Paul Rouquette

Claude Amra

Myriam Zerrad

Catherine Grèzes-Besset

Photo-induced thermal radiation within multilayer optics

Keywords: Emissivity, thermal radiation, temperature, thin films, optical coatings, microcavity

Exact models of photo-induced temperature and thermal radiation are presented for optical coatings submitted to spatiotemporal regimes. Results are useful to analyze and predict thermal processes in optical thin films at different temporal and spatial scales. They also open a door to control the emissivity patterns of multilayers.

INTRODUCTION

Optical coatings have been extensively characterized these last decades, due to more and more severe requirements. These characterizations include optical properties, uniformity, hardness, adhesion and stress, damage threshold, absorption, scattering and others… For applications involving mirrors for gyro-laser systems, detection of gravitational waves and space micro-optics, huge efforts were developed to overpass the "ppm barrier", though absorption and scattering still limit the energy balance to a few ppm losses.

Despite this state of the art, thermal radiation [START_REF] Rytov | Principles of Statistical Radiophysics 3: Elements of Random Fields[END_REF], [START_REF] Howell | Thermal Radiation Heat Transfer[END_REF] in optical coatings was rarely investigated in detail until now, while emissivity [START_REF] Howell | Thermal Radiation Heat Transfer[END_REF] plays a key role in numerous sectors related to energy and defense, space optics and MIR imaging... Thermal radiation in multilayers originates from both the ambient and the absorption-induced temperature, and has mostly been studied for opaque samples or for transparent samples supporting a constant temperature in time. Predictions can be found in the literature [START_REF] Howell | Thermal Radiation Heat Transfer[END_REF], [START_REF] Francoeur | Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green's functions and the scattering matrix method[END_REF], [START_REF] Wang | Direct and Indirect Methods for Calculating Thermal Emission From Layered Structures With Nonuniform Temperatures[END_REF], [START_REF] Zhuomin | Nano/Microscale Heat Transfer[END_REF] but often rely on approximations or assumptions, and are rarely connected to photo-induced effects in optical coatings.

The scope of this work is to provide an exact theory of thermal radiation in optical multilayers submitted to an arbitrary illumination (continuous, pulsed, modulated). In a first part, the modelling of the laser induced temperature in optical coatings is discussed and some examples are given for a quarter wave mirror submitted to nanosecond and millisecond pulses. Then the thermal radiation is linked to the temperature thanks to the fluctuation dissipation theorem. Finally, ideas to build planar optical coatings that radiate thermically at specific wavelength and angle are given.

PHOTO-INDUCED TEMPERATURE WITHIN MULTILAYERS OPTICS

Thermal radiation of optical coatings originates from both the ambient and the photo-induced temperature that occurs when the sample is submitted to an optical illumination. Hence, the first section of this article is dedicated to the modelling of this laser induced temperature.

Theory modelling

Figure 1 shows the geometry of an optical multilayer with p thin film layers. The media are considered isotropic and homogenous. For each layer (i), the thickness is noted ei, the refractive index ni, the thermal conductivity bi and the thermal diffusivity ai. The superstrate has the refractive index n0 whereas the substrate has the index nS. The illumination is coming from the superstrate. In this work we focus on the conduction process (heat diffusion), so that heat transfer by convection in the surrounding media is not considered. Hence the temperature in each layer (i) is governed by the classical heat equation:

∆𝑇 𝑖 (𝑟 ⃗, 𝑧, 𝑡) -( 1 𝑎 𝑖 ) 𝜕 𝑡 𝑇 𝑖 (𝑟 ⃗, 𝑧, 𝑡) = -( 1 𝑏 𝑖 ) 𝑆 𝑖 (𝑟 ⃗, 𝑧, 𝑡) (1)
with 𝑇 𝑖 is the photo-induced temperature, t is the temporal variable, 𝑟 ⃗ = (𝑥, 𝑦) the transversal space coordinates and z the perpendicular direction to the interfaces of the component. 𝑆 𝑖 is the thermal source and represents the absorption density created by the optical illumination in the media (i).

It is important to notice that this model is acceptable only if the pulse duration is longer than 10 picoseconds. To tackle the temperature induced by ultra-short pulses, one needs to consider the two-temperature model [START_REF] Taylor | Femtosecond laser polishing of germanium [Invited[END_REF].

The resolution of the heat equation has been widely studied in the literature. A lot of papers have considered finite difference algorithms [START_REF] Crank | A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type[END_REF], [START_REF] Masud Mansuripur | Laser-induced local heating of multilayers[END_REF], [START_REF] Hu | Laser-induced damage of a 1064-nm ZnS/MgF2 narrow-band interference filter[END_REF], [START_REF] Dai | Analysis of laser induced thermal mechanical relationship of HfO2/SiO2 high reflective optical thin film at 1064 nm[END_REF], [START_REF] Wang | Effect of defects on long-pulse laser-induced damage of two kinds of optical thin films[END_REF] and [START_REF] Li | Temperature field analysis of high reflection film induced by longpulse and short-pulse lasers under different irradiation angles[END_REF]. Others have used Hankel transform or Laplace transform [START_REF] Lax | Temperature rise induced by a laser beam[END_REF], [START_REF] Salazar | Flying spot thermography: Quantitative assessment of thermal diffusivity and crack width[END_REF].

In this article, we choose to rely on an analogy between optical propagation and heat diffusion [START_REF] Amra | Analogies between optical propagation and heat diffusion: applications to microcavities, gratings and cloaks[END_REF] that allows to reuse algorithms that have been developed for the optical propagation in multilayer systems.

and the double Fourier transform is written as follow:

𝑇 ̂(𝜈 ⃗, 𝑧, 𝑓) = ∫ ∫ 𝑇(𝑟 ⃗, 𝑧, 𝑡)𝑒 -𝑗2𝜋𝜈 ⃗ ⃗⃗.𝑟 ⃗ 𝑒 𝑗2𝜋𝑓𝑡 𝑑 2 𝑟 ⃗𝑑𝑡 (4)

From a mathematical perspective, equation ( 2) is similar for heat diffusion and optical propagation in absorbing media. The solution of (2) is the sum of the homogenous solution 𝑇 ̂ℎ,𝑖 and a particular solution 𝑇 ̂𝑔,𝑖 .

𝑇 ̂𝑖(𝜈 ⃗, 𝑧, 𝑓) = 𝑇 ̂ℎ,𝑖 (𝜈 ⃗, 𝑧, 𝑓) + 𝑇 ̂𝑔,𝑖 (𝜈 ⃗, 𝑧, 𝑓)

The homogenous solution is classically given by the following equation:

𝑇 ̂ℎ,𝑖 (𝜈 ⃗, 𝑧, 𝑓) = 𝑇 𝑖-1 + (𝜈 ⃗, 𝑓)𝑒 𝑗𝛼 𝑖 (𝑓,𝜈)𝑧 + 𝑇 𝑖-1 -(𝜈 ⃗, 𝑓)𝑒 -𝑗𝛼 𝑖 (𝑓,𝜈)𝑧 (6)
The constants 𝑇 𝑖-1 ± are determined thanks to the continuity of both the temperature and the heat flux at each interface of the component [START_REF] Masud Mansuripur | Laser-induced local heating of multilayers[END_REF].

