Guillermo López Quesada

Giorgos Tatsios
email: tatsios@mie.uth.gr

Marcos Rojas-Cárdenas
email: marcos.rojas@insa-toulouse.fr

Lucien Baldas
email: lucien.baldas@insa-toulouse.fr

Christine Barrot
email: christine.barrot@insa-toulouse.fr

Stéphane Colin
email: stephane.colin@insa-toulouse.fr

Dimitris Valougeorgis

Pumping effect due to temperature gradients imposed in a multistage assembly consisting of long tapered orthogonal channels

Keywords: Multistage Knudsen pump, thermal transpiration, rarefied gas flow, vacuum micropump, tapered channel, diode effect, kinetic modeling

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

EXTENDED ABSTRACT Introduction

Over the past two decades a large number of MEMS and micro-devices have been developed. These miniaturized systems, such as lab-on-chip sensors, gas chromatography analyzers, etc., require micro-pumps for air sampling through the testing stages of the device. Additionally, some microscale components such as radio frequency switches, microscopic vacuum tubes and other parts that depend on electron or ion optics, require certain vacuum environment for proper operation. Simply sealing the devices is not sufficient because leaks and outgassing are excessively detrimental in vacuum devices at the microscale level. Accordingly, such components may need vacuum pumping to maintain proper functionality. For years the thermal transpiration phenomenon has been extensively investigated, however, functional prototypes based on this phenomenon have been only recently developed. The Knudsen pump, which is one of the devices exploiting this well-known phenomenon, is able to generate a macroscopic gas flow by solely applying a tangential temperature gradient along a surface without imposing an external pressure gradient. Specific geometrical and operational configurations have been investigated to optimize the efficiency of the pump in terms of generated mass flow rate and pressure difference [START_REF] Colin | Single-phase gas flow in microchannels. Heat transfer and fluid flow in minichannels and microchannels[END_REF]. Thus, analytical and numerical solutions for different configurations have been provided [START_REF] Chen | Etude numérique et expérimentale des écoulements de gaz raréfiés générés par des gradients thermiques: application à la conception de pompes Knudsen[END_REF] and experimental measurements for thermal transpiration flow through channels have been reported [START_REF] Yamaguchi | Thermal transpiration flow through a single rectangular channel[END_REF]. However, when it comes to the particular point of fabricating a Knudsen pump, the progress in the field is limited mainly due to micro-fabrication difficulties and constraints linked to the control of local thermal gradients [START_REF] Gupta | Porous ceramics for multistage Knudsen micropumpsmodelling approach and experimental evaluation[END_REF][START_REF] Gupta | A Si-micromachines 48-stage Knudsen pump for onchip vacuum[END_REF].

In the current investigation, the effort is led towards integrating existent manufacturing and operational constraints into numerical simulations for designing a multistage Knudsen pump exploiting the diode effect. More specifically, the objective is to model a multistage cascade system consisting of a series of channels with linearly diverging or converging rectangular cross sections. Each stage of the multistage assembly consists of one converging and one diverging channel as shown in Fig. 1. The methodology introduced in the case of flow between tapered plates [START_REF] Tatsios | Computational investigation and parametrization of the pumping effect in temperature driven flows through long tapered channels[END_REF], is extended to parameterize the present assembly, i.e., to compute the deduced net mass flow rate and the associated pressure difference for various channel geometries, input pressures, the imposed temperature ratios and numbers of stages.

Numerical simulations

The analysis is based on the linearized Shakhov kinetic model subject to diffuse boundary conditions. The deduced kinetic coefficients are introduced in a mass balance equation to yield the flow rate and the pressure difference [START_REF] Sharipov | Data on internal rarefied gas flows[END_REF]. The computational investigation includes three cases: i) closed configuration with net mass flow rate equal to zero to achieve the maximum pressure difference (max ΔP) between the inlet and the outlet of the multistage assembly, ii) open configuration with overall pressure difference equal to zero to achieve maximum net mass flow rate and iii) the more general scenario of all intermediate flow cases achieving nonzero net mass flow rate and pressure difference. The latter scenario includes the development Following this approach, the influences of the inclination ratio (α=H in /H out ), aspect ratio (H m /W), length over height (L/H m ) and temperature ratio (T H /T C ) have been investigated. Additionally, the characteristic curves have been obtained for different configurations and the behavior of the multistage system has been analyzed for various operational conditions. The computational investigation has been performed taking into account existent manufacturing and operational constraints (geometry, size, materials, temperature control) provided by the Laboratoire d'Architecture et d'Analyse des Systèmes (LAAS, Toulouse, France) where the multistage device is planned to be manufactured. Typical results of the maximum pressure difference (ΔP) and net mass flow rate in terms of the number of stages at different operating pressures are presented in Fig. 2. The temperature difference between the inlet and the outlet is set to T H -T C =100 K, which can be easily achieved experimentally [START_REF] Rojas-Cárdenas | Temperature gradient induced rarefied gas flows[END_REF], while the channel dimensions (H m , α, W and L) have been provided by the LAAS. As expected, the maximum ΔP increases linearly with the number of stages of the device. It is noted however, that the maximum ΔP does not grow monotonically with the increase of the operating pressure. It can be seen that in this set-up the maximum ΔP is obtained at an operating pressure of 20 kPa. This behavior has been also observed in [START_REF] Rojas-Cárdenas | Temperature gradient induced rarefied gas flows[END_REF] and it is connected to the specific rarefaction level where the maximum performance is achieved. Regarding the maximum mass flow rate, it grows monotonically with the operating pressure. Also, the maximum mass flow rate is reduced as the number of stages is increased. It is seen however that it drops rather rapidly as the number of stages is increased from one up to ten and then, as the number of stages is further increased, it is reduced very slowly in an asymptotic manner. This is a useful observation for designing a multistage pump since by increasing the number of stages to more than ten the pressure difference grows linearly, while the mass flow rate remains almost constant.

The characteristic curves for a specific operating pressure, shown in Fig. 3, have been obtained by simulating the two limiting cases (closed configuration with maximum ΔP and open configuration with maximum net mass flow rate) and various flow configurations in between. These characteristic curves have been found out to be linear which implies that for mapping the performance of the Knudsen pump, the calculation of the two limiting cases are only required, reducing significantly the overall computational effort. Particular attention should be paid to the relation between H m and W. It is noted that the diode effect of the convergent-divergent design is effective only when H m W because the characteristic length controlling the intensity of the thermal transpiration phenomenon is the smallest dimension of the channel cross section. On the contrary, if H m W, the difference between the flows due to the varying height H in converging and diverging dimension becomes negligible.

