]Sec. 5.3]. Furthermore, its adjoint H * is given by the same expression but acts on the larger domain dom(H

* ) = H 1 (R -; C 2 ) ⊕ H 1 (R + ; C 2 )
. By a Dirac operator with a zero-range (or delta-type) interaction at the origin we mean any self-adjoint extension of H. Note that there already exist several papers discussing various parameterizations of such extensions, see e.g. [START_REF] Alonso | Delta-type Dirac point interactions and their nonrelativistic limits[END_REF][START_REF] Behrndt | Scattering matrices and Weyl functions[END_REF][START_REF] Benvegnù | Relativistic point interaction[END_REF][START_REF] Carlone | On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set[END_REF], but our approach will be slightly different.

By a direct computation one can check that (C 2 , Γ 1 , Γ 2 ) with

Γ 1 f = ( f 1 (+0) -f 1 (-0) f 2 (-0) -f 2 (+0) ) , Γ 2 f = 1 2 ( f 2 (-0) + f 2 (+0) f 1 (-0) + f 1 (+0)
) .

is a boundary triple for H * in the sense that for all f, g ∈ dom(H * ) one has

⟨f, H * g⟩ -⟨H * f, g⟩ = ⟨Γ 1 f, Γ 2 g⟩ -⟨Γ 2 f, Γ 1 g⟩,
and that the map dom(H

* ) ∋ f → (Γ 1 f, Γ 2 f ) ∈ C 2 × C 2 is surjective.
Here we have used the notation f j (±0) for lim ε↘0 f j (±ε) for any f j ∈ H 1 (R ± ).

Let us now recall a few facts about boundary triples, and refer to [START_REF] Brüning | Spectra of self-adjoint extensions and applications to solvable Schrödinger operators[END_REF] for a brief account, to [START_REF] Derkach | Generalized resolvents and the boundary value problems for Hermitian operators with gaps[END_REF][START_REF] Gorbachuk | Boundary value problems for operator differential equations[END_REF] for a detailed study, and to [START_REF] Schmüdgen | Unbounded self-adjoint operators on Hilbert space[END_REF] for a recent textbook presentation. One of the main interest of boundary triples is that they easily provide a simple description of all self-adjoint extensions of H and a tool for their spectral and scattering analysis. More precisely, let C, D ∈ M 2 (C) be 2 × 2 matrices, and let us denote by H CD the restriction of H * to the domain ) is exactly the above free Dirac operator H 0 , which is going to play the role of our reference operator. The other operators H CD will be interpreted as its perturbations.

dom(H CD ) := {f ∈ dom(H * ) | CΓ 1 f = DΓ 2 f } . Then,
Let us stress that the above parametrization is not unique in the sense that one can have

H CD = H C ′ D ′ for two different admissible pairs (C, D) and (C ′ , D ′ )
. This is the case if and only if C = KC ′ and D = KD ′ for a non-degenerate 2 × 2 matrix K. One may obtain a one-to-one parametrization between the 2 × 2 unitary matrices U and the self-adjoint extensions of H by setting C := 1 2 (1 -U ) and D := i 2 (1 + U ). We refer to the papers [START_REF] Alonso | Delta-type Dirac point interactions and their nonrelativistic limits[END_REF][START_REF] Benvegnù | Relativistic point interaction[END_REF] for alternative one-to-one parameterizations.

Remark 2.1. We note that the papers [START_REF] Behrndt | Scattering matrices and Weyl functions[END_REF][START_REF] Carlone | On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set[END_REF] also make use of boundary triples for studying perturbed Dirac operators, but with a different choice for the maps Γ 1 and Γ 2 , which leads to a rather complicated reference operator. Our choice is motivated by obtaining simpler expressions in the subsequent computations.

Weyl function and resolvent formula

Our aim is to obtain a resolvent formula for the self-adjoint extensions H CD . First, we compute explicitly two operator-valued maps playing a key role in the framework of boundary triples, namely the map

γ(z) := ( Γ 1 ker(H * -z) ) -1
and the Weyl function M (z) := Γ 2 γ(z). By a direct computation one obtains for any z /

∈ σ(H 0 ), ξ = ( ξ 1 ξ 2 ) ∈ C 2 and x ∈ R * that [ γ(z) ( ξ 1 ξ 2 )] (x) =     i m + z k ξ 2 + ξ 1 sgn(x) i k m + z ξ 1 -ξ 2 sgn(x)     e ik|x| 2 ≡ ξ 1 h 1 z (x) + ξ 2 h 2 z (x),
where we have set

h 1 z (x) = e ik|x| 2 ( sgn(x) i k m+z ) , h 2 z (x) = e ik|x| 2 ( i m+z k -sgn(x)
) .

For later use, note that the γ-function satisfies the following identity [5, Th. 1.23]

γ(z) * (H 0 -z)f = Γ 2 f, f ∈ dom(H 0 ), z / ∈ σ(H 0 ). (2.4)
Similarly, for the Weyl function we obtain

M (z) ( ξ 1 ξ 2 ) = 1 2     ik m + z ξ 1 i(m + z) k ξ 2     , i.e. M (z) is just the diagonal matrix M (z) = 1 2 diag ( ik m + z , i(m + z) k ) .
In particular, since for λ ∈ R the following limits hold:

k(λ + i0) ≡ lim ε↘0 k(λ + iε) = { sgn(λ) √ λ 2 -m 2 , |λ| ≥ m, i √ m 2 -λ 2 , |λ| < m, k(λ -i0) ≡ lim ε↘0 k(λ -iε) = { -sgn(λ) √ λ 2 -m 2 , |λ| ≥ m, i √ m 2 -λ 2 , |λ| < m.
it follows that for |λ| < m one has

M (λ ± i0) = 1 2 diag ( - √ m -λ m + λ , √ m + λ m -λ ) while for |λ| > m one has M (λ ± i0) = ± i 2 diag ( √ λ -m λ + m , √ λ + m λ -m ) .
With these various definitions a few additional relations between the operators H 0 and H CD can be inferred. For example, for z / ∈ σ(H 0 ) ∪ σ(H CD ) the matrix DM (z) -C is invertible, and the resolvent formula

(H 0 -z) -1 -(H CD -z) -1 = γ(z) ( DM (z) -C ) -1 Dγ(z) * (2.5) holds. In addition, a value λ ∈ (-m, m) is an eigenvalue of H CD if and only if det ( DM (λ + i0) - C ) = 0,

and then one has ker(H

CD -λ) = γ(λ) ker(DM (λ + i0) -C).
Due to the injectivity of the map γ(λ) : C 2 → H, the dimension of ker(DM (λ + i0) -C) corresponds to the multiplicity of the eigenvalue λ of H CD . Finally, for any ε ≥ 0, any λ ̸ ∈ [-m, m] and any admissible pair (C, D), let us mention the obvious equality M (λiϵ) = M (λ + iε) * and the identity (see for example [START_REF] Pankrashkin | Spectral and scattering theory for the Aharonov-Bohm operators[END_REF]Lem. 6]) :

[ ( DM (λ -iε) -C ) -1 D ] * = ( DM (λ + iε) -C ) -1 D. (2.6)
For the next statement, we need to introduce the set

Σ := (-∞, -m) ∪ (m, +∞).
as well as for each λ ∈ Σ the 2 × 2 matrix

B(λ) = 1 √ 2 diag ( 4 √ λ -m λ + m , 4 √ λ + m λ -m
) which clearly satisfies iB(λ) 2 = M (λ + i0). Furthermore, we also set

T CD ε (λ) := -2iB(λ) ( DM (λ + iε) -C ) -1 D B(λ).
We summarize some properties of this operator in the following assertion, whose proof is given in the Appendix.

Lemma 2.2. For any admissible pair (C, D), the operator T CD ε (λ) admits the limit

T CD 0 (λ) = -2iB(λ) ( DM (λ + i0) -C ) -1 D B(λ) (2.7)
in M 2 (C) as ε ↘ 0 locally uniformly in λ ∈ Σ. For any λ ∈ Σ the matrix 1 + T CD 0 (λ) is unitary, and the map Σ ∋ λ → T CD 0 (λ) ∈ M 2 (C) is continuous and admits limits at the boundary points of Σ, with T CD 0 (-∞) = T CD 0 (+∞).

Spectral analysis

In the next statement, we infer some spectral results for the operator H CD . ) , then

# σ p (H CD ) =      2 if ℓ 11 < 0 and ℓ 22 > 0, 0 if ℓ 11 ≥ 0 and ℓ 22 ≤ 0, 1 otherwise. (b) If dim[ker(D)] = 1 and (p 1 , p 2 ) is a unit vector spanning ker(D), then # σ p (H CD ) = 1 if p 1 p 2 ̸ = 0, if p 2 = 0 and tr(CD * ) > 0, or if p 1 = 0 and tr(CD * ) < 0. In the other cases, # σ p (H CD ) = 0. (c) If D = 0, then σ p (H CD ) = ∅.
Proof. Since the difference of the resolvents of H CD and H 0 is a finite rank operator, it clearly follows that σ ess (H

CD ) = σ ess (H 0 ) = σ(H 0 ) = (-∞, -m] ∪ [m, +∞).
Let f ∈ H and let µ CD f be the spectral measure associated with H CD and f . It is well known (see for example [START_REF] Jakšić | Topics in spectral theory[END_REF]Thm. 4.15]) that the singular part µ CD f,s of µ CD f is concentrated on the set

{ λ ∈ R | lim ε↘0 ℑ ⟨ f, (H CD -λ -iε) -1 f ⟩ = ∞ } . So, let us consider f ∈ C ∞ c (R; C 2
) and compute the above quantity for any |λ| > m. By (2.5) and for z = λ + iε, this reduces to an evaluation of the expressions

lim ε↘0 ℑ ⟨ f, (H 0 -z) -1 f ⟩ and lim ε↘0 ℑ ⟨ γ(z) * f, ( DM (z) -C ) -1 Dγ(z) * f ⟩ C 2 .
(2.8)

A simple computation which takes the expression (2.2) for the Green function into account shows that the first term in (2.8) is finite. Similarly, by using the explicit expression for γ(z), one easily obtains that the limits lim ε↘0 γ(λ ± iε) * f ∈ C 2 exist. On the other hand, still for |λ| > m one has ℑM (λ + i0) > 0, hence, by [22, Lem. 6], the limit

( DM (λ + i0) -C ) -1 ∈ M 2 (C) exists.
Therefore, the second expression in (2.8) is also finite, and thus the support of µ CD f,s does not intersect the set (-∞, -m) ∪ (m, +∞). Since f is an arbitrary element of a dense set in H, it means that H CD has no singular spectrum in (-∞, -m) ∪ (m, +∞), and in particular that the singular continuous spectrum is empty. Now, to see that ±m / ∈ σ p (H CD ) it is sufficient to observe that the only solutions of the ordinary differential equation Hf = ±mf with H given by the expression (2.1) are either constant or polynomially growing. In particular, this equation has no solution in H.