The particular solution is the convolution product (with respect to the z variable) between the thermal source and the Green function.

𝑇 ̂𝑔,𝑖 (𝜈 ⃗, 𝑧, 𝑓) = 𝐺 𝑖 (𝜈 ⃗, 𝑧, 𝑓) * 𝑧 ( -1 𝑏 𝑖 ) 𝑆 ̂𝑖(𝜈 ⃗, 𝑧, 𝑓) with 𝐺 𝑖 (𝜈 ⃗, 𝑧, 𝑓) = 1 2𝑗𝛼 𝑖 (𝑓, 𝜈) 𝑒 𝑗𝛼 𝑖 (𝑓,𝜈)|𝑧| ( 7)

The last step of the modelling is the determination of the thermal source 𝑆 𝑖 which is the absorption density 𝜕𝒜 𝑖 𝜕𝑣 of the incident laser beam in the layer (i). In the most general form, one can write:

𝜕𝒜 𝑖 𝜕𝑣 = 𝐸 𝑖 𝜕 𝑡 𝐷 𝑖 + 𝐻 𝑖 𝜕 𝑡 𝐵 = 𝑆 𝑖 (𝑟 ⃗, 𝑧, 𝑡) with 𝐷 𝑖 = 𝜀 𝑖 * 𝑡 𝐸 𝑖 and 𝐵 𝑖 = 𝜇 𝑖 * 𝑡 𝐻 𝑖 [START_REF] Masud Mansuripur | Laser-induced local heating of multilayers[END_REF] This expression can be greatly simplified by considering two reasonable assumptions. First, the laser is assumed to be quasi monochromatic around a central frequency f0. Then the divergence of the laser beam is assumed to be weak which means that the optical source is collimated. The incident laser beam occurs from one direction described with the spatial frequency 𝜈 0 = 𝑛 0 𝑠𝑖𝑛𝜃 0 /𝜆 0 where 𝜃 0 is the angle of incidence and 𝜆 0 the central wavelength of the laser. With these two assumptions one can write:

𝑆 𝑖 (𝑟 ⃗, 𝑧, 𝑡) = 𝜕 𝑧 𝒜 𝑖 (𝜈 0 , 𝑧, f 0 ) 𝑔 2 (𝑟 ⃗, 𝑧, 𝑡) (9)
with 𝒜 𝑖 (𝜈 0 , 𝑧, f 0 ) the monochromatic and monodirectional absorption and 𝑔 2 a function that describe the shape and the temporal envelope of the laser. The absorption is described by the classical formulation [START_REF] Dai | Analysis of laser induced thermal mechanical relationship of HfO2/SiO2 high reflective optical thin film at 1064 nm[END_REF] where (𝐸 ̂0 + , 𝐻 ̂0 + ) is the double Fourier transform of the incident electric and magnetic field:

𝜕 𝑧 𝒜 𝑖 (𝜈 0 , 𝑧, f 0 ) = 𝜋𝑓 0 {𝐼𝑚(𝜖 𝑖 ) | 𝐸 ̂𝑖(𝜈 0 , 𝑧, 𝑓 0 ) 𝐸 ̂0 + (𝜈 0 , 𝑓 0 ) | 2 + 𝐼𝑚(𝜇 𝑖 ) | 𝐻 ̂𝑖(𝜈 0 , 𝑧, 𝑓 0 ) 𝐸 ̂0 + (𝜈 0 , 𝑓 0 ) | 2 } ( 10
)
To go further, one can assume that the incident beam can be described by separating the temporal variable and the spatial one. If we choose a gaussian laser in time and space, then 𝑔 2 can be written as follow:

𝑔 2 (𝑟 ⃗, 𝑧 = 0, 𝑡) = 𝑔 1 (𝑟 ⃗)𝑔 2 (𝑡) with 𝑔 1 (𝑟 ⃗) = 𝑔 01 𝑒 - 2(𝑥 2 cos 2 𝜃 0 +𝑦 2 ) 𝐿 2 and 𝑔 2 (𝑡) = 𝑔 02 𝑒 - 2𝑡 2 𝜏 2 (11)
L represents the width of the beam at 𝑒 -2 and 𝜏 the pulse duration.

Finally, the amplitude of the two gaussian functions can be linked to the energy of the laser pulse W and the thermal source can be expressed, in the second Fourier plane, as:

𝑆 ̂𝑖(𝜈 ⃗ ⃗⃗, 𝑧, 𝑓) = 𝜕 𝑧 𝒜 𝑖 (𝜈 0 , 𝑧, f 0 )

2𝑊 ℜ{𝑛 ̃0} 𝑒 -𝜋 2 𝑓 2 𝜏 2 2 𝑒 - 𝜋 2 𝐿 2 2 ( 𝜈 𝑥 2 cos 2 𝜃 0 +𝜈 𝑦 2 ) ( 12
)
In the following, the illuminated surface on the top interface of the sample will be large and we will be interested in the depth variation of the temperature field. In these conditions, the incident wave can be assimilated to a plane wave and one can consider a surface energy density. The system becomes invariant in x, y, and the second Fourier transform is not necessary. Therefore, in the first Fourier plane, the heat equation becomes:

𝜕 𝑧 2 𝑇 ̃𝑖(𝑧, 𝑓) + 𝑗2𝜋𝑓 𝑎 𝑖 𝑇 ̃𝑖(𝑧, 𝑓) = -( 1 𝑏 𝑖 ) 𝑇 ̃𝑖(𝑧, 𝑓) (13)
with the thermal source:

𝑆 ̃𝑖(𝑧, 𝑓) = 𝜕 𝑧 𝒜 𝑖 (𝜈 0 , 𝑧, f 0 ) 2 ℜ{𝑛 ̃0} ∂W ∂S 𝑒 -(𝜋𝜏𝑓) 2 2 ( 14
)
∂W ∂S can be seen as the fluence of the illumination.

Numerical results

In this section, the laser-induced temperature of a mirror made of seven quarter wave layers is computed. The high index material is Nb2O5, and the low index is SiO2. The formula of the component is: Air/H(BH) 7 We first study the laser induced temperature in the nanosecond regime. The laser has an energy of one mJ over one nanosecond. The central wavelength of the laser is of 1064 nm. The illuminated surface is of 100 µm by 100 µm, which gives a fluence of 10 3 J/cm². Note here that laser-induced damage processes are not considered. We are interested in the link between temperature and incident energy. This link is linear and can be used to predict the temperature in other situations. The photo-induced temperature in this configuration is given in Figure 2.

The temperature is also studied in the millisecond regime. The properties of the laser and the materials are the same, except for the pulse duration chosen at one millisecond. Results are given in Figure 3. In the nanosecond regime, the temperature distribution along the direction of depth in the coating follows the normalized modulus squared of the tangential electric field (Figure 4) whereas in the millisecond regime the distribution profile is almost monotonic. This can be explained with the thermal diffusion length 𝐿 = √𝑎𝜏 which is equal, for the Nb2O5, to 20.7 nm in the ns regime, and to 20.7 µm in the ms regime. In the ns regime, the temperature field mostly stays in the layers and so follows the thermal source, which is proportional to the electric field. Conversely, in the ms regime the temperature field diffuses along the depth of the thin film which allows the temperature to stabilize.