Conclusion

In conclusion, an extensive computational study of a multistage Knudsen pump consisting of long rectangular tapered channels has been considered. The main output from this work is the characteristic curves that can be used in the design process of the Knudsen pump for determining the geometry and the number of stages needed to match particular pumping requirements. It is hoped that in the short future an early prototype, supported by the present design guidelines, will be fabricated in collaboration with LAAS and experimentally validated.

Figure 1 :

 1 Figure 1: Outline of one pumping stage of a multistage Knudsen type cascade pump consisting of a series of tapered (convergent-divergent) rectangular channels; T C and T H denote the temperatures at the cold and hot reservoirs, respectively, and P 1 , P 2 , P 3 denote the pressures in successive reservoirs; H in , H out and H m are the heights at the inlet, outlet and middle of the channel, respectively, while W and L are the width and length of the channel.

  Proceedings of the 5th European Conference on Microfluidics -µFlu18 3 rd European Conference on Non-Equilibrium Gas Flows -NEGF18 February 28-March 2, 2018 -Strasbourg, France 3 of the characteristic curves (net mass flow rate versus pressure difference) of the multistage Knudsen assembly.

Figure 2 :

 2 Figure 2: Maximum pressure difference (left) and maximum net mass flow rate (right) of a multistage convergentdivergent Knudsen pump in terms of number of stages at various operating pressures with H m =10 μm, W=15 μm, L/H m =20, T in =300 K, T out =400 K and α=7.

Figure 3 :

 3 Figure 3: Characteristic curves (pressure difference vs. mass flow rate) of a multistage convergent-divergent Knudsen pump for different numbers of stages at 1000 Pa; H m =10 μm, W=15 μm, L/H m =20, T in =300 K, T out =400 K, α=7.

Acknowledgements

This project has received funding from the European Union's Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie Grant Agreement No. 643095.


Irene Lara-Ibeas

Christina Andrikopoulou

Stéphane Colin

Lucien Baldas

Stéphane Le Calvé
email: slecalve@unistra.fr

µFLU-NEGF18-123 FABRICATION OF A SILICA-COATED MICRO GAS CHROMATOGRAPHY COLUMN FOR VOC SEPARATION

Keywords: MEMS technology, micro-column, micro-heaters, gas chromatography, VOC analysis

Portability and low energy consumption are of a great importance for on-site real time monitoring of indoor air quality. Therefore, the aim of this work is to miniaturize one of the bulkiest elements consuming the highest energy in analytical devices, i.e. a gas chromatography microcolumn. In this study, we report the design and fabrication on silicon wafers of three gas chromatography microcolumns of 1, 2 and 3-m long with integrated microheaters and temperature sensors. In a first stage, the microcolumn and the temperature control system have been developed separately to enable a faster optimization. The heating system has been tested and it allowed the device to reach 200 °C in 55 s with a power consumption of 13.8 W. A specific packaging has been designed to provide reliable fluidic connections to the MEMS-based columns. In a second stage, the latter will be integrated in a laboratory prototype to test their suitability for VOC separation.

Introduction

In the last decades, indoor air quality (IAQ) has gained more attention in the scientific communities due its influence on human health. Hazardous air pollutants, such as Volatile Organic Compounds (VOCs) are present in closed environments at levels that are usually 2 to 10 times higher than outdoors. Among these compounds, BTEX (Benzene, Toluene, Ethylbenzene and Xylenes) are the most frequently present in indoor air. Exposure to BTEX has adverse effects on human health and may lead to the development of serious pathologies such as asthma, lung cancer or cardiovascular diseases [START_REF] Kampa | Human health effects of air pollution[END_REF], [START_REF] Arif | Association between personal exposure to volatile organic compounds and asthma among US adult population[END_REF]. According to the World Health Organization (WHO), 4.3 million people per year die from the exposure to household air pollution. As a result, European Union initiated new regulations for air pollutants, i.e. a threshold of 5 µg m -3 (1.6 ppb) has been defined for benzene in ambient air (Directive 2008/50/EC). In France, this threshold has been decreased to 2 µg m -3 (0.6 ppb) from 1 st January 2016. To meet these new regulations, it is necessary to develop high-performance analytical instruments which allow to monitor BTEX in near real time even at low levels. In this context, ICPEES and In'Air Solutions recently developed a novel portable miniaturized GC for near real-time detection of BTEX at ppb levels [START_REF] Nasreddine | Development of a novel portable miniaturized GC for near real-time low level detection of BTEX[END_REF][START_REF] Nasreddine | Portable novel microdevice for BTEX real-time monitoring: Assessment during a field campaign in a low consumption energy junior high school classroom[END_REF]. A laboratory prototype of this device was tested under controlled conditions [START_REF] Nasreddine | Development of a novel portable miniaturized GC for near real-time low level detection of BTEX[END_REF] and during a field campaign (MERMAID project) [START_REF] Nasreddine | Portable novel microdevice for BTEX real-time monitoring: Assessment during a field campaign in a low consumption energy junior high school classroom[END_REF] showing its ability to monitor BTEX at low concentrations. This device has a detection limit of 1 ppb for benzene and an analysis cycle time of 10 min.

The objective of the present work is to reduce the time of analysis, the power consumption and the dimensions of this analyzer. To meet this goal, MEMS technology was employed to fabricate a micro gas chromatography column with integrated heaters and temperature sensors.

It is important to mention that, in this study, the development of the gas chromatography column and the heating system were investigated separately to achieve a faster optimization of both parts.

Materials and Methods

The heating system was designed to present a heating performance that allows the device to reach 200°C in 150 s. This system consists of six resistances deposited on a silicon substrate, four of them acting as heaters and the other two as internal temperature sensors. To fabricate the resistances, a layer of Ti with a thickness of 100 nm was deposited by e-beam evaporation. The connectors were fabricated employing the same technique to deposit an Au layer of 60 nm along with 10 nm of Ti as adhesion layer. The system was connected to a power supply by means of wires fixed with silver conductive adhesive for testing.

Five gas chromatography microcolumns were fabricated using 4-in. silicon wafer with a thickness of 400 µm. For all the columns, serpentine microchannel configuration was chosen since this design provides higher separation plate numbers [START_REF] Radadia | The effect of microcolumn geometry on the performance of micro-gas chromatography columns for chip scale gas analyzers[END_REF]. The cross section of the microchannels was 100 µm wide and 150 µm deep in all devices. Length was varied and columns of 1.01, 2.03 and 3.04 m long were etched by Deep Reactive-Ion Etching (DRIE) using an anisotropic standard Bosch process. Then, a 500 nm SiO 2 layer was grown on the surface of the channels by thermal oxidation, this layer being used as the stationary phase of the microcolumn. Finally, the microchannels were sealed with a 4-in Pyrex glass wafer by anodic bonding, and the chips were diced. A special package was designed to connect the chip's ports to tubes of 1/16 in. (see Figure 2). This package allows also to integrate three cartridge heaters and one external temperature sensor for temperature programming during the validation of the separation efficiency of the microcolumn.