It remains to count the eigenvalues of H CD in the interval (-m, m). For that purpose and as noted in the previous section, we first need to determine if the operator DM (λ + i0) -C has a 0-eigenvalue for some λ ∈ (-m, m). To simplify the notation, let us set t = √ m-λ m+λ . Then, this problem reduces in the study of the possible 0-eigenvalue of the operator 1 2 D diag(-t, 1/t) -C for t ∈ (0, ∞). a) We first consider the case det(D) ̸ = 0. For that purpose, we set Λ = D -1 C and study the equivalent admissible pair (Λ, 1). Now, observe that the matrix Λ is hermitian:

Λ =: ( ℓ 11 ℓ 12 ℓ 12 ℓ 22
) .

Thus, we are left with the study of the determinant of the operator 1 2 diag(-t, 1/t) -L with t ∈ (0, ∞). Let us still define the map (0, ∞) ∋ t → D Λ (t) := with ker(D) ⊥ , and let P denote its adjoint, i.e. P : C 2 → C is the composition of the orthogonal projection onto ker(D) ⊥ together with the identification of IC with C. Then, as shown in [22, Eq. ( 12)], the operator DM (λ + i0) -C is invertible if and only if the reduced operator P M (λ + i0)Iℓ is invertible, where ℓ := (DI) -1 CI ∈ R. By using this observation and the change of variable t = √ m-λ m+λ we see that the 0-eigenvalue of the operator 1 2 D diag(-t, 1/t) -C for t ∈ (0, ∞) coincides with the 0 of the map

] = √ -ℓ 11 ℓ 22 , D Λ (•) vanishes twice on (0, ∞) if sgn(ℓ 11 )[det(L) -1 4 ] > √ -ℓ 11 ℓ 22 ,
(0, +∞) ∋ t → d ℓ (t) := |p 2 | 2 t -|p 1 | 2 /t + 2ℓ ∈ R.
By elementary considerations we observe that this map vanishes once if p 1 p 2 ̸ = 0, if p 2 = 0 and ℓ > 0, or if p 1 = 0 and ℓ < 0. In the other cases the map d ℓ (•) never vanishes. It now only remains to relate these various results with the dimension of H p (H CD ). As mentioned before the statement of the lemma, for each λ ∈ (-m, m) such that 0 is an eigenvalue of DM (λ + i0) -C, one needs to determine the multiplicity of this eigenvalue. Equivalently, for each t ∈ (0, ∞) such that 0 is an eigenvalue of 1 2 D diag(-t, 1/t) -C, one needs to determine the multiplicity of this eigenvalue. An simple inspection in the previous computations shows that the multiplicity of the 0-eigenvalue is always 1, except in one special case already emphasized above, namely when ℓ 11 < 0, ℓ 11 ℓ 22 = -1 4 and ℓ 12 = 0, for which the multiplicity if 2. By collecting all these results, one finally obtains the statement of the lemma.

3 Scattering theory

Wave operators and scattering operator

In this section, we describe the scattering theory for the pair of the operators (H CD , H 0 ). Since (

H CD -i ) -1 -(H 0 -i) -
) := W + ( H CD , H 0 ) * W - ( H CD , H 0 ) (3.2)
is usually referred to as the scattering operator. Thus, the aim of the present section is to calculate these objects in terms of the Weyl function and of the parameters (C, D). To do that, we recall the so-called stationary expressions for the wave operators. Namely, for suitable f, g ∈ H we consider the operators W CD ± defined by

⟨W CD ± f, g⟩ = ∫ R lim ε↘0 ε π ⟨ (H 0 -λ ∓ iε) -1 f, (H CD -λ ∓ iε) -1 g ⟩ dλ.
Note that the precise choice for the elements f, g will be specified later on, and that the equality of W ± ( H CD , H 0 ) with W CD ± will follow from our computations. In the sequel, we concentrate on W CD -, and stress that the operator W CD + can be treated similarly.

For that purpose, consider for ε > 0 the function δ ε : R → R given by

δ ε (x) = 1 π ε x 2 + ε 2 . We clearly have δ ε ( H 0 -λ ) = ε π (H 0 -λ + iε) -1 (H 0 -λ -iε) -1 .
With this notation, the limit lim ε↘0 ⟨ δ ε ( H 0λ ) f, g ⟩ exists for a.e. λ ∈ R, and

∫ R lim ε↘0 ⟨ δ ε (H 0 -λ)f, g ⟩ dλ = ⟨f,
⟨(W CD --1)f, g⟩ = - ∫ R lim ε↘0 ⟨ ε π γ(λ -iε) * (H 0 -λ + iε) -1 f, ( DM (λ -iε) -C ) -1 Dγ(λ + iε) * g ⟩ C 2 dλ. (3.3)
Our next aim will be to put together some of these terms and to obtain some more coherent and simply understandable factors. For that purpose, the spectral representation of the operator H 0 will be needed.

Spectral representation of the free Dirac operator

In this section, we construct the spectral representation of H 0 , mimicking the construction provided in [START_REF] Ito | High-energy behavior of the scattering amplitude for a Dirac operator[END_REF]Sec. 2]. For any fixed p ∈ R, let us set

h(p) := ( m -ip ip -m ) ∈ M 2 (C).
The eigenvalues of this matrix are ± √ p 2 + m 2 and two normalized eigenfunction are defined by the expressions

ξ + (p) := 1 √ 2(p 2 + m 2 + m √ p 2 + m 2 ) ( m + √ p 2 + m 2 ip ) , ξ -(p) := 1 √ 2(p 2 + m 2 + m √ p 2 + m 2 ) ( ip m + √ p 2 + m 2
) .

(3.4)

For simplicity, the orthogonal projection on the subspace generated by ξ ± (p) will be denoted by

P ± (p) ∈ M 2 (C).
Then, for any λ ∈ R satisfying ±λ > m, let us define

H (λ) := ( P ± (- √ λ 2 -m 2 ) C 2 , P ± ( √ λ 2 -m 2 ) C 2
) ,

and observe that H (λ) is a two dimensional subspace of C 2 ⊕ C 2 . Let us also set

H := ∫ ⊕ Σ H (λ) dλ.
More precisely, any element φ ∈ H is of the form (φ 1 , φ 2 ) with

φ j ∈ L 2 (Σ; C 2 ), φ 1 (λ) collinear to ξ ± ( - √ λ 2 -m 2 ) and φ 2 (λ) collinear to ξ ± (√ λ 2 -m 2 )
for ±λ > m. One also defines the unitary operator U : H → H given for f ∈ H and ±λ > m by

[U f ](λ) := 4 √ λ 2 λ 2 -m 2 ( P ± (- √ λ 2 -m 2 ) f ( - √ λ 2 -m 2 ) , P ± ( √ λ 2 -m 2 ) f ( √ λ 2 -m 2 ) ) .
Note that its adjoint is provided by the following expression: for any φ ∈ H with φ = (φ 1 , φ 2 ) and for p ∈ R * one has

[U * φ](p) = 4 √ p 2 p 2 + m 2 { φ 1 ( - √ p 2 + m 2 ) + φ 1 (√ p 2 + m 2 ) if p < 0, φ 2 ( - √ p 2 + m 2 ) + φ 2 (√ p 2 + m 2 ) if p > 0.
Obviously, the above expressions have to be understood in the L 2 -sense, i.e. for almost every λ ∈ Σ or for almost every p ∈ R * . It is now a matter of a simple computation to check that for any λ ∈ Σ one has

[U h(•) U * φ](λ) = λ φ(λ).
In addition, if F denotes the Fourier transform on L 2 (R; C 2 ), then the operator F 0 := UF : H → H realizes the spectral representation of H 0 , namely

F 0 H 0 F * 0 = L 0 , (3.5)
where L 0 is the self-adjoint operator of multiplication by the variable λ in H .

Computing the wave operator : preliminary steps

In order to simplify the expression (3.3) we first use the identity (2.4), which gives, for z / ∈ σ(H 0 ),

ε π γ(z) * (H 0 -z) -1 = ε π Γ 2 (H 0 -z) -1 (H 0 -z) -1 = Γ 2 δ ε (H 0 -λ), γ(z) * = Γ 2 (H 0 -z) -1 .
By collecting these equalities and using (2.6), one infers that

⟨(W CD --1)f, g⟩ = - ∫ R lim ε↘0 ⟨ Γ 2 δ ε (H 0 -λ)f, ( DM (λ -iε) -C ) -1 DΓ 2 (H 0 -λ + iε) -1 g ⟩ C 2 dλ = 1 2 ∫ Σ lim ε↘0 ⟨ T CD ε (λ)B(λ) -1 Γ 2 δ ε (H 0 -λ)f, iB(λ) -1 Γ 2 (H 0 -λ + iε) -1 g ⟩ C 2 dλ.
Note that for the second equality, we have taken into account that the above integrant vanishes for almost every λ ∈ (-m, m) as ε ↘ 0; more precisely, it vanishes at any point λ ∈ (-m, m) which is not an eigenvalue of H CD .