PHOTO-INDUCED THERMAL RADIATION WITHIN MULTILAYER OPTICS

The aim of this section is to develop the model of the laser-induced thermal radiation of an optical thin film filter submitted to an arbitrary illumination.

Theory modelling

The previous section gives a map of the photo-induced temperature in the coating with respect to the time. This map is discretised in the space and the time domain. The steps of the discretisation are chosen to obtain areas where the temperature is considered as a constant. This discretisation is essential to satisfy the hypothesis of thermal equilibrium that is needed to apply the fluctuation dissipation theorem. Over a given period of time, we place ourselves in the situation of an optical filter where each layer has a different temperature. In this situation, the model of thermal radiation in an 1D multilayer structure can be found in [START_REF] Francoeur | Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green's functions and the scattering matrix method[END_REF], [START_REF] Wang | Direct and Indirect Methods for Calculating Thermal Emission From Layered Structures With Nonuniform Temperatures[END_REF], or [START_REF] Zhuomin | Nano/Microscale Heat Transfer[END_REF]. The major steps of the modelling are recalled here.

In each medium at thermal equilibrium (where a temperature can be defined), the random thermal movement of the charged particles can be modelled by a spatial time depend electric current density written 𝐽 𝑒 ⃗⃗ ⃗⃗ (𝑟 ⃗, 𝑧, 𝑡) [START_REF] Rytov | Principles of Statistical Radiophysics 3: Elements of Random Fields[END_REF], [START_REF] Novotny | Principles of Nano-Optics[END_REF]. The time average of this current density is zero. Rytov [START_REF] Rytov | Principles of Statistical Radiophysics 3: Elements of Random Fields[END_REF] has applied the fluctuation dissipation theorem and found a relation between the spatial correlation function of the current at two different locations and the temperature of a local isotropic medium:

〈𝐽 ̃𝑚,𝑒 (r ⃗, 𝑧, 𝑓)𝐽 ̃𝑛,𝑒 * (r ⃗ ′ , 𝑧 ′ , 𝑓)〉 = 4πϵ ′′ (𝑓)Θ(𝑓, 𝑇)δ 𝑛𝑚 δ(r ⃗ -r ⃗ ′ )δ(𝑧 -𝑧 ′ ) (13)
With:

- To compute the thermal radiation of the medium, one needs to introduce this thermal current density in the Maxwell equations. Most of the literature [START_REF] Francoeur | Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green's functions and the scattering matrix method[END_REF], [START_REF] Wang | Direct and Indirect Methods for Calculating Thermal Emission From Layered Structures With Nonuniform Temperatures[END_REF], [START_REF] Zhuomin | Nano/Microscale Heat Transfer[END_REF], [START_REF] Mulet | Modélisation du Rayonnement Thermique par une Approche Electromagnétique[END_REF] make the use of the Dyadic Green function to solve this problem. We choose to use the results of the theory of bulk scattering [START_REF] Amra | First-order vector theory of bulk scattering in optical multilayers[END_REF] that enable to compute the power spectrum of a bulk distribution of current.

〈
In this work we only consider non-magnetic mediums. To tackle the thermal radiation of magnetic component, an analogue relation of ( 13) that describes the presence of magnetic current must be considered. The imaginary part of the permittivity will be replaced by the imaginary part of the permeability.

Eventually for a component made of p layers at thermal equilibrium, the time average of the power of the thermal radiation that merges in the superstrate can be written as:

〈Φ 0 (𝑡)〉 = ∑ 2 ∫ ∫ ℜ{𝑛 ̃0}4𝜋𝜖 𝑖 ′′ (𝑓)Θ(𝑓, 𝑇 𝑖 )𝐹 𝑖 𝑑 2 𝜈 ⃗ +∞ 𝜈 ⃗ ⃗⃗=-∞ 𝑑𝑓 +∞ 𝑓=0 𝑝 𝑖=1 ( 14
)
where 𝑛 ̃0 is the effective index of the superstrate and 𝐹 𝑖 is a function that depends on the optical parameters (effective index, complex admittance) and the polarisation of the illumination.

It is useful to extract the common properties of thermal radiation such as intensity and emissivity [START_REF] Howell | Thermal Radiation Heat Transfer[END_REF] from equation ( 14).

The intensity is linked to the monochromatic surface flux density of thermal radiation with the following relation.

𝜕Φ 𝜕𝑆𝜕𝑓𝜕Ω = 𝐼(𝑓, Ω, 𝑇) cos 𝜃 ( 15
)
where S and T are respectively the emissive surface and the temperature of the component, Ω is the solid angle in which the radiation is propagating, (𝜃, 𝜙) are the spherical coordinates and f is the frequency of the radiation.

By considering equation ( 14) and ( 15), one can find for the intensity of thermal radiation of the layer (i) in the superstrate:

𝐼(𝑓, Ω, 𝑇 𝑖 ) = 2 𝑘 0 2
4𝜋 2 ℜ{𝑛 ̃0}4𝜋𝜖 𝑖 ′′ (𝑓)Θ(𝑓, 𝑇 𝑖 )𝐹 𝑖 [START_REF] Novotny | Principles of Nano-Optics[END_REF] with 𝑘 0 the wavevector in the superstrate.

It is usual to normalize this intensity by the intensity of the blackbody 𝐼 𝑏 (𝑓, 𝑇) = 2 𝑘 0 2 4𝜋 2 Θ(𝑓, 𝑇) which introduces the emissivity [START_REF] Howell | Thermal Radiation Heat Transfer[END_REF]. One can define, for each layer at thermal equilibrium, its emissivity:

𝜖 𝑖 (𝑓, Ω) = 𝐼(𝑓, Ω, 𝑇 𝑖 ) 𝐼 𝑏 (𝑓, 𝑇) (17)
To conclude this section, it is important to introduce the Kirchhoff's law that equalize the emissivity at a given frequency and angle of a given medium, and the absorption by this medium of an incident beam that originates from the same direction at the same frequency. This relation gives an indirect way to compute the thermal radiation by considering the absorption and allows to verify the computational codes [START_REF] Wang | Direct and Indirect Methods for Calculating Thermal Emission From Layered Structures With Nonuniform Temperatures[END_REF].

Numerical results

The thermal radiation of the optical component taken in the section 2 is now studied. The ns regime is chosen because of the higher temperature variations. The ambient temperature is chosen at 20°C (293.15K). The temporal, angular and wavelength pattern of thermal radiation are shown in Figure 5. Refractive index dispersion is not taken into account. The TE polarisation of the radiation is considered. In conclusion, the model is accurate to compute the thermal radiation of optical thin film illuminated by an arbitrary laser beam. It should be recall that the model depends on the knowledge of the optical indices of the materials that are poorly known in the infrared region.