Results

The heating system was tested at different voltages of power supply. Figure 1 presents the temperature increase achieved as function of time at the different voltages supplied. At a voltage of 17.3 V, the microcolumn reached 200 °C, starting from the ambient temperature of 20°C, in 55 s, thus an increase of temperature faster than the one defined in the initial requirements. The heating rates calculated for the set of experiments were 1.73, 2.26 and 3.15 °C/s at 12.6, 15.6 and 17.3 V, respectively. According to these results, the procedure followed for the design and fabrication of a heating system to be integrated in the microcolumn was validated. Furthermore, during the experiments, the current was measured to be 800 mA. Therefore, the corresponding power consumption of the heating system was 10.08, 12.48 and 13.84 W at the aforementioned voltages which is fully compatible with portable analyzers.

Once fabricated, the microcolumns were individually placed in the metallic packages (see Figure 3) and sealing of the assembled systems was verified to avoid leakage. The pressure drop along the microchannel was measured in columns of different lengths. Each measurement was repeated three times. The results obtained for 3 and 2 m-long column are presented in Figure 3. Figure 3 shows that flow rate increases linearly with the pressure drop, which is characteristic of a laminar regime. To verify this observation, the limit between the laminar and the transition flow regime was estimated to be 236 mL/min (corresponding to a Reynolds number of 2100). As the range of flow rates employed in the experiments was always lower than 236 mL/min, it can be confirmed that the regime inside the microcolumn was laminar. Furthermore, it is also demonstrated that the singular pressure drop in the inlet and the outlet of the microchannels is small compared to the linear pressure drop along the microchannel.

Figure 1 :

 1 Figure 1: Temperature variations of the microcolumn at different power supply voltage

Figure 2 :

 2 Figure 2: Scheme of the package integrating fluidic connections and cartridges heaters

Figure 3 :

 3 Figure 3: Flow rate as a function of the pressure drop in 3 m (red) and 2 m (blue) long microcolumns.

Acknowledgements

This work has received funding from the Clean Sky 2 Joint Undertaking under the European Union's Horizon 2020 research and innovation programme under grant agreement No 687014. This project was also supported through the ELCOD project which was implemented as a part of the INTERREG V Oberrhein/Rhin Supérieur program and was supported by the European Regional Development Fund (ERDF) and the co-financed project partners Region Grand Est in France and the countries of Baden-Württemberg and Rhineland-Palatinate.

Conclusions and Perspectives

In conclusion, a thin film Au/Ti heating system has been fabricated and tested at different voltages of power supply. At 17.3 V, the device reached 200 °C, starting from the ambient temperature of 20°C, in 55 s with a power consumption 13.84 W. Therefore, our heating system is appropriate to be integrated in a microfabricated gas chromatography column. On the other hand, a gas chromatography column has been fabricated using MEMS technology and a specific package has been designed to serve the purposes of fluidic connections and temperature control. The next step of this work will focus on testing VOC separation in the microfabricated columns. Once the separation optimized, microchannels and heating system will be fabricated on the same wafer and integrated in a laboratory prototype for field testing.


H D Cornean
email: cornean@math.aau.dk

P Duclos
email: duclos@univ-tln.fr

B Ricaud
email: ricaud@cpt.univ-mrs.fr

On the skeleton method and an application to a quantum scissor

Keywords: 1991 Mathematics Subject Classification. Primary 81Q05, 81Q10; Secondary 31A10, 31A35 Mathematical Quantum Mechanics, Spectral Theory

In the spectral analysis of few one dimensional quantum particles interacting through delta potentials it is well known that one can recast the problem into the spectral analysis of an integral operator (the skeleton) living on the submanifold which supports the delta interactions. We shall present several tools which allow direct insight into the spectral structure of this skeleton. We shall illustrate the method on a model of a two dimensional quantum particle interacting with two infinitely long straight wires which cross one another at angle θ: the quantum scissor.

Introduction

Let us consider the following one dimensional model of N quantum particles interacting through delta potentials. In suitable units, the corresponding Hamiltonian reads as (1.1)

- N i=1 ∆ i 2m i + 1≤i<j≤N Z i Z j δ(x i -x j ), acting in L 2 (R N ),
where m i and Z i denote respectively the mass and the charge of the i'th particle. When the particles are identical (i.e. all the m i 's and Z i 's are equal), it is a well known fact that this model is indeed exactly solvable [LL, McG]; for a quick and fairly complete review, see [vD]; see also the introduction of [AlGH-KH-E]. However, it is not known whether the model is exactly solvable if the particles are distinct, and we strongly suspect that it is not. We have shown in [START_REF] Cornean | Three quantum charged particles interacting through delta potentials[END_REF] that one can nevertheless expect partial exact results, at least. To explore this eventual solvability we have developed a mathematical tool, that we call the skeleton method, which requires to work with a system of integral operators.

The main issue of this article is to give a thorough exposition of this skeleton method, see sections 2 and 3. Finally we shall demonstrate the power of this tool by the spectral analysis of bound states in a model of leaky wires that we call a quantum scissor, see [BEPS] for this terminology. 1.1. Leaky wires. We shall consider the problem (1.1) only in the case N = 3, and

(1.2) m 1 = m 2 > 0, Z 1 = Z 2 < 0 and Z 3 > 0 with the center of mass removed. Then the Hamiltonian expressed in the relative Jacobi coordinates acts in L 2 (R 2 ). After rescaling ( see [START_REF] Cornean | Three quantum charged particles interacting through delta potentials[END_REF] for more details) we have

(1.3) H := - 1 2 ∆ x - 1 2 ∆ y -δ(A ⊥ 1 • (x, y)) -δ(A ⊥ 2 • (x, y)) + λδ(A ⊥ 3 • (x, y))
where A i , i = 1, 2, 3 are three normalized vectors as shown in Figure 1 where the angles θ i,j 's and λ ≥ 0 depend on the original parameters m i 's and Z i 's. Here A ⊥ i denotes A i rotated clockwise by π/2 and the dot in A ⊥ i • (x, y) stands for the scalar product in R 2 . Thus H in (1.3) may be interpreted as the Hamiltonian of a quantum particle confined to a two dimensional plane, which interacts with three straight and infinitely long leaky wires directed by the vectors A i . The "leaky wire" expression appears probably for the first time in [EI] . Another suitable expression for such a quantum model is "leaky graph" which appears in [EN].