By looking at the previous equality inside the spectral representation of H 0 , one has thus obtained that for suitable φ, ψ ∈ H :

⟨F 0 (W CD --1)F * 0 φ, ψ⟩ H = 1 2 ∫ Σ lim ε↘0 ⟨ T CD ε (λ)B(λ) -1 Γ 2 F * 0 δ ε (L 0 -λ)φ, iB(λ) -1 Γ 2 F * 0 (L 0 -λ + iε) -1 ψ ⟩ C 2 dλ. (3.6)
For the next statement, one needs to be a little bit more cautious about the set of suitable elements of H . For that purpose, we introduce the following space:

S := { η = ( η 1 η 2 ) | η j ∈ C ∞ c (Σ) } .
Our interest in this set comes from its dense embedding into H . Indeed, define

J : S → H , [Jη](λ) := ( η 1 (λ) ξ ± ( - √ λ 2 -m 2 ) , η 2 (λ) ξ ± ( √ λ 2 -m 2 ) ) for ± λ > m.
It clearly follows that JS is dense in H and that J extends to a unitary operator from L 2 (Σ; C 2 ) to H . We then set

L := J * L 0 J, (3.7)
i.e. L is simply the operator of multiplication by the variable in L 2 (R, C 2 ).

Let us finally introduce for each λ ∈ Σ the unitary 2 × 2 matrix N (λ) defined by

N (λ) = 1 √ 2 ( 1 1 -i i ) if λ < -m and N (λ) = 1 √ 2
( -i i 1 1 ) if λ > m.
The operator of multiplication by the function N defines a unitary operator in L 2 (Σ; C 2 ), and it will be denoted by the same symbol N .

Remark 3.1. Let us stress that the precise form of this unitary transformation N is not really relevant here. Indeed, this transformation highly depends on our choice for the functions ξ ± in the spectral representation of H 0 . In fact, only the product N J * U really matters, as it can be inferred from formula (3.10) (recall that F 0 = UF).

Lemma 3.2. For any η ∈ S and λ ∈ Σ one has

lim ε↘0 √ πB(λ) -1 Γ 2 F * 0 δ ε (L 0 -λ)Jη = N (λ) η(λ).
Proof. For shortness, let us set p m := √ p 2 + m 2 . Then, a simple computation gives

Γ 2 F * 0 δ ε (L 0 -λ)Jη = 1 √ 2π ∫ 0 -∞ 4 √ p 2 p 2 + m 2 [ δ ε (-p m -λ)η 1 (-p m )ξ + (p) + δ ε (p m -λ)η 1 (p m )ξ -(p) ] dp + 1 √ 2π ∫ ∞ 0 4 √ p 2 p 2 + m 2 [ δ ε (-p m -λ)η 2 (-p m )ξ + (p) + δ ε (p m -λ) η 2 (p m )ξ -(p) ] dp = 1 √ 2π ∫ -m -∞ 4 √ µ 2 µ 2 -m 2 δ ε (µ -λ) [ η 1 (µ)ξ + ( - √ µ 2 -m 2 ) + η 2 (µ)ξ + ( √ µ 2 -m 2 ) ] dµ + 1 √ 2π ∫ ∞ m 4 √ µ 2 µ 2 -m 2 δ ε (µ -λ) [ η 1 (µ)ξ -( - √ µ 2 -m 2 ) + η 2 (µ)ξ -( √ µ 2 -m 2 ) ] dµ.
By taking the limit as ε ↘ 0 one obtains for ±λ > m

lim ε↘0 Γ 2 F * 0 δ ε (L 0 -λ)Jη = 1 √ 2π 4 √ λ 2 λ 2 -m 2 [ η 1 (λ)ξ ∓ ( - √ λ 2 -m 2 ) + η 2 (λ)ξ ∓ ( √ λ 2 -m 2 ) ] .
Finally, by using the expressions (3.4) for ξ ± and by considering separately the case ±λ > m, a short computation leads directly to the statement.

In addition, one can also show: Lemma 3.3. For any η ∈ S and λ ∈ Σ one has

B(λ) -1 Γ 2 F * 0 (L 0 -λ + iε) -1 Jη = 1 √ π ∫ Σ B(λ) -1 (µ -λ + iε) -1 B(µ) N (µ) η(µ) dµ.
Proof. By a computation similar to the previous proof, one obtains that

Γ 2 F * 0 (L 0 -λ + iε) -1 Jη = 1 √ 2π ∫ -m -∞ 4 √ µ 2 µ 2 -m 2 (µ -λ + iε) -1 [ η 1 (µ)ξ + ( - √ µ 2 -m 2 ) + η 2 (µ)ξ + ( √ µ 2 -m 2 ) ] dµ + 1 √ 2π ∫ ∞ m 4 √ µ 2 µ 2 -m 2 (µ -λ + iε) -1 [ η 1 (µ)ξ -( - √ µ 2 -m 2 ) + η 2 (µ)ξ -( √ µ 2 -m 2 ) ] dµ = 1 √ 2π ∫ -m -∞ (µ -λ + iε) -1 1 √ 2 diag ( 4 √ µ -m µ + m , 4 √ µ + m µ -m ) ( 1 1 -i i ) ( η 1 (µ) η 2 (µ) ) dµ + 1 √ 2π ∫ ∞ m (µ -λ + iε) -1 1 √ 2 diag ( 4 √ µ -m µ + m , 4 √ µ + m µ -m ) ( -i i 1 1
) ( η 1 (µ) η 2 (µ)

) dµ = 1 √ π ∫ Σ (µ -λ + iε) -1 B(µ) N (µ) η(µ) dµ
which leads directly to the expected result.

For ε > 0, let us finally define the integral operator Θ ε on S which kernel is

Θ ε (λ, µ) := i 2π B(λ) -1 (µ -λ + iε) -1 B(µ).
A straightforward computation leads then to the following equality for any η ∈ S and λ ∈ Σ:

[ lim ε↘0 Θ ε η ] (λ) ≡ [Θ 0 η](λ) = i 2π B(λ) -1 P.v. ∫ Σ 1 µ -λ B(µ) η(µ) dµ + 1 2 η(λ).
Starting from (3.6) and by taking Lemmas 3.2 and 3.3 into account, one can already guess that the singular integral operator Θ 0 is going to play a central role in the expression for the wave operator. However, let us observe that the maps λ → B(λ) and λ → B(λ) -1 are not bounded as λ → ±m. Therefore, it is not very easy to deal with the above kernel. For that reason, our last task is to get a better understanding of this integral operator by looking at it in another unitarily equivalent representation.

The upside-down representation

Let us finally define the unitary operator V :

L 2 (Σ; C 2 ) → L 2 (R; C 2 ) given for η ∈ L 2 (Σ; C 2 ) and x ∈ R by [ Vη ] (x) := √ 2m e x/2 e x -1 η ( m e x + 1 e x -1
)

The special feature of this representation is that the values ±m are sent to ±∞ while any neighbourhood of the points ±∞ is then located near the point 0. The adjoint of the operator V is provided for ζ ∈ L 2 (R; C 2 ) and λ ∈ Σ by the expression

[ V * ζ ] (λ) = √ 2m √ λ + m λ -m 1 λ + m ζ ( ln [ λ + m λ -m ])
.

We shall now compute the kernel of the operator VΘ 0 V * , and observe that this new kernel has a very simple form.

For that purpose, let us use the standard notation X for the self-adjoint operator on L 2 (R) of multiplication by the variable, and by D the self-adjoint operator on the same space corresponding to the formal expression -i d dx . For a measurable (matrix-valued) function K : R → M 2 (C) we denote by K(X) the operator of pointwise multiplication by the matrix K(•) in L 2 (R; C 2 ), and by K(D) we denote the operator F * K(X)F, where F is the Fourier transformation in L 2 (R; C 2 ).

One checks, by a direct substitution, that for any measurable function ρ : Σ → M 2 (C) one has

Vρ(L)V * = ρ ( m e X + 1 e X -1
) .

(3.8)

In particular, such a relation holds for ρ = T CD 0 . Furthermore, for any

ζ = (ζ 1 , ζ 2 ) ∈ C ∞ c (R \ {0}; C 2 ) and x ∈ R * \ {0}, it can be obtained straightforwardly that [VΘ 0 V * ζ](x) = i 8π P.v. ∫ R ( -1 sinh((y-x)/4) + 1 cosh((y-x)/4) 0 0 -1 sinh((y-x)/4) + -1 cosh((y-x)/4) ) ζ(y) dy + 1 2 ζ(x) = i 8π ( g + ⋆ ζ 1 g -⋆ ζ 2 ) (x) + 1 2 ζ(x), where g ± (x) = 1 sinh(x/4) ± 1 cosh(x/4) ,
and ⋆ means the (distributional) convolution product. Using then the identity

g ⋆ f = [Fg](D)f
and the explicit expressions for the Fourier images of g ± from [13, Table 20.1] we obtain VΘ 0 V * = R(D) * with R(•) defined for all x ∈ R by

R(x) := 1 2 [ ( tanh(2πx) -i cosh(2πx) -1 0 0 tanh(2πx) + i cosh(2πx) -1 ) + 1 ] , (3.9)
We are now ready to prove the existence of the wave operator and a new representation for it.

Proposition 3.4. The wave operator W CD exists and is equal to the operator W - ( H CD , H 0 ) defined in (3.1). In addition, the following equality holds in L 2 (R; C 2 ):

V N J * F 0 (W CD --1) F * 0 J N * V * = R(D) T CD 0 ( m e X + 1 e X - 1
) .