TAILORING THE THERMAL RADIATION

The thermal radiation of solid object at a given temperature is usually quasi-isotropic and spread over a large wavelength bandwidth. Huge efforts have been made to tailor the thermal radiation with the help of nanostructures [START_REF] Zhuomin | Nano/Microscale Heat Transfer[END_REF]. Some articles show how to confine the thermal radiation in small wavelength bandwidth or in a small directional aperture thanks to 2D multilayers structures [START_REF] Zhao | Thermophotovoltaic emitters based on a two-dimensional grating/thin-film nanostructure[END_REF] or gratings [START_REF] Greffet | Coherent emission of light by thermal sources[END_REF]. Some works are also dedicated to the control of thermal radiation with 1D multilayer structure such as metallic Fabry Perot [START_REF] Wang | Direct and Indirect Methods for Calculating Thermal Emission From Layered Structures With Nonuniform Temperatures[END_REF], photonic crystals coated with SiC [START_REF] Lee | Coherent thermal emission from modified periodic multilayer structures[END_REF] or Ag [START_REF] Lee | Surface waves between metallic films and truncated photonic crystals observed with reflectance spectroscopy[END_REF]. In this article, we demonstrate that we can theoretically achieve coherent thermal radiation with the use of Zero Admittance Layer (ZAL) [START_REF] Amra | Energy density engineering via zero-admittance domains in all-dielectric stratified materials[END_REF], [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF] in 1D dielectric multilayer structures under total internal reflection.

The ZAL layer allows to make a resonant mode of the structure at a specific frequency. At this specific frequency, coherence of thermal radiation can be recovered thanks to the narrow bandwidth. The physical aspects under this phenomenon are similar to those involved in a Fabry Perot filter or in planar micro-microcavities [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF], [START_REF] Amra | Electromagnetic power provided by sources within multilayer optics: free-space and modal patterns[END_REF], [START_REF] Amra | Mutual coherence and conical pattern of sources optimally excited within multilayer optics[END_REF].

A multilayer, made of a quarter wave stack with Nb2O5 and SiO2 for, respectively, the high and low index material, and a ZAL layer of Nb2O5 has been simulated. The wavelength is chosen at 1µm. The structure is assumed to be at thermal equilibrium at a constant temperature of 300K. Optical parameters of the medium are given in Table 1. Results are shown in Figure 6. We can see that the thermal radiation is confined in a very narrow angular and wavelength region.

To compare with the literature, the full width at half maximum of the emissivity peak in wavelength (Δ𝜆) and in angle (Δ𝜃) are computed. We find Δ𝜆 = 0.4 nm, which defined a Q factor of 𝑄 = 𝜆 𝑐 /Δ𝜆 = 2500, and Δ𝜃 = 0.014° which defines a coherent length of 𝐿 𝑐 = 𝜆/Δ𝜃 = 71.4𝜆. The Q factor is 10 times higher than those of the structures found in [START_REF] Wang | Direct and Indirect Methods for Calculating Thermal Emission From Layered Structures With Nonuniform Temperatures[END_REF], [START_REF] Greffet | Coherent emission of light by thermal sources[END_REF], [START_REF] Lee | Coherent thermal emission from modified periodic multilayer structures[END_REF] and [START_REF] Lee | Surface waves between metallic films and truncated photonic crystals observed with reflectance spectroscopy[END_REF]. The coherence length is also higher. Dahan et al [START_REF] Dahan | Extraordinary coherent thermal emission from SiC due to coupled resonant cavities[END_REF] have made a coupled resonant cavity structure that have a larger coherence length but a smaller Q factor. It should be recalled that these results are theoretical ones. The Q factor and the coherence length will probably be reduced in practice. In conclusion, planar multilayer structures made of dielectric are useful to tailor thermal radiation. The Q factor of the structure and the spatial coherence length of the radiation are similar or improved compared to the literature. It is important to highlight the fact that our results were achieved in the near infrared. If one wants to move to the medium infrared region, the materials will have to change to Ge, ZnS and so on. Indeed, the optical materials are likely to be opaque in this region preventing the interference effects to occur.

Figure 1 :

 1 Figure 1: geometry of an optical multilayer (see text)

Figure 2 :

 2 Figure 2: Laser-induced temperature of a quarter wave stack illuminated by a nanosecond pulse (see text). (left) time dependence of the temperature at the top surface of the mirror and of the gaussian shape of the laser pulse. (right) temperature at fixed time with respect to the stack depth z.

Figure 3 :

 3 Figure 3: Laser-induced temperature of a quarter wave stack illuminated by a millisecond pulse (see text). (left) time dependence of the temperature at the top surface of the mirror and of the gaussian shape of the laser pulse. (right) temperature at fixed time with respect to the stack depth z.

Figure 4 :

 4 Figure 4: Variation with stack depth z of the normalized modulus squared of the tangential electric field Because of the confinement of the temperature field, the ns regime may give the opportunity to solve different inverse problems such as the extraction of the thermal parameters of the different layers from the measurements.

  . 〉 is a statistical average -𝑚, 𝑛 ∈ {𝑥, 𝑦, 𝑧} are the coordinates of the current density -𝐽 ̃𝑚,𝑒 (r ⃗, 𝑧, 𝑓) = ∫ 𝐽 𝑚,𝑒 (r ⃗, 𝑧, 𝑡)𝑒 𝑗2𝜋𝑓𝑡 𝑑𝑡 -𝜖 ′′ is the imaginary part of the dielectric permittivity of the medium energy of Planck's oscillator at the frequency f and temperature T in thermal equilibrium δ 𝑛𝑚 is the Kronecker symbol -𝛿 is the Dirac function

Figure 5 :

 5 Figure 5: The temporal, angular and wavelength pattern of the thermal radiation of a quarter wave stack illuminated by a nanosecond pulse (see text). (left) wavelength dependence of the intensity at normal incidence and at fixed time. (right) angular dependence at 10µm and at fixed time. Refractive index dispersion is not considered. Patterns in TE polarisation.

Figure 6 :

 6 Figure 6: Emissivity of a quarter wave stack with a ZAL layer. (left) wavelength dependence at fixed angles. (right) angular dependence at fixed wavelength

Table 1 :

 1 /BK7. The optical and thermal properties chosen for the materials are summarized in Table1. Optical and thermal parameters used for the simulations of the laser-induced temperature

	Materials	Nb2O5	SiO2	Air	BK7
	Refractive index @ 1µm	2.25 + 10 -4 j	1.45 + 10 -6 j	1	1.52
	Thermal conductivity (W/m/K)	1	0.5	2.5*10 -2	1.14
	Thermal diffusivity (m²/s)	4.3*10 -7	1.84*10 -7	2.05*10 -5	6.2*10 -7


Radhia Ayechi

Ilhem Boukhris

Julien Royer

ENERGY DECAY FOR A SYSTEM OF SCHR ÖDINGER EQUATIONS IN A WAVE GUIDE

Keywords: 2010 Mathematics Subject Classification. 47N50, 47A10, 35B40, 47B28, 47B44, 35J05, 35G45, 35P10 Systems of PDEs, damped Schrödinger equation, wave guide, eigenvalue asymptotics, Riesz basis. 1

We prove exponential decay for a system of two Schrödinger equations in a wave guide, with coupling and damping at the boundary. This relies on the spectral analysis of the corresponding coupled Schrödinger operator on the one-dimensional cross section. We show in particular that we have a spectral gap and that the corresponding generalized eigenfunctions form a Riesz basis.

Introduction

Let d ě 2, ą 0 and Ω " R d´1 ˆs0, r. All along the paper, a generic point in Ω is denoted by px, yq with x P R d´1 and y Ps0, r. We consider on Ω a system of Schrödinger equations # iB t u `∆u " 0, iB t v `∆v " 0, on R `ˆΩ.