1.2. Physical applications. Hamiltonians of the type (1.1) are not only convenient mathematical models, but they do also describe physical systems when some physical parameters are pushed to a limit. It has been recognized long time ago, see e.g. [Spr], that atoms in a strong homogenous magnetic field can be modelled by (1.1), see [BaSoY, BD] for a recent mathematical treatment of this problem. Quasiparticles on carbon nanotubes like excitons can be modelled by a system of charged quantum particles living at the surface of an infinitely long cylinder, see [P]. When the radius of the cylinder tends to zero, it has been shown in [START_REF] Cornean | One dimensional models of excitons in carbon nanotubes[END_REF][START_REF] Cornean | Effective models for excitons in carbon nanotubes[END_REF] that a model of the type (1.1) is a good effective Hamiltonian for these quasiparticles. Not only does the quantum world provide us with such models. For example, in classical optics, photonic crystals with a high contrast in the dielectric constant between the (thin) crystal and air, can also be modelled by such a Hamiltonian, see [START_REF] Kuchment | Differential operators on graphs and photonic crystals. Modeling and computation in optics and electromagnetics[END_REF]§2] for more details.

The skeleton

Most of the content of this section could be obtained as a by-product of [BEKS]. However we think it is worth to make public this more "operator theoretical" version. For any normalized vector A in R 2 we introduce τ A :

H 1 (R 2 ) → L 2 (R) as the continuous restriction map (2.1) H 1 (R 2 ) ∋ ψ → τ A ψ ∈ L 2 (R), τ A ψ(s) := ψ(sA).
Let g be a diagonal 3 × 3 matrix with the diagonal entries {g i } 3 i=1 := (-1, -1, λ). The Hamiltonian H in (1.3) is properly defined as the unique self-adjoint operator associated to the closed and bounded from below quadratic form:

(2.2) H 1 (R 2 ) ∋ u → 1 2 ∇u 2 + 3 i=1 g i τ Ai u 2 ∈ R, Let us set τ i := τ Ai and τ := (τ 1 , τ 2 , τ 3 ) : H 1 (R 2 ) → ⊕ 3 i=1 L 2 (R)
. Then H may be rewritten as

(2.3) H = H 0 + τ ⋆ gτ, H 0 := - 1 2 ∆, dom H 0 := H 2 (R 2 ).
Notice that the above sum defining H must be understood in the sense of quadratic forms, and as a matter of fact dom H = dom H 0 . Thanks to the particular values of the coupling constant g i 's and by an application of the HVZ theorem one gets Lemma 2.1. For all λ ≥ -1, the essential spectrum of H is [-1 2 , ∞). We want to show that the eigenvalue problem HΨ = EΨ for E < -1 2 , i.e. below the essential spectrum can be reduced to a one-dimensional eigenvalue problem involving integral operators. Using Krein's formula with R(z) := (H -z) -1 , R 0 (z) := (H 0 -z) -1 we get at once:

(2.4) R(z) = R 0 (z) -R 0 (z)τ ⋆ (g -1 + τ R 0 (z)τ ⋆ ) -1 τ R 0 (z), z ∈ ρ(H 0 ) ∩ ρ(H).
By classical Sobolev trace theorems the following operators are continuous:

τ R 0 (z) : L 2 (R 2 ) → 3 i=1 H 3 2 (R), τ R 0 (z)τ ⋆ : 3 i=1 H s (R) → 3 i=1 H s+1 (R) for all z /
∈ spect H 0 and all s ∈ R. This allows to consider g -1 + τ R 0 (z)τ ⋆ as a bounded operator on S := ⊕ 3 i=1 L 2 (R) when z / ∈ R + .

Definition 2.2. We shall call S(k)

:= g -1 + τ R 0 (-k 2 )τ ⋆ the skeleton of H at energy -k 2 . Theorem 2.3. E < -1 2 is an eigenvalue of H iff ker(g -1 + τ R 0 (E)τ ⋆ ) = {0}.
If P is the orthogonal projector on this kernel, then the multiplicity of E is equal to the dimension of P . In addition, the operator P τ R 2 0 (E)τ ⋆ P is invertible on the range of P , and the eigenprojector of H associated to E is given by

R 0 (E)τ ⋆ P τ R 2 0 (E)τ ⋆ P -1 τ R 0 (E).
Proof. 1. We start by showing that ∀E := k 2 < -1 2 the essential spectrum of S(k) obeys: for all λ ≥ 0 0

/ ∈ spect ess S(k) = spect ac S(k) = [-1, -1 + 1 √ 2k ] ∪ [λ -1 , λ -1 + 1 √ 2k ]. Indeed if one sets T θi,j := τ i R 0 (z)τ ⋆ j then (2.5) S(k) =   -1 + T 0 0 0 0 -1 + T 0 0 0 0 λ -1 + T 0   +   0 T θ1,2 T θ2,3 T θ1,2 0 T θ2,3 T θ2,3 T θ2,3 0  
Since the diagonal of the first matrix consists of multiplication operators (in the Fourier representation see (3.2)) and the entries of the second matrix are all trace class operators (see Theorem 3.3), we are done. That 0 / ∈ spect ess S(k) is now obvious. 2. Assume that E < -1 2 is an eigenvalue of H, but 0 is not an eigenvalue of S(k). Then S(k) has a bounded inverse (after an easy application of the Fredholm alternative). Since R 0 (E) and τ R 0 (E) are bounded operators, it means that R(z) is also bounded at z = E. This contradicts the fact that E is an eigenvalue of H. We conclude that S(k) cannot be invertible (injective) if -k 2 coincides with an eigenvalue of H. 3. Now let us prove that all singularities of S(k) -1 correspond to eigenvalues of H. One has the identity

(2.6) (g -1 + τ R 0 (z)τ ⋆ ) -1 = g -gτ R(z)τ ⋆ g
valid for z where at least one and therefore two members of this identity exists. Now assume that for some E < -1/2, the operator

g -1 + τ R 0 (E)τ ⋆ is not invertible (i.e.
not injective in our case). Assume also that E is not in the (discrete) spectrum of H. Then (2.6) implies that in a small disc around E we have that (g