(3.10)

Proof. Starting from the equality (3.6) and by taking Lemmas 3.2 and 3.3 into account, one easily deduces that the following equalities hold for any η, η ′ in the dense subset S of L 2 (Σ; C 2 ):

⟨F 0 (W CD --1)F * 0 Jη, Jη ′ ⟩ H = 1 2 ∫ Σ lim ε↘0 ⟨ T CD ε (λ)B(λ) -1 Γ 2 F * 0 δ ε (L 0 -λ)Jη, iB(λ) -1 Γ 2 F * 0 (L 0 -λ + iε) -1 Jη ′ ⟩ C 2 dλ = ∫ Σ ⟨ T CD 0 (λ)N (λ)η(λ), [Θ 0 N η ′ ](λ) ⟩ C 2 dλ.
Now, the existence of the limit (inside the integral sign) for any λ ∈ Σ and any η, η It only remains to link the multiplication operator T CD 0 (L) in L 2 (Σ; C 2 ) with the scattering operator S CD := S(H CD , H 0 ) introduced in (3.2). For that purpose, recall that since the operator S CD commutes with H 0 , the operator F 0 S CD F * 0 commutes with L 0 and, therefore, corresponds to an operator of multiplication in H . For that reason, one usually writes F 0 S CD F * 0 = S CD (L 0 ), where S CD (λ) is a unitary operator in H (λ) for almost every λ ∈ Σ which is called the scattering matrix at energy λ. Lemma 3.5. For almost every λ ∈ Σ, the following equality holds:

S CD (λ) = 1 + N (λ) * T CD 0 (λ) N (λ). ( 3

.11)

Proof. We proceed by using the intertwining relation and the invariance principe. It is well known that if α : Σ → R is smooth and has a positive derivative, then the scattering operator S CD is the strong limit of the operators e itα(H 0 ) W CD -e -itα(H 0 ) as t → ∞, see for example . Clearly, this function is smooth on Σ with a positive derivative, which gives

s -lim t→∞ e itα(H 0 ) ( W CD --1 ) e -itα(H 0 ) = S CD -1.
We also observe that due to (3.8) we have

e itα(L) V * = V * e -itX . (3.12)
Now, by using successively relations (3.5), (3.10), (3.8) (3.7) and the commutativity of e -iα(L) with N and T CD 0 (L), (3.12), and the usual relation between the operators X and D, one infers that

S CD (L 0 ) -1 =F 0 (S CD -1)F * 0 =s -lim t→∞ F 0 e itα(H 0 ) ( W CD --1 ) e -itα(H 0 ) F * 0 =s -lim t→∞ e itα(L 0 ) F 0 ( W CD --1 ) F * 0 e -itα(L 0 ) =s -lim t→∞ e itα(L 0 ) J N * V * R(D) T CD 0 ( m e X + 1 e X -1 ) V N J * e -itα(L 0 )
=slim Once in the suitable C * -algebraic framework, the proof of this statement is quite standard. We refer to [START_REF] Kellendonk | On the structure of the wave operators in one dimensional potential scattering[END_REF][START_REF] Richard | New formulae for the wave operators for a rank one interaction[END_REF] for the construction of the framework and for the related proofs.

As a final remark, let us comment of the contribution of each term Γ CD j for j ∈ {1, 2, 3, 4} in the l.h.s. of (4.14). Clearly, the contribution of Γ CD 4 is trivial, while the contributions of Γ CD where relation (3.11) has been taken into account. Thus, when θ varies from -∞ to +∞, one easily observes that the contribution due to Γ CD 2 is provided by two distinct contributions, the one coming from det [ S CD (λ) ] as λ runs from m to +∞, and the one coming from det [ S CD (λ) ] as λ goes from -m to -∞. Note that the difference of relative orientation for the two contributions was already noticed in the literature, see for example [START_REF] Clemence | Low-energy scattering and Levinson's theorem for a one-dimensional Dirac equation[END_REF][START_REF] Klaus | On the Levinson theorem for Dirac operators[END_REF]. Note also that even if S CD (-∞) ̸ = S CD (+∞), the equality det c) The remaining case D = 0 is trivial.

Proposition 2 . 3 .

 23 Let (C, D) be an admissible pair. Then, one has σ ac (H CD ) = (-∞, -m] ∪ [m, +∞) and σ sc (H CD ) = ∅. Moreover, σ p (H CD ) ⊂ (-m, m), and the number of eigenvalues can be explicitly described as follows: (a) If det(D) ̸ = 0 and D -1 C := ( ℓ 11 ℓ 12 ℓ 12 ℓ 22

  while D Λ (•) does not vanish in the remaining case. However, note that there is a very explicit set of solutions of the relation sgn(ℓ 11 ) ℓ 22 , namely when ℓ 11 < 0, ℓ 11 ℓ 22 = -1 4 and ℓ 12 = 0. In addition, a simple computation shows that the two conditions ℓ 11 ℓ 22 < 0 and sgn(ℓ 11 )[det(L) -1 4 ] > √ -ℓ 11 ℓ 22 hold whenever ℓ 11 < 0, ℓ 22 > 0, and either ℓ 11 ℓ 22 ̸ = -1 4 or ℓ 12 ̸ = 0, while the two conditions ℓ 11 ℓ 22 < 0 and sgn(ℓ 11 )[det(L) -1 4 ] < √ -ℓ 11 ℓ 22 hold whenever ℓ 11 > 0 and ℓ 22 < 0. Now, if ℓ 11 = 0 then D Λ (•) vanishes once on (0, ∞) if ℓ 22 > 0 and does not vanish if ℓ 22 < 0; if ℓ 22 = 0 and ℓ 11 < 0 then D Λ (•) vanishes once while if ℓ 11 > 0 then D Λ (•) does not vanish on (0, ∞). Finally, if ℓ 11 = ℓ 22 = 0 there is no solution for D Λ (•) = 0. b) We now consider the case dim[ker(D)] = 1 and follow the construction described in [22, Sec. 3]. Let (p 1 , p 2 ) be a unit vector spanning ker(D). Let I : C → C 2 be the identification of C

  Finally, in order to obtain the statement of the lemma, let us recall a useful relation between ℓ and (C, D), namely sgn(ℓ) = sgn[tr(CD * )], see [14, Sec. 3.B]. c) If D = 0, then H CD = H 0 and hence H CD has no eigenvalue.

  [START_REF] Yafaev | Mathematical scattering theory. General theory[END_REF] Sec. 2.6]. Let us consider the function α : Σ → R defined by α(λ) := ln ( λ-m λ+m )

Theorem 4 . 1 .

 41 t→∞ e itα(L 0 ) J N * V * R(D) V T CD 0 (L) N J * e -itα(L 0 ) =slim t→∞ J N * e itα(L) V * R(D) V e -itα(L) T CD 0 (L) N J * =slim t→∞ J N * V * e -itX R(D) e itX V T CD 0 (L) N J * =slim t→∞ J N * V * R(D + t) V T CD 0 (L) N J * .Finally, since slim t→∞ R(D + t) = 1, it directly follows from the relation (3.7) between L andL 0 that S CD (L 0 ) -1 = J N * T CD 0 (L) N J * = N * (L 0 ) T CD 0 (L 0 ) N (L 0). The statement is then a consequence of the pointwise identification of these two multiplication operators.with orientation of chosen clockwise. Here, T denotes the set of complex numbers of modulus 1, and det denotes the usual determinant on M 2 (C). Then, the following topological version of Levinson's theorem holds: For any admissible pair (C, D) one has wind[Γ CD ] = -# σ p (H CD ).(4.14)

1 and Γ CD 3 are

 3 directly related to the threshold effects at ±m. These effects do depend on the choice of the pair (C, D). For the contribution of Γ CD 2 , let us observe that det [

)

  shown in the above computations. composition of the orthogonal projection onto ker(D) ⊥ together with the identification of IC with C. Then, as shown in[START_REF] Pankrashkin | Spectral and scattering theory for the Aharonov-Bohm operators[END_REF] Eq. (12)], one has (DM (z) -C) -1 D = I ( m(z)ℓ ) -1 P , where ℓ := (DI) -1 CI ∈ R and m(z) = P M (z)I ∈ C. One easily checks that m(λ+iε) = iβ(λ)+k(λ, ε) with k(λ, ε) → 0 as ε ↘ 0 locally uniformly in λ ∈ Σ, with β(λ) = P B(λ) 2 I ∈ R, and β(λ) > 0 for λ ∈ Σ. The continuity statement now follows from the continuity of the maps Σ ∋ λ → β(λ) ∈ R and Σ × [0, +∞) ∋ (λ, ε) → m(λ + iε) ∈ C. By using P * = I, we then obtain( + i0)ℓ ) -1 ( m(λ + i0) *ℓ ) -1 B(λ)I ( P B(λ) 2 I -m(λ + i0)m(λ + i0) *2iunitarity. The existence of the limits at the boundary can be checked explicitly.

  the operator H CD is self-adjoint if and only if the matrices C and D satisfy the following

	conditions:
	(i) CD * is self-adjoint, Moreover, any self-adjoint extension of H in H is equal to one of the operator H CD . For sim-(2.3) (ii) det(CC * + DD * ) ̸ = 0. plicity, a pair (C, D) of elements of M 2 (C) satisfying relations (2.3) will be called an admissible pair. Note that with this notation, the operator H 10 ≡ H

* | ker(Γ 1

  ′ ∈ S already shows the existence of the stationary wave operator W CD -, see [28, Def. 2.7.2]. In addition, its equality with W - ( H CD , H 0 ) follows from [28, Thm. 5.2.4]. Finally, relation (3.10) can be deduced by a conjugation with the unitary operators N and V.

In[START_REF] Lin | Friedel sum rule, Levinson theorem, and the Atiyah-Singer index[END_REF] the analogy between Levinson's theorem for Dirac operators and the Atiyah-Singer index theorem is mentioned, but nothing is deduced from this observation.

Remark 3.6. By taking the relation W CD + = W CD -(S CD ) * into account, an explicit expression for W CD + , similar to the one obtained in (3.10) for W CD -, could also be derived. Remark 3.7. As a consequence of Lemma 2.2 and of the explicit formula (3.9), the maps

) are continuous on the whole real line and admit limits at ±∞. This is an essential feature of these functions, and it plays an essential role in the subsequent algebraic construction.

Topological results

In this section, we briefly deduce the main corollary of the explicit formula (3.10), and refer to [START_REF] Kellendonk | Levinson's theorem and higher degree traces for Aharonov-Bohm operators[END_REF] and [START_REF] Richard | New formulae for the wave operators for a rank one interaction[END_REF] for a thorough description of the underlying algebraic framework.