(1.1)

Given a ą 0 and b P R ˚, damping and coupling are given by the boundary conditions $ ' & ' % B ν upt; x, 0q " iaupt; x, 0q `ibvpt; x, 0q, B ν vpt; x, 0q " ´ibupt; x, 0q, B ν upt; x, q " B ν vpt; x, q " 0, @t ą 0, @x P R d´1 .

(1.2)

We could similarly consider the problem with damping and/or coupling on both sides of the boundary. This problem is completed with the initial conditions u| t"0 " u 0 , v| t"0 " v 0 ,

where u 0 , v 0 P L 2 pΩq.

We will check that the problem (1.1)-(1.3) is well posed. If U " pu, vq is a solution, then for t ě 0 we consider the energy Ept; U q " }uptq} 2 L 2 pΩq `}vptq} 2 L 2 pΩq . A straightforward computation shows that E is a non-increasing function of time:

d dt Ept; U q " ´2a ż R d´1
|upt; x, 0q| 2 dx ď 0.

In this paper we are interested in the decay of this energy when the time t goes to `8.

There is already a large litterature about the energy decay for the Schrödinger equation or for the closely related damped wave equation, on compact or noncompact domains, with damping in the domain or at the boundary.

For the damped wave equation on a compact domain, it is known that as soon as we have effective damping in an open subset of the domain or of its boundary, then the energy goes to 0 [START_REF] Haraux | Stabilization of trajectories for some weakly damped hyperbolic equations[END_REF][START_REF] Lebeau | Équation des ondes amorties[END_REF]. The decay is uniform with respect to the initial condition (and hence exponential) if and only if all the trajectories of the corresponding classical problem (the rays of light) go through the damping region (see [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF] for the damping in the domain and [START_REF] Bardos | Microlocal ideas in control and stabilization[END_REF] for the damping at the boundary). Otherwise we have at least logarithmic decay with a loss of regularity [START_REF] Lebeau | Équation des ondes amorties[END_REF][START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF]. There are intermediate rates of decay when the set of undamped classical trajectories is small and the classical flow is unstable near these trajectories (see for instance [START_REF] Burq | Energy decay for damped wave equations on partially rectangular domains[END_REF][START_REF] Anantharaman | Sharp polynomial decay rates for the damped wave equation on the torus[END_REF]).

Similar results have been proved for the (undamped) Schrödinger and wave equations in unbounded domains. In this case we look at the energy on a compact subset. It goes to zero if the energy escapes to infinity. The contribution of high frequencies behaves as above. The local energy always goes to 0, at least with logarithmic decay and loss of regularity, and uniformly with decay faster than any negative power of t if and only if all the classical trajectories go to infinity. On the other hand the local energy of the contribution of low frequencies always decays uniformly, with a polynomial rate of decay. We refer for instance to [START_REF] Lax | Scattering Theory[END_REF][START_REF] Ralston | Solution of the wave equation with localized energy[END_REF][START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF][START_REF] Bouclet | Low frequency estimates and local energy decay for asymptotically Euclidean laplacians[END_REF][START_REF] Bony | Local Energy Decay for Several Evolution Equations on Asymptotically Euclidean Manifolds[END_REF].

We can also consider the damped Schrödinger or wave equation in an unbounded domain. If we are interested in the local energy decay, then the geometric condition for high frequencies is that all classical trajectories should go either to infinity or through the damping region. The contribution of low frequencies behaves as in the undamped case if the damping is localized (see for instance [AK02, AK07, AK10, BR14, Roy18]), while it behaves like the solution of some heat equation if the damping is effective at infinity (see [START_REF] Matsumura | On the asymptotic behavior of solutions of semi-linear wave equations[END_REF][START_REF] Malloug | Energy decay in a wave guide with damping at infinity[END_REF][START_REF] Joly | Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation[END_REF] and references therein).

Here we are interested in the global energy for damped Schrödinger equations on a wave guide with damping at infinity. The case of a single Schrödinger equation was discussed in [START_REF] Royer | Exponential decay for the Schrödinger equation on a dissipative wave guide[END_REF]. In that case it was already remarkable that the energy decays exponentially without the geometric control condition for the contribution of high frequencies. Here we have two equations, and only the first is damped (and again, the classical trajectories parallel to the boundary never see the damping). Moreover, the coupling itself is supported by the boundary and does not satisfy the geometric control condition. However, we observe that the energy of both components u and v goes to 0, and furthermore the decay is uniform and hence exponential. Our main result in this paper is the following.

Theorem 1.1. Let a ą 0 and b P R ˚. There exist γ ą 0 and C ą 0 such that for pu 0 , v 0 q P L 2 pΩq ˆL2 pΩq and t ą 0 we have

}uptq} L 2 pΩq `}vptq} L 2 pΩq ď Ce ´γt `}u 0 } L 2 pΩq `}v 0 } L 2 pΩq ˘,
where pu, vq is the solution of (1.1)-(1.3).

The proof of Theorem 1.1 is based on the spectral properties of the corresponding coupled Schrödinger operator. For a, b P R we consider on H " L 2 pΩq ˆL2 pΩq » L 2 pΩ, C 2 q the operator P a,b " ˆ´∆ 0 0 ´∆˙, (1.4) defined on the subspace DompP a,b q of functions U " pu, vq in H 2 pΩ; C 2 q such that, for all x P R d´1 , $ ' & ' % B ν upx, 0q " iaupx, 0q `ibvpx, 0q, B ν vpx, 0q " ´ibupx, 0q, B ν upx, q " B ν vpx, q " 0.

(1.5)

We will check in Section 2 that if a ě 0 then P a,b is a maximal dissipative operator on H , so by the usual Lummer-Phillips Theorem it generates a contractions semigroup pe ´itP a,b q tě0 on H . Then, given U 0 " pu 0 , v 0 q P DompP a,b q, we see that

U " pu, vq satisfies (1.1)-(1.3) if and only if # iB t U ´Pa,b U " 0, U | t"0 " U 0 . (1.6)
Moreover, if H is endowed with the natural norm, Theorem 1.1 is equivalent to the uniform exponential decay in LpH q of the propagator e ´itP a,b when t Ñ `8.

Theorem 1.2. Let a ą 0 and b P R ˚. There exist γ ą 0 and C ą 0 such that for t ą 0 we have › › e ´itP a,b › › LpH q ď Ce ´γt . By the Gearhart-Prüss Theorem (see for instance Theorem V.1.11 in [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]), we get uniform exponential decay for e ´itP a,b if we can prove that the resolvent of P a,b is well defined and uniformly bounded in an open upper half-plane which contains the real axis. Thus, Theorem 1.2 can be deduced from the following spectral result.

We denote by ρpP a,b q and σpP a,b q the resolvent set and the spectrum of P a,b , respectively.

Theorem 1.3. Let a ą 0 and b P R

˚.

(i) There exists γ 1 ą 0 such that any z P C with Impzq ą ´γ1 belongs to ρpP a,b q. (ii) There exist m P N ˚and C 1 ą 0 such that for z P ρpP a,b q we have

› › pP a,b ´zq ´1› › LpH q ď C 1 distpz, σpP a,b qq rms ,
where for s ą 0 we have set s rms " minps, s m q.

As usual in a wave guide, we use separation of variables and deduce the spectral properties of P a,b on Ω from the analogous properties for the corresponding operators on R d´1 and on the cross section s0, r. We will split the Laplacian P a,b on Ω as the sum of the usual Laplace operator on the first pd ´1q variables, and a Laplacian with boundary conditions on s0, r.