-1 + τ R 0 (z)τ ⋆ ) -1 is uniformly bounded, which means that g -1 + τ R 0 (E)τ ⋆ is invertible by Neumann series, contradiction.
4. Now let us investigate the dimension of the spectral subspace associated to an eigenvalue. Assume that 0 is an eigenvalue of g -1 +τ R 0 (E)τ ⋆ and let P be the finite dimensional associated eigenprojector. We have shown that E is also an eigenvalue of H, and denote by P (E) its finite dimensional projection. We want to prove here that dim(P ) = dim(P (E)). Since (g -1 + τ R 0 (E)τ ⋆ )P = 0 and using the resolvent identity:

(2.7) (g -1 + τ R 0 (z)τ ⋆ )P = (z -E)τ R 0 (z)R 0 (E)τ ⋆ P.
Using (2.6), and knowing that near E we have

(2.8) (z -E)R(z) = -P (E) + O((z -E)),
it follows that

P = (z -E)(g -1 + τ R 0 (z)τ ⋆ ) -1 τ R 0 (z)R 0 (E)τ ⋆ P (2.9) = gτ P (E)τ ⋆ gτ R 0 (z)R 0 (E)τ ⋆ P + O((z -E)).
Taking the limit z → E we obtain (2.10)

P = gτ P (E)τ ⋆ gτ R 0 (E) 2 τ ⋆ P.
If Ran(P (E)) is spanned by the eigenvectors {ψ j } dim(P (E)) j=1

, then (2.10) says that Ran(P ) is spanned by {gτ ψ j } dim(P (E)) j=1

, therefore dim(P ) ≤ dim(P (E)).

We now want to prove the reverse inequality. Denote by Q := id -P , so that in matrix notation (use (2.7) and its adjoint)

g -1 +τ R 0 (z)τ ⋆ = (z-E) P τ R 0 (z)R 0 (E)τ ⋆ P P τ R 0 (z)R 0 (E)τ ⋆ Q Qτ R 0 (z)R 0 (E)τ ⋆ P (z -E) -1 Q(g -1 + τ R 0 (z)τ ⋆ )Q
.

To invert this matrix we use the Feshbach method. One has (i) the operator

Q(g -1 + τ R 0 (E)τ ⋆ )Q has a bounded inverse on the range of Q, thus (z -E) -1 Q(g -1 + τ R 0 (z)τ ⋆ )Q is bounded invertible for z in a neighbourhood of E,
:= P τ R 0 (z)R 0 (E)τ ⋆ P -P τ R 0 (z)R 0 (E)τ ⋆ Q(z -E) •(Q(g -1 + τ R 0 (z)τ ⋆ )Q) -1 Qτ R 0 (z)R 0 (E)τ ⋆ P.
Notice that this operator is nothing but a finite dimensional matrix, acting in Ran (P ). The Feshbach formula says that the above operator A(z) is invertible if and only if (g -1 +τ R 0 (z)τ ⋆ )/(z-E) is invertible. Moreover, for z in a neighborhood of E this formula gives:

(z -E)P (g -1 + τ R 0 (z)τ ⋆ ) -1 P = A(z) -1 , z = E.
Using again (2.6) and (2.8), we obtain

A(z) -1 = P gτ P (E)τ ⋆ gP + O(z -E), z = E.
This inverse is bounded near E, and A(z) is continuous at z = E, hence A(E) is invertible and

(2.11)

A(E) -1 = P τ R 0 (E) 2 τ ⋆ P -1 = P gτ P (E)τ ⋆ gP.
Summarizing, via the Feshbach formula, we obtain that

(2.12) (z -E)(g -1 + τ R 0 (z)τ ⋆ ) -1 = A(z) -1 + O((z -E)).
Multiply (2.6) with (z -E), use (2.8), (2.12), and take the limit z → E. This gives:

(2.13)

P (E) = R 0 (E)τ ⋆ A(E) -1 τ R 0 (E) = R 0 (E)τ ⋆ P gτ P (E)τ ⋆ gP τ R 0 (E). Now assume that {φ j } dim(P ) j=1
are eigenvectors spanning the range of P . Then (2.13) says that the range of

P (E) is spanned by {R 0 (E)τ ⋆ φ j } dim(P ) j=1
, which implies dim(P (E)) ≤ dim (P ) and we are done.

The T θ operators

In this section we shall establish various properties of the T θi,j operators.

3.1. Generalities. Let A and B be two normalized vectors of R 2 . We shall consider τ A R 0 (-k 2 )τ ⋆ B where τ A , τ B are defined by (2.1). We can obtain explicit formulas for their integral kernels using the Fourier transform that we denote by a hat. We summarize the results in the following technical lemma:

Lemma 3.1. The operator τ A R 0 (-k 2 )τ ⋆
B depends only on the angle θ between the vectors A and B. When det(A, B) = 0, the Fourier transform of τ A R 0 (-k 2 )τ ⋆ B is an integral operator with kernel

(3.1) Tθ (t, s; k) = 1 2π| sin(θ)| 1 t 2 -2 cos(θ)ts+s 2 2 sin 2 (θ) + k 2 .
When A = B, the Fourier transform of τ A R 0 (-k 2 )τ ⋆ A is the multiplication operator given by the function

(3.2) T0 (s; k) := 1 √ s 2 + 2k 2 .
The proof of this lemma is elementary and left to the reader.

Remark 3.2. (a) One has:

∀θ ∈ (-π, π), T θ (k) ≤ 1 √ 2k .
This is clear for the case θ = 0 since then T0 (k) is an explicit multiplication operator.

For θ = 0 we use

τ A R 0 τ ⋆ B 2 ≤ τ A R 1 2 0 2 R 1 2 0 τ ⋆ B 2 = τ A R 0 τ ⋆ A τ B R 0 τ ⋆ B = T 0 2 .
One can also compute explicitly the Hilbert-Schmidt norm of T θ : (c) Let Π : L 2 (R) → L 2 (R) denote the parity operator Πϕ(s) = ϕ(-s). Then [Π, Tθ ] = 0 so that one can decompose

T θ (k) 2 HS = 1 2π sin(θ)k 2 , θ = 0 mod π.
Tθ = T + θ ⊕ T - θ , with T ± θ := 1 ± Π 2 Tθ .
(d) By a simple inspection of (3.1) we have the reflection properties:

∀θ ∈ (0, π), T ± π-θ = ±T ± θ 3.2.
Rank one operator decomposition of Tθ . Let us first consider T π 2 ; we have the formula

(3.3) T π 2 (p, q) = 1 π 1 p 2 + q 2 + 2 = 1 π ∞ 0
e -2s e -sp 2 e -sq 2 ds which shows that T π 2 is a "sum" of positive rank one operators so that T π 2 ≥ 0. Since (p, q) → T π 2 (p, q) is continuous and

R T π 2 (p, p)dp = 1 2 < ∞
this shows in view of [START_REF] Simon | Trace ideals and their applications[END_REF]th 2.12], that T π 2 is trace class and that its trace and therefore its trace norm are 1/2. We are indebted to R. Brummelhuis who showed us the trick (3.3). The above derivation can be generalized to any angle 0 < θ < π as follows.