Let us start by defining the following map: For x, y ∈ R one sets

where R(•) has been introduced in (3.9) and T CD 0 (•) has been computed explicitly in (2.7). It follows from Remark 3.7 that Γ CD can be continuously extended to a function on

with Γ CD (x, y) provided by (4.13). The asymptotic values of this function can then be easily computed, namely

) 2 is related to the scattering operator through the relation (3.11). Note also that the precise value of T CD 0 (±m) ∈ M 2 (C) could be explicitly computed in terms of C and D, but that this is not our concern here (a similar computation has been performed for example in [START_REF] Pankrashkin | Spectral and scattering theory for the Aharonov-Bohm operators[END_REF]Prop. 14] for the Aharonov-Bohm operator).

Let us now observe that the boundary of consists in the union of four parts . Note that the property Γ CD (θ) ∈ U (2) for any θ ∈ can be checked explicitly, but also directly follows from the unitarity of the image of the wave operators in the Calkin algebra.

One of the main result of the C * -algebraic framework which has been developed for example in [START_REF] Kellendonk | Levinson's theorem and higher degree traces for Aharonov-Bohm operators[END_REF] and [START_REF] Richard | New formulae for the wave operators for a rank one interaction[END_REF] is to relate the function Γ CD to the number of bound states of H CD . More precisely, let us define the winding number wind[Γ CD ] of the map

Appendix

Proof of Lemma 2.2. We first observe that for any λ ∈ Σ we can write

with K(λ, ε) → 0 as ε ↘ 0 locally uniformly in λ ∈ Σ. The scheme of the following argument is similar to the one already used in the proof of Proposition 2.

In addition, we get after some elementary algebra that

which shows the unitarity of 1 + T CD 0 (λ). The existence of the limits can be checked directly. In particular, the equality T CD 0 (-∞) = T CD 0 (+∞) follows from M (-∞ + i0) = M (+∞ + i0) = i 2 . b) We now consider the case dim[ker(D)] = 1 and proceed as in [START_REF] Pankrashkin | Spectral and scattering theory for the Aharonov-Bohm operators[END_REF]Sec. 3]. Let I : C → C 2 be the identification of C with ker(D) ⊥ , and let P denote its adjoint, i.e. P : C 2 → C is the


B O Flower

Marin-Lamellet Jean-Louis
email: louis.marin-lamellet@univ-lyon2.fr

Gender and imagined purity of at the turn of the 20th century: the example of

Keywords: Benjamin O. Flower, feminism, purity crusade, spirituality, anti-Catholicism Benjamin O. Flower, féminisme, croisade pour la pureté, spiritualité, anticatholicisme

Gender and imagined purity of at the turn of the 20th century: the example of B.O. Flower, reformer Boston progressive editor Benjamin O. Flower (1858-1918) pushed for a wide range of women-friendly reforms and publicized many turn-of-the-century feminists. He saw "female purity" as the engine of progress -the moral purification women inspired was the backbone of a radical regeneration of the country that would lead to individual, social, economic, political and family transformations. This article purposes to explore this social imagination of purity, its scope and its evolution. It examines Flower's role in the "Purity Movement," a crusade that aimed at putting an end to prostitution -within and without marriage -and to low age of consent laws. For Flower, male immorality, urban poverty, and ignorance about sex accounted for women's degradation. Realistic literature and journalistic exposures were therefore the necessary tools to regenerate society. This article also analyzes the ambiguous trajectory of Flower's vision of "female purity." He saw in Mary Baker Eddy, the founder of Christian Science, the embodiment and the logical outcome of his feminist struggles. However, this new movement only led to social apathy. His exhortations were also predicated upon essentialization (men as beasts or pure intellect and reforming energy, women as victims or as models of moral idealism) and the traditional postulates of Victorian culture (the centrality of home). Finally, in his last years, Flower embarked on a crusade against the "menace" Catholicism posed to women. This episode reveals the nativist potential of imagined purity as Flower tolerated nothing short of an ethereal "feminine" idealism.

Résumé

Ambiguïtés de l'imaginaire de la pureté au tournant du XXème siècle: l'exemple d'un réformateur, B.O. Flower Au tournant du XXe siècle, Benjamin O. Flower (1858-1918), journaliste progressiste de Boston, se vantait de diriger la revue la plus ouverte aux luttes féministes et aux réformatrices de son époque. Il considérait la « pureté féminine » comme le moteur du progrès : la purification morale inspirée par les femmes, clé de voûte de la régénération du pays, conduirait à des transformations individuelles, sociales, économiques, politiques et familiales. Cet article se propose d'explorer cet imaginaire social de la pureté, son champ d'application et son évolution. Il étudie le rôle de Flower dans la « croisade pour la pureté » qui cherchait à mettre un terme à la prostitution, dans la rue ou au sein du mariage, et à relever l'âge de la majorité sexuelle. Pour Flower, la dégradation des femmes s'expliquait par l'immoralité masculine, la misère urbaine et la méconnaissance des questions sexuelles ; il voyait dans le réalisme en littérature et le journalisme d'investigation les outils indispensables pour régénérer la société. La trajectoire de sa vision de la « pureté féminine » permet également d'en analyser les ambiguïtés. Flower voyait dans Mary Baker Eddy, la fondatrice de la Science Chrétienne, l'incarnation et l'aboutissement de ses combats féministes. Cependant, ce nouveau mouvement ne mena qu'à l'apathie sociale. En outre, ses exhortations se fondaient sur l'essentialisation des hommes (bestialité incarnée ou pur intellect et puissance réformatrice) et des femmes (victimes ou modèles d'idéalisme moral), ainsi que sur les prémisses traditionnelles du la culture victorienne, notamment la centralité du foyer. Enfin, les dernières années de Flower, parti en croisade contre la « menace » que le catholicisme posait aux femmes, révèlent les potentialités nativistes de cet imaginaire de la pureté ne tolérant rien en-deçà d'un idéalisme « féminin » éthéré.

Outline

The "era of woman" "The constantly broadening sphere of woman's influence" Flower's "threefold crusade" Purity and "evolutionary republicanism" Flower (1858-1918) promoted his magazine, The Arena, as "more hospitable to women than any other a great review published in the civilized world" (cf. figures 1 and 2). 1 Dozens of feminists contributed to this monthly -Elizabeth Cady Stanton; WCTU president Frances Willard; novelist and woman suffrage champion Helen Gardener; home economics advocate and urban poverty specialist Helen Campbell to name but a few examples. 2 In the 1890s, female reformers wrote one-fourth of all articles in the magazine. [START_REF] Nations | Benjamin Orange Flower[END_REF] Women were central to reform efforts as they were both the victims of social ills and the agents of regeneration. Flower and the feminists he publicized shared the Victorian belief in inherent female purity: women were "pure as the glistening snow-clad peaks in the midst of the moral degradation which taints manhood." They saw "female purity" as a moral standard and the engine of political action and social progress. They embodied an "urban-moral control impulse" that would tame the industrial city. 4 This article purposes to explore this social imagination of purity, its scope and its evolution.

Figure 1. Benjamin Flower at the Purity Congress in Baltimore in 1895 1 The quote comes from an advertisement for The Arena on the back end-paper of Fashion's Slaves, a pamphlet that reproduced Flower's series of articles on dress reform. Flower, Benjamin Orange, Fashion's Slaves, Boston, The Arena Publishing Company, 1892, unnumbered page. 2 Gardener also co-edited The Arena for a few months [START_REF] Nations | Benjamin Orange Flower[END_REF] The "era of woman" "The constantly broadening sphere of woman's influence"

The US, for Flower, was entering "the era of woman": the "constantly broadening sphere of woman's influence" showed that society was outgrowing "barbarous conception[s], born of a cowardly and brutal childhood age," namely the Edenic legend of female inferiority and Saint Paul's "theory of woman's sphere." 5 Flower for instance praised Mrs. French-Sheldon who, like the famous Henry Stanley, explored Africa, climbed the Kilimanjaro and lived alone with natives 6 (cf. figure 3). He took pains to put men and women on an equal footing -elevating civilization depended on "arousing earnest men and women to action" and on "ennobling manhood and womanhood." 7 Even though Flower did not relegate women to the private sphere, he did not completely challenge the separate spheres theory, as he held that women would carry over the domestic virtue of purity into society at large. He underlined that empowering women would not "unsex" them-reformers like Willard or Campbell were examples of "noble, fine, true womanhood."8 Men and women were inherently different but complementary -each would, in their own way, lead the country on the path of progress. Like some turn-of-the century reformers who believed segregation could be progressive, Flower and many feminists he promoted thought men and women could be "separate but equal." Moral uplift implied the enlistment of both male and female activists in order to elevate the country as a whole.9 Figure 3. Mrs. French-Sheldon, the "intrepid woman" who explored East Africa Flower's "threefold crusade" Flower publicized the urgent need to embark on "a threefold crusade" for moral reform. First, reformers ought to push for "a childhood resulted from an awakened conscience, the fruit of intelligence and love"in other words for a protective, caring and nurturing home, which implied marriages of love, the necessary condition for proper child-rearing. Then, he pressed for "absolute justice for woman," focusing on women's right to vote, to own independent property and to control their bodies, notably through dress reform. Finally, regeneration meant fighting "for a purer, simpler, and less sensuous and extravagant life, with a determined warfare on those things which stimulate passion and lower the moral ideal, chief among which are intoxicants and opium" -yet another example of the interplay between temperance and feminism at the time. 10 Flower, Stanton and Willard were fighting for a broad radical agenda and did not focus -like their counterparts at the beginning of the 20 th century -on one issue, usually suffrage or temperance. [START_REF]For instance, Willard asserted that "Our policy is 'The Do-everything-policy, and do it all the time[END_REF] They considered moral regeneration to be the mainstay of reforms -it would radiate out and lead to individual, social, economic, political and family transformations. Most of the reform organizations Flower publicized were big-tent women's organizations (working girls' clubs, the WCTU, the White Ribbon). 12 He modeled his own Arena clubs on women's clubs. This Sunday afternoon program is an example of the wide range of reforms Flower's Arena clubs supported -these clubs were optimistic, morally tinged, eclectic; they were anchored both in contemporary economic and political debates as well as concrete reform work (cf. figure 4). [START_REF] Flower | For more details about Arena clubs and Flower's "association campaign[END_REF] Male and female crusaders who took up the cause of women were in the vanguard of civilization. Moral reform was rooted in an evolutionary perspective that saw female pioneers as the future of the "race," what historian Beryl Satter calls "evolutionary republicanism," i.e. republicanism blending with a "reform Darwinist perspective." Women's "exaltation mean[t] the elevation of the race" and their "splendid spirit of altruism" would solve "fundamental problems"-female virtue and spirituality would steer society towards a higher moral ground and put an end to materialism and male desire that only led to Social Darwinists' acceptance of laissez faire and the manly, brutish laws of the jungle. [START_REF] Satter | Each Mind a Kingdom: American Women, Sexual Purity, and the New Thought Movement, 1875-1920[END_REF] Flower took the jungle metaphor literally -in his racialized and gendered rhetoric, Flower associated male immorality with primitivism and barbarism. Men, when they indulged in their most bestial instincts, were no longer real men -"manhood is eclipsed" as Flower put it -and that endangered womanhood and country as a whole. Flower used the example of the Carmencita ball at Madison Square garden in 1891 to illustrate his point.