The damping and the coupling are encoded in the domain of the transverse operator. For a, b P R we consider on S " L 2 p0, ; C 2 q the operator

T a,b " ˆ´B 2 0 0 ´B2 ˙, (1.7) defined on the domain DompT a,b q " U P H 2 p0, ; C 2 q : U 1 p0q `iM a,b U p0q " 0, U 1 p q " 0 ( , (1.8)
where we have set

M a,b " ˆa b ´b 0 ˙.
(1.9)

Theorem 1.4. Let a ą 0 and b P R ˚. The spectrum of T a,b consists of a sequence pλ k q kPN ˚of eigenvalues. Moreover, (i) there exists m P N ˚such that all the eigenvalues have algebraic multplicity smaller than or equal to m, (ii) there exists γ 1 ą 0 such that Impλ k q ď ´γ1 for all k P N, (iii) there exists a Riesz basis of S which consists of generalized eigenfunctions of T a,b .

Here we are mostly interested in the dissipative case, but we will see in Section 2 that the adjoint of T a,b is T ´a,b , so T a,b is selfadjoint if a " 0 (then the eigenvalues are real and there exists an orthonormal basis of eigenfunctions), and for a ă 0 the properties are the same as for a ą 0, except that the eigenvalues have positive imaginary parts. To be complete we will also include in the intermediate results the already understood case a ą 0, b " 0. This corresponds to two independant equations, one with Neumann boundary conditions and one with damping at the boundary (see [START_REF] Royer | Exponential decay for the Schrödinger equation on a dissipative wave guide[END_REF]).

Theorem 1.4 is what we need on the transverse operator to prove Theorem 1.3, but we can give more precise spectral properties for T a,b . For example, we will see that T a,b satisfies a Weyl Law. If P 0 is the usual Dirichlet Laplacian on a bounded domain Ω 0 of R d , then the standard Weyl Law says that the number N 0 prq of eigenvalues of P 0 (counted with multiplicities) not greater than r satisfies

N 0 prq " rÑ`8 r d 2 ω d |Ω 0 | p2πq d ,
where ω d is the volume of the unit ball in R d and |Ω 0 | is the volume of the domain Ω 0 . This result has been improved and extended in many directions.

When a " b " 0 we have two decoupled Schrödinger equations with Neumann boundary conditions, so the eigenvalues of T 0 " T 0,0 are the n 2 π 2 { 2 , n P N, and these eigenvalues have multiplicity 2. It is easy to deduce that the number of eigenvalues of T 0 (counted with multiplicities) smaller than r ą 0 is of the form 2 ? r{π `Op1q.

For r ą 0 we denote by N a,b prq the number of eigenvalues of T a,b (counted with multiplicities) with real part smaller than r.

Theorem 1.5 (Weyl Law). Let a, b P R. We have

N a,b prq " 2 ? r π `O rÑ`8 p1q.
In Theorem 1.4 we have said that the sequence of multiplicities of the eigenvalues of T a,b is bounded. In fact, the maximum multiplicity m given there is the parameter m which appears in Theorem 1.3. We can be more precise about these multiplicities.

Proposition 1.6. Let a, b P R.

(i) All the eigenvalues of T 0,0 have geometric and algebraic multiplicities 2.

(ii) If pa, bq ‰ p0, 0q then all the eigenvalues of T a,b are geometrically simple.

(iii) If a 2 ą 4b 2 then all the eigenvalues of T a,b are algebraically simple.

(iv) If a 2 " 4b 2 ‰ 0 then all the eigenvalues of T a,b have algebraic multiplicity 2.

(v) If a 2 ă 4b 2 then the eigenvalues of T a,b can have algebraic multiplicity 1 or 2. More precisely, there exists a countably infinite subset Θ in R 2 such that all the eigenvalues of T a,b are simple if and only if pa, bq R Θ. If pa, bq P Ωztp0, 0qu then T a,b has one eigenvalue of algebraic multiplicity 2, and all the others are simple. In particular, in Theorem 1.3 we can choose m " 1 if a 2 ą 4b 2 or a 2 ă b 2 and pa, bq R Θ, and m " 2 otherwise.

Finally, we notice that we can see our system of two equations on a line segment as a problem on a graph with two edges and non-standard non-selfadjoint conditions at the common vertex (if we have coupling at both ends, then this gives a graph with two edges which have the same ends). Little is known for general non-selfadjoint quantum graphs (see [START_REF] Hussein | Maximal quasi-accretive Laplacians on finite metric graphs[END_REF][START_REF] Hussein | Non-self-adjoint graphs[END_REF] for general properties of nonselfadjoint quantum graphs). We also refer to [START_REF] Rivière | Spectrum of a non-selfadjoint quantum star graph[END_REF] for a non-selfadjoint Robin Laplacian on a star-shapped graph. In terms of non-selfadjoint quantum graphs, our analysis concerns a very particular example but provides much more precise spectral properties.

Organization of the paper. After this introduction, we give in Section 2 the basic properties for the operator P a,b on Ω and for the transverse operator T a,b on s0, r. We discuss in Section 3 the localization of the large eigenvalues of T a,b . This will give in particular the first two statements of Theorem 1.4, Theorem 1.5 and Proposition 1.6 for large eigenvalues. In Section 4 we finish the proof of Theorem 1.4 by proving the Riesz basis property and, in Section 5, we prove Theorem 1.3 from which Theorem 1.2 and hence Theorem 1.1 follow. Finally, we give in Section 6 more results about the eigenvalues of the transverse operator, in particular about low frequencies.

General properties of the coupled Schrödinger operators

In this section we give the basic properties of the operators P a,b and T a,b defined by (1.4)-(1.5) and (1.7)-(1.8), respectively. We will use the following version of the Trace Theorem.

Lemma 2.1. Let ε ą 0. There exists C ą 0 such that for all u P H 1 p0, q we have }u} 2 L 8 p0, q ď ε }u 1 } 2 L 2 p0, q `C }u} 2 L 2 p0, q . Proof. Let u P H 1 p0, q and let x 0 P r0, s be such that min |u| " |upx 0 q|. For any x P r0, s we have

|upxq| 2 ď |upx 0 q| 2 `ˇˇˇż x x 0 2upsqu 1 psq ds ˇˇˇď |upx 0 q| 2 `ε }u 1 } 2 L 2 p0, q `}u} 2 L 2 p0, q ε .
Since |upx 0 q| 2 ď }u} 2 L 2 p0, q , the conclusion follows. We recall that an operator A on a Hilbert space H is said to be sectorial if there exist γ 0 P R and θ P " 0, π 2 " such that Aϕ, ϕ H belongs to the sector Σ γ 0 ,θ " tζ P C, |argpζ ´γ0 q| ď θu for all ϕ P DompAq with }ϕ} H " 1. Then it is said to be maximal sectorial if some (and hence any) ζ P CzΣ γ 0 ,θ belongs to the resolvent set of A. We similarly define dissipative and maximal dissipative operators by replacing the sector Σ γ 0 ,θ by the half-space tζ P C, Impζq ď 0u. Proof. ' For U " pu, vq P DompP a,b q we have