Theorem 3.3. For all θ ∈ (0, π) one has in the trace norm ( • 1 ) topology

Tθ = 2 -1 2 sin(θ) π n∈N Γ(n + 1 2 ) Γ(n + 1) cos n (θ) ∞ 0 ds s -1 2 e -2 sin 2 (θ)s P n,s (3.4) T + θ = 2 -1 2 sin(θ) π ∞ n=0 Γ(2n + 1 2 ) Γ(2n + 1) cos 2n (θ) ∞ 0 ds s -1 2 e -2 sin 2 (θ)s P 2n,s T - θ = 2 -1 2 sin(θ) π ∞ n=0 Γ(2n + 3 2 ) Γ(2n + 2) cos 2n+1 (θ) ∞ 0 dss -1 2 e -2 sin 2 (θ)s P 2n+1,s
where P n,s denotes the rank one orthogonal projector on the vector g n,s defined by

(3.5) g n,s (p) := (2s) n+ 1 2 Γ(n + 1 2 )
p n e -p 2 s .

Accordingly one has

(3.6) ∀θ ∈ (0, π 2 ], T ± θ ≥ 0 and ∀θ ∈ [ π 2 , π), ± T ± θ ≥ 0.
It follows that T θ and T ± θ are trace class and

tr T + θ = T + θ 1 = cos θ 2 + sin θ 2 2 √ 2 sin(θ) tr T - θ = cos θ 2 -sin θ 2 2 √ 2 sin(θ) , T - θ 1 = cos θ 2 -sin θ 2 2 √ 2 sin(θ) tr T θ = 1 2 √ 2 sin θ 2 , T θ 1 = max cos θ 2 , sin θ 2 √ 2 sin(θ)
Proof. To find the rank one operator decomposition of Tθ we simply expand its kernel as follows. Let A := p 2 + q 2 + 2 sin 2 θ and B := 2pq cos(θ), one can easily check that A > 0 and |B/A| < 1 for all 0 < θ < π. Thus one has 1 2π sin(θ)

1 p 2 -2 cos(θ)pq+q 2 2 sin 2 (θ) + 1 = sin θ π 1 A -B = sin(θ) π ∞ n=0 A -1 B A n = sin(θ) π ∞ n=0 B n ∞ 0 ds s n e -sA n! = sin(θ) π ∞ n=0 2 n cos n (θ) n! ∞ 0 ds e -2s sin 2 (θ)
s n e -s(p 2 +q 2 ) (pq) n .

To arrive at (3.4) one needs to normalize in L 2 (R) the vector p → p n e -sp 2 which gives the vector g n,s in (3.5). Since P n,s = P n,s 1 the convergence in the trace norm topology of the r.h.s. of (3.4) is true since the terms in the following sum are all positive and one has explicitly:

2 -1 2 sin(θ) π n∈N Γ(n + 1 2 ) Γ(n + 1) | cos n (θ)| ∞ 0 ds s -1 2 e -2 sin 2 (θ)s = 1 2 1 -| cos(θ)| .
This shows that Tθ is trace class and that (3.4) is valid in the trace norm topology. Since g n,s has the parity of n, one gets T ± θ by selecting the even and odd values of n in (3.4) resp.. The rest is now obvious up to some tedious explicit computations of sums.

We shall draw some other useful properties from the above theorem. (ii) If one labels the eigenvalues of T + θ by descending order:

E + 1 (θ) ≥ E + 2 (θ) ≥ . . . ≥ E + n (θ) ≥ . . . then each function (0, π) ∋ θ → E + n (θ)
is continuous and decreasing on (0, π 2 ] and increasing on [ π 2 , π).

If one labels the eigenvalues of T -

θ by descending order on (0, π 2 ) and ascending order on ( π 2 , π), then each function (0, π) ∋ θ → E - n (θ) is continuous and decreasing on (0, π).

Proof. (i) is a direct consequence of the convergence of the r.h.s. of (3.4) on D. To prove (ii) we shall consider another s.a. family of operators which is the image of {T θ } θ under the scaling p → sin(θ)p, 0 < θ < π:

(3.7) ♯ T θ (p, q) = 1 π 1 p 2 + q 2 -2pq cos(θ) + 2 .
Then proceeding as in the previous theorem we get

♯ T + θ = 1 π n∈N (2 cos θ) 2n B 2n ♯ T - θ = 1 π n∈N (2 cos θ) 2n+1 B 2n+1
where B n denotes the positive operator with kernel

B n (p, q) := (pq) n (p 2 + q 2 + 2) n+1 = ∞ 0 s n n!
e -2s (pq) n e -sp 2 e -sq 2 .

If we label the eigenvalues of T + θ , i.e. the eigenvalues of ♯ T + θ in descending order they are all continuous in θ and in view of the elementary dependence of ♯ T + θ on θ, they are decreasing on (0, π/2] and increasing on [π/2, π). We skip the analogous reasoning for T - θ .

3.3. T ± θ are ergodic. The reader can find the definition of an ergodic operator in [START_REF] Reed | Methods of modern mathematical physics IV. Analysis of operators[END_REF]§XIII.12].

Proposition 3.5. (i) For all θ ∈ (0, π), T + θ is ergodic and sup T + θ is a simple eigenvalue of T + θ .

(ii) For all θ ∈ (0, π/2) ∪ (π/2, π), sign(π/2 -θ)T - θ is ergodic and sup sign(π/2θ)T - θ is a simple eigenvalue of T - θ . Proof. We have previously seen that T + θ ≥ 0. Also T + θ is self adjoint and compact, thus

T + θ = sup T +
θ is an eigenvalue of T + θ . Clearly since T + θ (p, q) > 0 for all p and q, one has (T + θ f, g) > 0 whenever f and g are positive. Thus T + θ is ergodic. By applying [START_REF] Reed | Methods of modern mathematical physics IV. Analysis of operators[END_REF]Th. XIII.43] we get that sup T + θ is a simple eigenvalue of T + θ . The proof for T - θ is analogous.