After wealthy capitalists took their wives home, they came back to the ball and rushed into debaucheryif the actual details remain euphemized, the savagery and sinfulness of the scene were conveyed by Flower's racially charged comparison of men ominously jumping up and down like the "wild Sioux in a ghost dance." [START_REF] Flower | Civilization's Inferno[END_REF] On the contrary, civilized men "had evolved the capacity to suppress their unmanly passions." Virility implied rationality, self-control and republican virtue. As Mrs. French-Sheldon had done in Africa, real men could conquer savagery, here the "Dark Continent" within themselves. 16 This progressive vision was understood within an apocalyptic framework. Flower naturalized purity, claiming that, like oxygen, it was literally vital for the country: the "fate of home, nation and civilization" was predicated upon the protection of women and their purifying influence on society. Flower drew an explicit parallel with the fate of Greece and Rome -ancient civilizations died when they gave up on their republican virtues and pandered to men's baser instincts. 17 Spencerian arguments combined with nationalistic feelings -the US had to give women their proper place in order to keep its status as an exceptional nation and remain at the top of the evolution scale. The "Purity Movement" in particular was supposed to enable a laggard America to catch up with the onward march of history.

Flower and the "Purity Movement"

"Prostitution within the marriage bond" Moral reform implied the protection of women and an attack on the "well-springs of immorality." 18 In a paper prepared for the Social Purity Congress at the Chicago World's Fair and in another American Purity Alliance conference in Baltimore in 1895, Flower expounded the causes of immorality: prostitution -within and without marriage -, low age of consent laws, economic inequalities, ignorance about sex and alcoholism 19 (cf. figure 5). His reform agitation aimed at putting an end to "the conventionality that tolerates immorality in men." [START_REF] Flower | Is This You Son[END_REF] He did not believe in the inherent depravity of "fallen women." He shifted the blame instead on male lust and exploitation, opposing it to women's spiritual superiority. Women were described as asexual beings, while men's deviant sexuality was "rampant" and "hard to control." [START_REF] Lindley | Have Stept Out of Your Place: A History of Women and Religion in America[END_REF] Flower kept repeating that "the ethical standard for man must be raised, or the degradation of woman will follow." This symbiotic relationship between male and female morality demanded equality between and women in order to advance "the race." His moral standard was nothing less than "absolute purity" and "the triumph of love over lust, the moral over the animal, the soul over the body." 22 For Flower, the bond between a man and a woman was sacred and foreordained by God. As a result, loveless marriages were not real and divorce, he claimed, "more perfectly secure [d] the divine than a continuation of the compact." A "bitter medicine: divorce" could lead to "enhanced purity." Flower saw the idea of indissoluble marriage as pernicious because it forced married women into sexual intercourse and compelled them to have children with husbands who were "addicted to drink or the victims of revolting diseases" -a corruption of the holy marriage compact that he condemned as the "unholiest prostitution." Flower saw in Tolstoy's Kreutzer Sonata the perfect illustration of this problem. [START_REF] Flower | The Postmaster-General and the Censorship of Morals[END_REF] Male "gross sensualism" led to "prostitution within marriage" and "crimes against the unborn" because fathers' lust would "poison" their offspring, even if their wives were "pure." As a result, children would be predisposed towards immorality, poverty, and criminality -a "curse to the race." [START_REF] Flower | Leprosy of the Soul[END_REF] Discourses on purity: science and radical traditions Flower kept underlining the "mysterious influence of the little understood law of heredity" and the "measureless influence" of home, "the fountain" of morality. In his rhetoric, the scientific discourses of heredity and environmental determinism prevalent at the time blended with fringe beliefs like the importance of pre-natal influences -children were affected by their mothers' "mental impressions." Flower followed the advice of Dr. Sydney B. Elliot: mothers should elevate their fetuses by exposing them to Mozart or Robert Burns for instance. [START_REF] Flower | Home Influence and The Child[END_REF] Using the same kind of printing metaphor he used to explain the role of photography in his reform work (illustrations were used "to impress terrible facts vaguely believed but not realized by the great majority of our thoughtful people"; italics mine), mothers had to impress the plastic mind of children with uplifting images -their very womb therefore became, with playgrounds or model tenements, a key locus of environmental meliorism. 26 Flower's moral reading of medical and scientific discourses prevalent then carried on abolitionist ideals into the home -like slaveholders usurped the place of God when they dominated slaves, husbands usurped the place of God when they dominated their wives. Self-sovereignty was paramount in both contexts. Purity crusaders opposed the regulation of prostitution and called for its abolition; many were, for that matter, former abolitionists. Flower's conception of marriage was also part of the "free love" tradition, which, at the time, meant the sanctity of spiritual bonds between individuals: no religious and civil authorities "[had] in their power to determine matters related to the heart." [START_REF] Morita | Unseen (and unappreciated) matters: Understanding the reformative nature of 19th-century Spiritualism[END_REF] "Free love" was radical because it challenged Calvinist ideas of marriage as an indissoluble contract and favored divorce. Its advocates were also part of radical circles that included former abolitionists, freethinkers and spiritualists. Ironically, the romantic love fantasy they promoted has become mainstream today.

The exaltation of purity: romantic love and art "Prostitution within marriage" was a symptom of an immoral society that tolerated archaic inequalities between men and women. The solutions for Flower were first "to exalt marriage," in other words to elevate pure, romantic love as a model for family and interpersonal relationships, and secondly to push for the moral right of women to own independent property and to decide what to do with their bodies and their sexuality:

If wives were given absolute control of their bodies, and the right to say when they would become mothers, if at the altar they became possessed of one half the property interests of their husbands, it would be infinitely better for humanity, and the servitude and dependence which now compel numbers of women to become slaves to their husbands' passion, would disappear; while husbands who now take advantage of the privileges accorded by the wife's dependent condition would come to treat their companions in deed and fact as equals. [START_REF] Flower | Well-Springs and Feeders of Immorality[END_REF] Sexual and financial independence would emancipate women and lead to gender equality.

Art had a key role in the exaltation of purity since it could "impress" uplifting images on the mind of people. Flower in particular promoted Helen Gardener's realistic novels. [START_REF] Gardener | Is this your son, my lord? a novel[END_REF] Is this your son, my lord? "boldly unmask[ed] masculine immorality" and called for the same moral standards for men and women. Gardener asserted she wanted to do for the "fallen woman" what Harriet Beecher Stowe had done for slaves. The novel tells the story of the ruin of Minnie Kent, a 15-year-old innocent working class girl, by two outwardly respectable men -Mansfield, a rich and pious mill owner and the owner of the department store where Minnie works, and his son Preston. [START_REF] Odem | Delinquent Daughters: Protecting and Policing Adolescent Female Sexuality in the United States, 1885-1920[END_REF] The crusade against white slavery: prostitution at work and urban poverty Gardener's novel was part and parcel of Flower's campaign in favor of "the white slaves of New York." Framing his crusade in abolitionist terms, he considered the victims of prostitution at work -the many Minnie Kents that peopled American urban workplaces -as new slaves. Narratives of seduction were a commonplace of reform literature then: "men of status and wealth took advantage of poor, innocent young women," usually white working-class girls lost in the city. In the end, they fell into prostitution and met a disastrous end. [START_REF]This narrative of seduction had long been popular in nineteenth-century melodrama and romance novels, but reformers adapted it to their own social context and political purposes[END_REF] Because they did not vote, voiceless women had to suffer the burden of the feminine condition in the workplace. Women had to accept the dreadfully low wages that often led them to "starve or sin." Flower particularly admired the powerful "feminine character" that resisted "seductive temptation." His campaigns prompted other reformers to carry on the struggle. Rev. Louis Banks wrote his own study of "white slavery" after reading Flower's articles. 32 Like Jacob Riis in New York, Flower studied poverty in Boston and used the power of photography to document the life of the underprivileged in a book entitled Civilization's Inferno, or, Studies in the Social Cellar. 33 His study -part sociological survey based on fieldwork and statistical analysis, part mise en scène of poverty and a modern rewriting of Dante's circles of hell to shock readers into action -revealed the gendered nature of urban poverty at the time. His description of the case of a widow and two children working in an underground tenement was, for him, "typical" of problems at the time (cf. figure 6). 34 Flower, like many middle-class reformers, was shocked by the discovery of women's poverty and the dangers it posed to their purity. For Flower, the written word failed to describe the "social cellar" -only "a few glimpses of the horrors of the abyss" could be caught. 35 The word "abyss" and the darkness metaphor evoke all the unspeakable dark mysteries of the industrial city and the seemingly bottomless immorality it produced, the metaphor encapsulating in one vague, ominous term the various moral, religious, physical, racial, gender and imperial aspects of a complex phenomenon: the inexplicable squalor in a country of abundance, female degradation, male immorality, crime, and the explicit analogy between investigators' "studies in the social cellar," and the exploration, in the wake of Stanley, of the "Dark Continent" and its savagery. 36 Figure 6. An example of "typical" poverty of the time: "widow and two children in under-ground tenement"

Helen Campbell also discussed the "social question" in terms that revealed the interlocking of economic and feminist issues. She published a six-part series of papers on "women wage-earners" in The Arena where she described vignettes of female poverty in New York. She concluded that women could only 33 He published a series of illustrated articles in The Arena between February 1891 and March 1893. He then reprinted the articles in a book with the telltale title: Civilization's Inferno, or, Studies in the Social Cellar. 34 Flower, "Society's Exiles," p. 44. 35 improve their working conditions by organizing workers' associations and Consumers' Unions. 37 Even though Flower wanted women to directly take part in public debates and in social work, his analysis nonetheless testifies to a gendered reading of economic and social problems. Female reformers were seen as maternal figures -Campbell "possessed the mother heart." 38 It was logical for Flower: reform work was the extension, in the public sphere, of the alleged selfless, pure love and noble virtues mothers exhibited within the home. Poor women were supposed to be vulnerable mothers and were accordingly the main recipients of the relief campaign organized by The Arena. 39 They would give birth to future citizens, that's why the degrading factory and tenement environments were especially a cause of concern.