P a,b U, U H " ´∆u, u L 2 pΩq ` ´∆v, v L 2 pΩq " }∇u} 2 L 2 pΩq `}∇v} 2 L 2 pΩq ´ia }up¨, 0q} 2 L 2 pR d´1 q ´2bIm up¨, 0q, vp¨, 0q L 2 pR d´1 q .
In particular, if a ě 0 we have

Im P a,b U, U H " ´a }up¨, 0q} 2
L 2 pR d´1 q ď 0, (2.1) so P a,b is dissipative. In general, with C given by Lemma 2.1 applied with ε "

1 2p1`|a|`|b|q we have Re P a,b U, U H ě }∇u} 2 L 2 pΩq `}∇v} 2 L 2 pΩq ´|b| `}up¨, 0q} 2 L 2 pR d´1 q `}vp¨, 0q} 2 L 2 pR d´1 q ě 1 2 }∇u} 2 L 2 pΩq `1 2 }∇v} 2 L 2 pΩq ´C |b| `}u} 2 L 2 pΩq `}v} 2 L 2 pΩq ˘. and ˇˇIm P a,b U, U H ˇˇď 1 2 }∇u} 2 L 2 pΩq `C |a| }u} 2 L 2 pΩq , so Re `Pa,b `Cp|a| `|b|q ˘U, U H ě ˇˇIm `Pa,b `Cp|a| `|b|q ˘U, U H ˇˇ.
This proves that P a,b is sectorial with γ 0 " ´Cp|a| `|b|q and θ " π 4 . ' Let λ ă γ 0 . For U P DompP a,b q we have

}pP a,b ´λqU } 2 H " }pP a,b ´γ0 qU } 2 H `pγ 0 ´λq 2 }U } 2 H `2pγ 0 ´λqRe pP a,b ´γ0 qU, U H ě pγ 0 ´λq 2 }U } 2 H . (2.2)
In particular, pP a,b ´λq is injective. Now let F " pf, gq P H . For U " pu, vq and Φ " pϕ, ψq in

H 1 pΩ; C 2 q we set QpU, Φq " ∇U, ∇Φ H ´λ U, Φ H ´i M a,b U p¨, 0q, Φp¨, 0q L 2 pR d´1 ;C 2 q .
This defines a sesquilinear form on H 1 pΩ; C 2 q. The computation above ensures that it is coercive. By Lemma 2.1, it is also continuous. Then, by the Lax-Milgram Theorem, there exists U P H 1 pΩ; C 2 q such that @Φ P H 1 pΩ; C 2 q, QpU, Φq " F, Φ H .

(2.3) Applied with Φ in C 8 0 pΩ; C 2 q, this shows that U belongs to H 2 pΩ; C 2 q and p´∆ ´λqU " F in the sense of distributions. Then, after an integration by parts,

@Φ P H 1 pΩ; C 2 q, B ν U p¨, 0q ´iM a,b U p¨, 0q, Φp¨, 0q L 2 pR d´1 ,Cq " 0.
This implies that U belongs to DompP a,b q, so pP a,b ´λqU " F and pP a,b ´λq is surjective. With (2.2) we see that pP a,b ´λq ´1 is bounded on H , so λ belongs to the resolvent set of P a,b . This proves that P a,b is maximal sectorial, and maximal dissipative if a ě 0.

' By direct computation we see that for U P DompP a,b q and U ˚P DompP ´a,b q we have P a,b U, U ˚ H " U, P ´a,b U ˚ H , so DompP ´a,b q Ă DompP å,b q and P å,b coincides with P ´a,b on DompP ´a,b q. On the other hand, with the same kind of argument as above, we check that if for some U ˚P H there exists F P H such that

@U P DompP a,b q, P a,b U, U ˚ H " U, F H , then U ˚belongs to DompP ´a,b q.
This proves that DompP å,b q Ă DompP ´a,b q. Finally we have proved that P å,b " P ´a,b and the proof of the proposition is complete.

We also need similar properties for the transverse operator T a,b .

Proposition 2.3. For a, b P R the operator T a,b is maximal sectorial on S, its spectrum consists of a sequence of isolated eigenvalues with finite multiplicities and T å,b " T ´a,b . If a ě 0 then T a,b is also maximal dissipative. If a ą 0 and b ‰ 0, then all the eigenvalues have negative imaginary parts.

Proof. The facts that T a,b is maximal sectorial, maximal dissipative if a ě 0, and that T å,b " T ´a,b are proved as for P a,b in Proposition 2.2. Since T a,b is maximal sectorial, its resolvent set is not empty. And since DompT a,b q is compactly embedded in S , its spectrum consists of isolated eigenvalues with finite multiplicities. Now assume that a ą 0 and b ‰ 0. By dissipativeness, the eigenvalues of T a,b have non-positive imaginary parts. Now assume that λ P R and U " pu, vq P DompT a,b q are such that T a,b U " λU . In particular, as in (2.1) we have

0 " Im T a,b U, U S " ´a |up0q| 2 .
This gives up0q " 0 and hence v 1 p0q " 0. We have ´v2 " λv on r0, s and v 1 p0q " v 1 p q " 0, so there exists n P N such that λ " n 2 π 2 { 2 . Then we have ´u2 " pn 2 π 2 { 2 qu, u 1 p q " 0 and up0q " 0. This implies that u " 0. Then u 1 p0q " 0, so vp0q " 0, which implies that v " 0. Thus, λ is not an eigenvalue of T a,b .

Transverse eigenvalues

Let a, b P R. In this section, we give more precise properties about the localization and the multiplicities of the eigenvalues of T a,b . When there is no ambiguity we omit the subscripts a, b of all the involved quantities.

As usual in this kind of context, it is easier to discuss the square roots of these eigenvalues. We set

Z " Z a,b " z P C : z 2 is an eigenvalue of T a,b ( . (3.1) Let ν " π .
As said in introduction, when a " b " 0 we have two decoupled Schrödinger equations with Neumann boundary conditions, so Z 0,0 " νZ, and for all n P N the eigenvalue ν 2 n 2 has multiplicity 2. An orthonormal basis of eigenfunctions is given by ˆ1 ?

2

A j e nν ˙nPN,jPt1,2u ,

where pA 1 , A 2 q is any orthonormal basis of C 2 and e nν pxq " 2 cospnνxq for all n P N and x P r0, s.

Let

µ ˘" µ ˘,a,b " a ˘δ 2
, where δ "

# ? a 2 ´4b 2 if a 2 ě 4b 2 , i ? 4b 2 ´a2 if a 2 ď 4b 2 .
For z P C and x P r0, s we set (i) z 2 is an eigenvalue of T if and only if φ ´pzqφ `pzq " 0, (3.5)

where we have set φ ˘pzq " φ ˘,a,b pzq " pz ´µ˘q e 2iz ´pz `µ˘q " zpe 2iz ´1q ´µ˘p e 2iz `1q.