3.4. T θ is injective. This question was brought to us by T. Dorlas.

Lemma 3.6. For all 0 < θ < π, one has ker T θ = {0}.

Proof. We find it more convenient to work with ♯ T θ , see (3.7), which is unitarily equivalent to T θ . We recall that Π denotes the parity operator, see Remark 3.2(c). Using the formula derived in the proof of Corollary 3.4, we get with

ϕ ∈ Ran Π + , that ♯ T + θ ϕ = 0 implies ( ♯ T + θ ϕ, ϕ) = 0 ⇒ 1 π ∞ n=0 (2 cos(θ)) 2n (2n)! ∞ 0 s 2n e -2s (C 2n,s ϕ, ϕ)ds = 0
where C n,s with kernel C n,s (p, q) := (pq) n e -sp 2 e -sq 2 is a positive rank one operator. If θ = π 2 it follows that: ∀n ∈ N, ∀s > 0, (C 2n,s ϕ, ϕ) = 0 since (2 cos(θ)) 2n and s 2n e -s are strictly positive. But

(C 2n,s ϕ, ϕ) = 0 ⇐⇒ R |p 2n e -sp 2 ϕ(p)| 2 dp = 0
which shows that ϕ ⊥ p 2n e -p 2 2 for all n ∈ N by choosing s = 1/2. Clearly {p 2n e -p 2 2 , n ∈ N} is total in Ran Π + since they generate the even Hermite functions. Thus ϕ = 0. A similar argument shows that if ϕ ∈ Ran Π -and ♯ T θ ϕ = 0 then ϕ = 0; notice that it is understood here that θ = π 2 since otherwise ♯ T θ = 0. Finally we consider the case θ = π/2 and ϕ ∈ Ran Π + . Here we get as above ∀s > 0, (ϕ, e -sp 2 ) = 0 and by differentiating indefinitely this identity with respect to s in s = 1 2 we find ϕ ⊥ p 2n e -p 2 2 , ∀n ∈ N, which implies as above that ϕ = 0.

3.5. Some properties of (2

-1 2 -T0 ) -1/2 T - θ (2 -1 2 -T0 ) -1/2
. From the rank one operator decomposition of T - θ , see Theorem 3.3, one gets

T - θ := (2 -1 2 -T0 ) -1/2 T - θ (2 -1 2 -T0 ) -1/2 = 2 -1 2 sin(θ) π ∞ n=0 Γ(2n + 3 2 ) Γ(2n + 2) cos 2n+1 (θ) ∞ 0 dss -1 2 e -2 sin 2 (θ)s P 2n+1,s (3.8) where P 2n+1,s := (•, g 2n+1,s ) g 2n+1,s with g 2n+1,s := (2 -1 2 -T0 ) -1/2 g 2n+1,s .
It turns out that g 2n+1,s belongs to L 2 (R) and the r.h.s. (3.8) is convergent in the trace norm. More precisely

P 2n+1,s 1 = g 2n+1,s 2 = √ 2 4n + 8s + 4 √ s U -1 2 , -2n, 4s + 1 4n + 1 ≤ 8n √ s Γ(2n) + (4n + 16s + 1)Γ 2n + 1 2 √ 2 Γ 2n + 3 2
where U denotes the confluent hypergeometric function, see [AS,13.1.3]. The last estimate is obtained by integration of the r.h.s. of the bound

|g(2n + 1, s)(p)| 2 ≤ (2s) 2n+ 3 2 Γ(2n + 3 2 ) p 4n e -2sp 2 ( √ 2p 2 + 2|p| + 4 √ 2).
which is more convenient in view of the summations over s and n. Then the integration over s gives sin(θ)

∞ 0 e -2s sin 2 (θ) √ s Γ(2n + 3 2 ) Γ(2n + 2) g(2n + 1, s) 2 ds ≤ √ πΓ 2n + 1 2 2Γ(2n + 2) + 2n √ πΓ 2n + 1 2 Γ(2n + 2) + 2 √ πΓ 2n + 1 2 sin 2 (θ) Γ(2n + 2) + √ 2Γ(2n + 1) sin(θ) Γ(2n + 2) =: a 1 + a 2 + a 3 + a 4 .
In the summation over n of these four terms, only the second one causes problems to arrive at a convenient final formula: ( the sums are computed with 0 < θ < π/2)

∞ n=0 a 1 cos 2n+1 (θ) √ 2π = 1 2 cos θ 2 -sin θ 2 ∞ n=0 a 2 cos 2n+1 (θ) √ 2π = cos 3 (θ) 2 F 1 5 4 , 7 4 ; 5 2 ; cos 2 (θ) 4 √ 2 ∞ n=0 a 3 cos 2n+1 (θ) √ 2π = 2 sin 2 (θ) cos θ 2 -sin θ 2 ∞ n=0 a 4 cos 2n+1 (θ) √ 2π = tanh -1 (cos(θ)) π(sin(θ)
One replaces a 2 by the the following bound valid for all n ∈ N:

2nΓ 2n + 1 2 Γ(2n + 2) ≤ Γ 2n + 1 2 Γ(2n + 1) = a ′ 2 which gives ∞ n=0 a ′ 2 cos 2n+1 (θ) √ 2π = 1 2 cos(θ) sin(θ) cos θ 2 + sin θ 2 .

Summing up gives

Lemma 3.7. For all θ ∈ (0, π), T - θ is trace class and for all 0 < θ < π/2 one has:

T - θ 1 ≤
4 sin(θ) tanh -1 (cos(θ)) + π 9 cos θ 2 -cos 5θ 2 -9 sin θ 2 + sin 5θ 2 4π sin(θ) .

In particular:

(3.9) In particular the lowest one is simple and monotonically decreasing and pass by -1 for θ = 2π/3 in view of (3.10).

T - 2π 3 1 = T - π 3 1 ≤ - 4 3 + 5 √ 3 + log(3) √ 3π ∼
3.6. Exact eigenvalues and eigenvectors. We collect here some exact results about these Tθ operators. They can be checked by direct inspection. (3.11) and (3.12) are simply obtained by translating known exact eigenfunctions in the skeleton frame work; the first one comes from the exactly solvable quantum scissor, see (4.1), with angle π/2. The second one comes from the Mc Guire bound state eigenfunction of its three particle system, see [McG, §IV.D]. The last one seems to be new. Since T0 (0) -T0 (p) ∼ 2 -5 4 |p| as p → 0, this function has a cusp at 0.