Protecting women meant protecting the future; women, in other words, were progress embodied.

The campaign to raise the age of consent "Has law ever touched a lower depth of degradation?" -such was the indignant rhetorical question that started Flower's crusade to raise age of consent laws. In 1891, in 36 US states, the age of consent was under 15, in 10 states it was 10 and in Delaware it was 7 years old. This shameful, barbarous legislation was enacted by "fathers, husbands, and brothers for the furtherance of animal lust," hence the necessity of suffrage to moralize society. He stressed that all reforms were interdependent: in states where women had the right to vote, like Wyoming and Kansas, age of consent had risen from 14 to 18. 41 Flower popularized the movement on a national scale and followed the model of English editor William T. Stead: he had used his newspaper, the Pall Mall Gazette, to tear "the mask of hypocrisy" and forced politicians to pass a sterner law. For Flower, if the people knew, they would not tolerate "social evils"; as a result, exposing problems ought to lead naturally to reforms. Only underhand scheming by corrupt men could stop progress, like the "secret sessions" to discuss age of consent legislation that were so "unfit and demoralizing" for girls. 42 37 Campbell, Helen, "Association in Clubs with Its Bearing on Working Women," vol. 5, December 1891, p. 61-6; "Women wage-earners; their past, their present, and their future. First paper," vol. 7, January 1893, p. 153-172; "Women wage-earners; their past, their present, and their future. II. Rise and growth of trades up to the present time," vol. 7, February 1893, p. 321-336; "Women wage-earners; their past, their present, and their future. III. Present wage rates in the United States," vol. 7, March 1893, p. 433-440; "Women wage-earners; their past, their present, and their future. IV. General conditions for English and continental workers," vol. 7, May 1893, p. 668-679; "Women wage-earners; their past, their present, and their future. V. General Conditions in the Western States," vol. 8, June 1893, p. 32-38; "Women wage-earners; their past, their present, and their future. VI. Specific evils and abuses in factory life and in general trades," vol. 8, July 1893, p. 172-184. 38 Flower, Progressive Men, p. 124-125. 39 Flower, "Our Poor Fund," vol. 6, July 1892, p. lvi. 40 Flower, "The Age of Consent," vol. 3, February 1891, p. 381-382. 41 Flower, Well-Springs and Feeders of Immorality," vol. 11, January 1895, p. 192-215. 42 Flower, "More Facts and What They reveal," vol. 3, February 1891, p. 382-384. "Demoralizing" should be understood in its archaic sense of corrupting morals as well as shattering hope. Like journalism, art had to play a role in the crusade. Flower published Gardener's Pray You, Sir, Whose Daughter? and considered it the Uncle Tom's Cabin for the "white ribbon army," i.e. the purity crusade (cf. figure 7). A department store manager takes Ettie Berton, his employee, to Coney Island; he seduces her but since she is over the age consent he is blameless while she gets fired: she is then ostracized and dies alone. Her moral decline is contrasted with another story: Gertrude Foster, a college-educated daughter from a respectable family, saves another shop girl, Frances King, from the same fate. She embodies the work of "friends of purity" -she influences legislators and sets up working girls' clubs, where women could find recreation and moral instruction. 43 Flower was one of these "friends of purity." In January 1895, he organized a symposium on what he called the "the shame of America" with male and female contributors. His "nine defenders of home" included American Purity Alliance leader Aaron M. Powell and Frances Willard (cf. figure 8). Flower also promoted the "Petition of the Woman's Christian Temperance Union for the Protection of Woman" and its "black list of states" where "a little girl may consent to her ruin" (cf. figure 9). 44 Agitation turned into political lobbying: Flower urged readers to flood their representatives with letters. 45 For all his talk about a radical and broad purification of society, a single-issue campaign finally did prove more effective -by the end of 1895, the age of consent was raised in six states. 46 Figure 8. "Some defenders of the home" 45 Letter from Flower to Henry Demarest Lloyd, June 12, 1895, reel 6, The Papers of Henry Demarest Lloyd (microfilm edition), State Historical Society of Wisconsin, 1970. 46 Flower did not oppose purity to the open and free discussion of sex. On the contrary, he denounced "mock modesty" that led to women's ignorance and eventually prostitution, and called for "purity through knowledge, not innocence through ignorance."47 Not only was the "ostrich method" dangerous for women because ignorance made them "easy prey of vice" but Flower also contended that debauched men and reactionary bigots were objective allies. For him, "conservatism and sensualism forged an unhallowed alliance" -fanatics like anti-obscenity activist Anthony Comstock actually "protect[ed] vice by assailing all who seek to purify life." There was nothing to fear from a true-to-life, photography-like representation of reality. 48 To moralize society meant to purify, in other words to "expose" or "boldly unmask" injustice. Flower's language blended journalism and photography with surgery in one metaphor of revelation: "they who dwell in stagnant depths of conservatism (…); who believe in allowing the social body to rot with eating, cancerous ulcers rather than uncover the loathsome sores that the surgeon's knife may be applied and the body saved." 49 Flower condemned "mock modesty which sees immorality where a pure mind beholds only beauty and grace." The author's pure intentions redeemed realistic descriptions of "morbid pruriency." [START_REF] Flower | The Postmaster-General and the Censorship of Morals[END_REF] What mattered was to reach the deeply rooted psychological cause of immorality, the image buried deep down, and then replace it with exalting images that would modify people's behavior accordingly. This kind of transparency fantasy had actually already materialized in the radical reform work of the ultimate model of purity -Jesus. Jesus assailed the very thought of sin, "the thought which prompted the deed." [START_REF] Flower | Home Influence and The Child[END_REF] Regenerating America therefore implied changing people's representations and intentions, hence Flower's constant emphasis on journalistic exposures, realistic novels and "thought-molders." He took term literally: not only did he believe in the power of mind to mold other minds but he thought mind could also influence matter. [START_REF] Flower | A Transition Period[END_REF] Accordingly, he was very much interested in scientific research and religions that could explain the mechanics of the inner self and the power of mind, for example "psychical research," hypnotism, Christian Science and New Thought. [START_REF] Satter | Each Mind a Kingdom[END_REF] Flower published a utopian novel in 1893 that developed this vision -Unveiling a Parallel: A Satirical Romance. [START_REF]The "two women" in question were two housewives from Iowa, Alice Ilgenfritz Jones and Ella Merchant. Home could therefore be the locus of challenges to the social order. 55 Flower, "Books of the Day[END_REF] The novel reversed gender roles on Mars and "unmask[ed] double standard of morals," laying bare "how horrible impurity is in either man or woman." The narrator visits two societies: Caskia and Paleveria. In Paleveria, women are reduced to their bestial passions. They are described as "spiritually dead" and bodies without a soul. In Caskia, "exalted morality" has led to gender equality and, consequently, to an egalitarian society. Gender equality is shown as intrinsically linked to spirituality. As Flower remarked, "the spiritual world is to them [Caskians] a real world." The Caskians had put into practice Flower's ideal of psychological and moral transparency -in their society, the very thought of lust was already adultery and was censured accordingly in order to protect children from inheriting a "curse of lust." 55 Marriage was pure; even sexuality was spiritualized: children were "conceived in immaculate purity." [START_REF]Two Women of the West, Unveiling a Parallel[END_REF] In the coming civilization, women would be "pure and exalted" and "the servant of the spirit." In short, Flower radicalized Victorian assumptions about purity, which euphemized female sexuality, and disembodied women.

The ambiguous trajectory of "female purity"