A quantum scissor

We consider the Hamiltonian (2.3) in the particular case λ = 0: (4.1)

H θ := - ∆ 2 -δ(A ⊥ 1 •) -δ(A ⊥ 2 •) = - ∆ 2 -τ 1 τ ⋆ 1 -τ 2 τ ⋆ 2
which described a two dimensional particle in a scissor-shaped waveguide, a name borrowed from [BEPS]. We assume without loss of generality that θ := θ 1,2 belongs to

(4.2) θ ∈ [ π 2 , π);
where θ denotes the angle made by the two vectors A 1 and A 2 which generate the supports of the delta interactions, see Figure 1. We note that the case θ = π is exactly solvable, and that the angles θ ∈ (0, π/2] are covered by the cases (4.2) since H θ and H π-θ are unitarily equivalent. Thanks to Lemma 2.1, the essential spectrum of H θ is [-1/2, ∞). The skeleton associated to H θ in the Fourier representation is

-1 + T0 (k) Tθ (k) Tθ (k) -1 + T0 (k) ∼ k -1 -k + T0 Tθ Tθ -k + T0
where the unitarily equivalent second expression is obtained through the scaling s → ks, see Remark 3.2(b). It acts on S := L 2 (R) ⊕ L 2 (R). Thus, according to Theorem 2.3, -k 2 < -1/2 is an eigenvalue of H θ iff k is an eigenvalue of T θ := T0 Tθ Tθ T0 .

Notice that in view of Corollary 3.4, {T θ } θ∈D is bounded selfadjoint family of analytic operators and since T θ is trace class ( see Theorem 3.3) one has

spect ess T θ = spect ac T θ = spect T 0 = [0, 2 -1 2 ].
4.1. Reduction by symmetries. We use (x, y) for the coordinates in R 2 and we recall that Π stands for the parity operator on L 2 (R), see Remark 3.2(c). Let Π y := Π ⊗ 1, Π x := 1 ⊗ Π acting in L 2 (R 2 ) denote respectively the reflection with respect to the y and x axis.

H θ fulfills [H θ , Π x ] = [H θ , Π y ] = 0.
This allows to reduce H θ as

H θ = α,β∈{±1} H α,β
θ , where H α,β θ := Π α x Π β y H θ and Π α x := 1 2 (id + αΠ x )), Π β y := 1 2 (id + βΠ y )) denote the eigenprojectors of Π x and Π y resp.. We also stress that H α,β θ is unitarily equivalent to the operator acting in L 2 (R + × R + ) with same symbol as H θ but with additional Dirichlet boundary conditions on x = 0 if α = -1 or on y = 0 if β = -1 and Neumann boundary condition in the opposite case, see Table 1.

Similarly T θ enjoys the following symmetries

[T θ , Π] = [T θ , E] = [Π ⊕ Π, E] = 0.
where Π := Π ⊕ Π : S → S is the parity operator and E : S → S is the exchange of components operator: E(φ 1 ⊕ φ 2 ) := φ 2 ⊕ φ 1 . Thus we may consider separately

Π α E β T θ , α = ±1, β = ±1
where Π α and E β denote the spectral projectors of Π and E resp.:

Π α := 1 2 (id + αΠ) E α := 1 2 (id + αE).
One has the following elementary result the proof of which is left to the reader: Let R(z) α,β := Π α x Π β y (H θ -z) -1 and similarly for R 0 := (H 0 -z) -1 then using (2.4) and Definition 2.2 we get since T - θ ≥ -1, see Remark 3.8(b) and (c). Thus T -,- θ ≤ 2 -1 2 which implies that it cannot have an eigenvalue larger than 2 -1 2 . To prove (ii) it is sufficient to show that there exists k > 2 -1 2 so that inf spect (k -T -,- θ ) < 0 for all θ > 2π/3. We first establish that the lowest eigenvalue E - 1 (θ) of T - θ is strictly smaller than -1 for all 2π/3 < θ < π. Let ϕ denote the normalized eigenvector of T -

Figure 1 .

 1 Figure 1. The three supports of the δ leaky wires.

  (b) If we perform the scaling s → ks then clearly Tθ (k) becomes k -1Tθ (1). Since in the sequel we shall use this property and work only with Tθ (1), we denote Tθ := Tθ (1).

  Corollary 3.4. (i) θ → T θ is a selfadjoint analytic family as a map from D := {θ ∈ C, | cos(θ)| < 1} with values in the ideal of trace class operators.

  1.75532 Remark 3.8. (a) If we do not replace a 2 by a ′ 2 we get a better bound: T -We shall see below that -1 is an eigenvalue of T 2π 3 , see (3.13). Since the trace norm of T - 2π 3 is less than 2, see (3.9), it follows that this eigenvalue is simple and is the lowest eigenvalue of T 2π 3 . Thus we may conclude that (3.10) inf T 2π 3 = -1. (c) The statements in Corollary 3.4(ii) and Proposition 3.5 for eigenvalues of T - θ works as well for those of T - θ .

Lemma 4. 1 .

 1 For all α, β in {±1}, Π α E β T θ is unitarily equivalent toT For all α, β in {±1}, -k 2 < -1 2 is a discrete eigenvalue of H α,β θ iff k > 2 -1 2 is a discrete eigenvalue of T αβ,β θ .Proof. Due to the chosen orientation of the two normalized vectors, see Figure1, we have the relations between mappings fromH 1 (R 2 ) to L 2 (R) τ Π y = Eτ, τ Π x = ΠEτ so that 4τ Π α x Π β y = τ (id + αΠ x )(id + βΠ y ) = (id + αΠE)τ (id + βΠ y ) = (id + αΠE)(id + βE)τ = (id + αβΠ)(id + βE) = 4Π αβ E β τ.

odd with respect to this second axis when 2π/3 < θ < π. The number of such bound states is bounded above by T - θ 1 , and therefore by the bound given in Lemma 3.7. (iii) It has at least one bound state for all 0 < θ < π which is even with respect to both axis of the scissor and unique for θ in [π/3, 2π/3]. (iv) All bound states of H θ are bounded below by -2 and monotonously decreasing with respect to θ on [π/2, π).

Concluding remarks and open problems

We are far from having found the answers to all the questions about the quantum scissor of § 4. Let us review these questions; most of them are already in [BEPS, §III]:

(1) Every bound state is even w.r.t. the scissor axis: done, see Th 4.6(i).

(2) With respect to the second axis the bound states can have both parities (done, see Th 4.6(ii) and (iii)) which are alternating if the bound states are arranged according to their energies: not done. (1) can we enlarge the list of exact spectral results, see § 3.6? (2) Are all eigenvalues of T ± θ simple? (3) Is it true that θ → T ± θ are monotonous?