Purity and Christian Science At the beginning the the 20 th century, Flower saw in Mary Baker Eddy, the founder of Christian Science, the embodiment and the logical outcome of his feminist struggles (cf. figure 10). She personified his spiritual conception of feminism and the culmination of decades of progress for women. Eddy was "the greatest religious leader in the New World" and "the most famous woman in the world." Christian Science was the first American religion and the first religion to be established by a woman. Most of its devotees were women. In Christian Science, since faith could cure and matter was unreal, bodies were spiritualized, which enabled the "emancipation of the soul." The climax of Flower's discourse of imagined purity, which had always belittled bodily desires, therefore left women completely disembodied. [START_REF] Flower | [END_REF] Figure 10. Mary Baker Eddy For some of his reformist friends, Flower spent too much time defending Christian Science and not enough fighting for progressive practical reforms; for Flower, both were part of the same struggle. His friends did have a point though. With Christian Science, women's concrete progress remained subservient to religious regeneration and, after his involvement with Christian Science, Flower no longer embarked in another progressive feminist crusade -since spiritual purification was the solution to both individual and social problems, then why try to reform society structurally and push for women-friendly legislation? 58 Flower's essentialist reading of feminism For all his talk about the "era of woman" and his support for crusades to achieve "absolute justice and perfect equality," Flower never subverted the "root causes" of female inferiority, that is to say Victorian mentalities. In his discourse, the boundaries of gender stereotypes were not shifted. Firstly, his exhortations were predicated upon essentialization. Women and children -the "mothers of the race and the oncoming citizenship of the land" -were seen as vulnerable victims as well as the embodiment of natural purity; women could also be the helpless receptacles of male corruption. Women could radiate out purity -they worked in terms of influence, not of direct action. That meant they could be a force for good while remaining passive. Male desire, when it was not liberated from bestiality by spiritual purification, forbidden by the government. The subject of convents was crucial for Flower as it revealed the threat Catholicism posed to women. [START_REF] Flower | The Rupture Between the French Republic and the Vatican[END_REF] Flower perpetuated a long nativist tradition that saw convents as dens of vice for pure Protestant women. Anti-Catholic writers took up the "white slavery" trope that was prevalent then in anti-prostitution literature. Convents were seen as prison-like brothels and sinister priests, much like debauched husbands or the male bosses that took advantage of their females employees, seduced vulnerable nuns, notably at the confessional. The seduction narratives that pervaded purity literature in the 1890s were adapted to the new crusade, blending with the old captivity tale tradition and the escaped nun motif. [START_REF]Catholic immigration was in full swing and anti-Catholic hostility was rampant (the 1834[END_REF] Flower never wrote explicitly fanatical pieces like ex-priest Jeremiah Crowley's book that made the Pope the "chief of white slaves" but he nonetheless publicized it in the pages of his new newspaper, the aptly named The Menace. [START_REF] Crowley | The Pope -Chief of White Slaves, High Priest of Intrigue[END_REF] Flower's tone was more measured. However, the fact that Catholics refused to open the doors of their convents for state inspections and that they were "terror-stricken" at the idea of an "impartial investigation" cast doubts over their intentions and was suspicious to say the least. If convents were not defiling female purity, his argument went, then why object to inspections that would lift suspicions? If Flower was cautious about stories of sexual abuse, he nonetheless described women as victims of violence. He denounced the Houses of the Good Shepherd -these convents for juvenile offenders turned out to be brutal prisons for girls. [START_REF] Flower | The Patriot's Manual, dealing with the irrepressible conflict between two mutually exclusive world theories of government; a compendium of facts, historical data, reasons and present-day chronicles, showing why every friend of fundamental democracy must oppose politico-ecclesiastical Romanism in its un-American campaign to make America "dominantly Catholic[END_REF] In another parallel with his crusade against prostitution, Flower denounced Catholics who used the antiobscenity Comstock laws to stop fearless investigators like The Menace from revealing Romanist wrongdoings. In January 1916, Menace editors were prosecuted in Joplin, MO.: they were charged with publishing an indecent article about the alleged seduction of a young girl and an extract from Crowley's book which contained lewd questions prescribed for girls' confessions. [START_REF] Maury | The Wars of the Godly[END_REF] The Joplin trial outraged Flower and confirmed his anti-Catholic convictions. [START_REF]The Free Press Defense League[END_REF] Even if Flower never explicitly blamed Catholics for defiling Protestant girls, he saw in Catholic-inspired prosecutions an ominous admission of guilt and considered the Menace journalists as defenders of purity: "that journal did what decent journals must do if the purity of society is to be maintained, viz., expose what appear to be immoral and vice-breeding conditions." 70 The Menace was eventually acquitted and Flower noted that, while the jury was deliberating, "the Protestant women of Aurora, Mo., the home of the defendants, were holding prayer meetings in the churches, praying for the acquittal of the editors." 71 Flower wanted to enlist both sexes in the fight, but his description of womanhood took stereotypical forms. [START_REF]The Free Press Defense League[END_REF] Girls who entered convents were supposed to be weak and naïve. Female activists remained paradoxically passive -they prayed, radiating out their inherent purity, while men would "[bear] the brunt of the battle." Men travelled on the lecture circuit: in Flower's list of the "most widely known patriotic speakers," only 2 out of 22 were women and they were ex-nuns, i.e. victims, telling about their experiences, not intellectuals analyzing a trend (Anna Lowry's Martyr In Black and Helen Jackson's Convent Cruelties). The Menace was a masculine world where women were employed in menial jobs like stenographers or secretaries. Men were described as fighters who were part of an "army of patriots" who opposed "militant enemies" on the "firing line." 73 In short, they conformed to "gendered expectations of proper manly roles of vigilance and military mobilization in defense of women and children." 74

Conclusion

In his study of the "paranoid style" in American culture, historian Richard Hofstadter claimed that anti-Catholicism had always been "the pornography of the Puritan." [START_REF] Hofstadter | The Paranoid Style in American Politics, and other essays[END_REF] Was Flower's final crusade his own stint at pornography? Such an obsession with purity could hide repressed desires that found expression in his support for lurid anti-Catholic literature. Such an assessment of Flower's strange trajectory is selfdefeating though, as that would imply peering into the depths of his own intentions -such psychohistorical considerations would essentialize his contingent trajectory. Flower's own brand of feminism was central in his evolution because it set him against Catholicism from the beginning and because the growing popularity of "feminist" ideas like woman suffrage and temperance in the 1910s reoriented his reform energy towards anti-Catholicism. Flower was a "reluctant modernist" and a member of the "radical middle-class," in other words a "genteel radical." [START_REF]What historian Cline calls "radicalism of a genteel type[END_REF] His stance on "the woman question" reflected this ambivalence -he pushed for radical reforms, like women's right to divorce and to control their bodies for example, while clinging to Victorian gender stereotypes. 49 281 characters, spaces, footnotes and bibliography included Illustrations Figure 1. Benjamin Flower at the Purity Congress in Baltimore in 1895 Figure 2. Advertisement for symposia on "Women's Clubs" and "Woman's Dress" in The Arena Figure 3. Mrs. French-Sheldon, the "intrepid woman" who explored East Africa Figure 4. Example of Sunday afternoon program of Women's Arena Club of Philadelphia, in Hirschler, Diana, "Union of Philadelphia" The Arena, vol. 9, March 1894, p. 551 Figure 5. Flower's speech at the 1895 American Purity Alliance conference in Baltimore Figure 6. An example of "typical" poverty of the time: "widow and two children in under-ground tenement" Figure 7. Pray You Sir, Whose Daughter? book cover Figure 8. "Some defenders of the home" Figure 9. Blacklisting states with a low age of consent Figure 10. Mary Baker Eddy

Flower

  and the "Purity Movement" "Prostitution within the marriage bond" Discourses on purity: science and radical traditions The exaltation of purity: romantic love and art The crusade against white slavery: prostitution at work and urban poverty The campaign to raise the age of consent Purity, the free discussion of sex and spirituality The ambiguous trajectory of "female purity" Purity and Christian Science Flower's essentialist reading of feminism Catholicism vs. feminism Conclusion Boston reformist editor B.O.

  Payne, Alma J, "Woman militant in The Arena of Benjamin Orange Flower," in Popular literature in America; a symposium in honor of Lyon N. Richardson, ed. James C. Austin et Donald A. Koch, Bowling Green, Ohio, Bowling Green University Popular Press, 1972, p. 173.

Figure 2 .

 2 Figure 2. Advertisement for symposia on "Women's Clubs" and "Woman's Dress" in The Arena

Figure 4 .

 4 Figure 4. Example of Sunday afternoon program of Women's Arena Club of Philadelphia, in Hirschler, Diana, "Union of Philadelphia," The Arena, vol. 9, March 1894, p. 551

Figure 5 .

 5 Figure 5. Flower's speech at the 1895 American Purity Alliance conference in Baltimore

Figure 7 .

 7 Figure 7. Pray You Sir, Whose Daughter? book cover

  Flower, "Social Purity Congress at the World's Fair," p. xxi. "The Purity Conference," New York Times, October 17, 1895. Powell, The National Purity Congress. Engs, Ruth C., The Progressive Era's Health Reform Movement: A Historical Dictionary, Greenwood Publishing Group, 2003, p. 273.

Figure 9 .

 9 Figure 9. Blacklisting states with a low age of consent

Flower, "The Era of Woman," The Arena, vol. 4, August 1891, p. 382 (unless otherwise stated, magazine articles referenced in this paper are all extracted from The Arena). Boyer, Paul, Urban Masses and Moral Order in America, Cambridge, Harvard University Press, 1978, p. 153, 162.

Flower, "The Era of Woman," p. 382-383; "The Next step Forward for Women; Or, Thoughts on the Movement for Rational Dress," vol.

6, October 1892, p. 641.6 Flower, "The Next Step Forward for Women," p. 640-64; M. French-Sheldon, "The First Circumnavigation of Lake Chala," vol. 6, July 1892, p. 133-147.7 Flower, "Fronting the Future," vol. 3, December 1890, p. 125; Civilization's Inferno, or, Studies in the Social Cellar, Boston, The Arena Publishing Company, 1893, p. 3, 149. Flower devoted a whole chapter to "woman and the larger life" in his memoirs: Flower, Progressive Men, Women, and Movements of the Past Twenty-five Years, Boston, The New Arena, 1914, p. 143-150.

Flower, "Notes and Announcements," vol. 7, March 1893, p. xxvi.

Flower, "Masculine Immorality," vol. 3, February 1891, p. 381.

One can also notice here middle-class reformers' fear of the fateful influence of urban modern environments on girls, whether it be modern amusement parks or modern workplaces like department stores. Odem, Delinquent Daughters, p. 17-18.

Gardener, "II. What Shall the Age of Consent Be?" vol. 11, January 1895, p. 196-198; "A Battle for Sound Morality, or The History of Recent Age-of-Consent Legislation in the United States, Part II. The Victory in Colorado, Nebraska, and Missouri," vol. 14, September 1895, p. 1-32; "A Battle for Sound Morality. Part III," vol. 14, October 1895, p. 205-220; "A Battle for Sound Morality. Final Paper," vol. 14, September 1895, p. 401-419.

Flower, "Some Causes of Present Day Immorality," in Powell, The National Purity Congress, p. 317.

Flower, "The Postmaster-General and the Censorship of Morals," p. 547, 552; "Conservatism and Sensualism an Unhallowed Alliance," p. 127-128; "Masculine Immorality," 379; "Home Influence and The Child," p. 506.

Flower, "Is This You Son, My Lord," vol. 2, November 1890, p. 760; Flower, "The Postmaster-General and the Censorship of Morals," p. 542.