(3.6)

Moreover, if a 2 ‰ 4b 2 the functions φ ´and φ `have no common zero. (ii) If φ ˘pzq " 0 we have kerpT ´z2 q " tA 1 e z , A 1 P kerpM ´µ˘q u (3.7) and ker `pT ´z2 q 2 " tA 2 e z `A1 ẽz , A 1 P kerpM ´µ˘q and pM ´µ˘q A 2 " η ˘pzqA 1 u , (3.8)

where

η ˘pzq " η ˘,a,b pzq " µ ˘`i pz 2 ´µ2 ˘q 2z 2 . (3.9)
Proof. ' Let z P C ˚and U P H 2 ps0, r, C 2 q. Then we have ´U 2 " z 2 U if and only if there exist A, Ã P C 2 such that @x Ps0, r, U pxq " Ae izx `Ãe ´izx .

(3.10)

Then U P DompT q if and only if à " e 2iz A and `p1 `e2iz qM `p1 ´e2iz qz ˘A " 0.

(3.11)

There exists a non trivial solution A of (3.11) if and only if z 2 p1 ´e2iz q 2 `azp1 ´e2iz qp1 `e2iz q `b2 p1 `e2iz q 2 " det `p1 `e2iz qM `p1 ´e2iz qz ˘" 0. (3.12)

Assume that b ‰ 0. We have e 2iz ‰ 1 if z is a solution of (3.12). Then (3.12) can be seen as a second order equation in z with z-dependant coefficients. The solutions are given by z ˘" µ ˘e2iz `1 e 2iz ´1, (3.13) and (3.5) follows. We conclude similarly when b " 0. Moreover, since e 2iz `1 ‰ 0, we see that if φ ´pzq " 0 " φ `pzq we have µ `" µ ´and hence a 2 " 4b 2 . ' If b ‰ 0 we have z ˘‰ µ ˘by (3.13). Then we can write p1 ´e2iz ˘ q " ´2µ z˘´µ˘a nd p1 `e2iz ˘ q " 2z z˘´µ˘, so (3.11) with z ˘holds if and only if A P kerpM ´µ˘q . This gives (3.7). Let σ P t´, `u and U 2 P kerppT ´z2 σ q 2 q. Let U 1 " pT ´z2 σ qU 2 P kerpT ´z2 σ q. By (3.7) there exists A 1 P kerpM ´µσ q such that U 1 " A 1 e zσ . Then, by (3.4) there exist A 2 , Ã2 P C 2 such that, for all x P r0, s,

U 2 pxq " A 1 ẽzσ pxq `A2 e izσx `Ã 2 e ´izσx .
Since U 1 2 p q " 0 we necessarily have Ã2 " e 2izσ A 2 , so that

U 2 " A 1 ẽzσ `A2 e zσ . (3.14)
Finally the condition U 1 2 p0q `iM U 2 p0q " 0 gives pM ´µσ qA 2 " η σ pz σ qA 1 .

(3.15)

Conversely, if U 2 satisfies (3.14) and (3.15) then it belongs to kerppT ´z2 σ q 2 q. The proof is similar when b " 0. Now we apply (3.5) to prove that the large eigenvalues of T " T a,b are close to the eigenvalues of the Neumann decoupled operator T 0 " T 0,0 . Proposition 3.2. Let a, b P R.

(i) Let ε ą 0. There exists R ε ą 0 such that for s P r0, 1s and z P Z sa,sb with |z| ě R ε we have distpz, νZq ă ε. In particular there exists n 0 P N ˚such that for all s P r0, 1s we have

Z sa,sb Ă Dp0, Rq Y ď |n|ěn 0 D ´nν, ν 6 ¯, where R " ˆn0 ´1 2 ˙ν.
(ii) For n ě n 0 the functions φ ´and φ `have exactly one zero in the disk D `nν, ν 6 ˘.

Proof. ' Assume by contradiction that the first statement does not hold. Then there exist sequences ps m q mPN in r0, 1s, pz m q mPN in C and pσ m q mPN in t˘u such that φ σm,sma,smb pz m q " 0, distpz m , νZq ě ε for all m P N and |z m | Ñ `8. In particular z m R νZ so s m b ‰ 0 and z m ‰ µ m , where we have set µ m " µ σm,sma,smb . Since µ m " s m µ σm,a,b is bounded we have

e 2izm " z m `µm z m ´µm ÝÝÝÝÑ mÑ`8
1, so distpz m , νZq Ñ 0 and we get a contradiction. The second statement follows by choosing ε " ν 6 (we can take R ν{6 larger to ensure that it is of the required form). ' For z P C we set φ 0 pzq " φ ˘,0,0 pzq " pe 2iz ´1qz.

(3.16)

Then, for all n P N ˚, φ 0 has one simple zero in the disk D `nν, ν 6 ˘and does not vanish on its boundary. On the other hand φ ˘does not vanish on the circle C `nν, ν 6 ˘for n large enough. Choosing n larger if necessary we also have on this circle |φ ˘pzq ´φ0 pzq| " |µ ˘| ˇˇe 2iz `1ˇˇă |φ 0 pzq| .

By the Rouché Theorem, φ ˘has exactly one simple zero in D `nν, ν 6 ˘.

For n ě n 0 we denote by z n,˘" z n,˘,a,b the unique solution of φ ˘pzq " 0 in D `nν, ν 6 ˘. If a 2 " 4b 2 we can simply write z n instead of z n,`o r z n,´.

The following proposition gives a rough localization of the high frequency eigenvalues and in particular the Weyl Law for T . Proposition 3.3. Let a, b P R. Let n 0 P N ˚be given by Proposition 3.2. The spectrum of T is contained in

D " Dp0, R 2 q Y ď něn 0 D ˆn2 ν 2 , nν 2 2 ˙.
The sum of the algebraic multiplicities of the eigenvalues of T in Dp0, R 2 q is 2n 0 and, for n ě n 0 ,

(i) if a 2 ‰ 4b 2 , then T has exactly two distinct simple eigenvalues in D ´n2 ν 2 , nν 2 2 ¯,
given by z 2 n,´a nd z 2 n,`,

(ii) if a 2 " 4b 2 , then T has a unique double eigenvalue in D ´n2 ν 2 , nν 2 2 ¯, given by z 2 n . In particular, for r ą 0 we have

2 ? r ν ´1 ď N prq ď max ˆ2n 0 , 2 ? r ν `3˙.
Proof. Let n ě n 0 . The operator T s " T sa,sb is analytic with respect to the parameter s in the sense of Kato (family of type B, see [START_REF] Kato | Perturbation Theory for linear operators[END_REF]). By Proposition 3.2, the circle C `n2 ν 2 , nν 2 2 ˘is included in the resolvent set of T s for all s P r0, 1s, so the number of eigenvalues (counted with multiplicities) of T s in the disk D n " D `n2 ν 2 , nν 2 2 ȋs independant of s P r0, 1s. Since T 0 has exactly one double eigenvalue in D n , the number of eigenvalues (counted with multiplicities) of T " T 1 in D n is also equal to 2.

If a 2 ‰ 4b 2 , we already know that T has two distinct eigenvalues z 2 n,´a nd z 2 n,ì n D n . They are necessarily simple. If a 2 " 4b 2 , then z 2 n is the unique eigenvalue of T in D n . It has algebraic multiplicity 2 (we know from Proposition 3.1 that it is geometrically simple if pa, bq ‰ p0, 0q).

Similarly, T has the same number of eigenvalues as T 0 in Dp0, R 2 q, that is 2n 0 (counted with multiplicities). Now let r ą 0. If r ď R 2 " `n0 ´1 2 ˘2ν 2 , then N pRq ď 2n 0 . Otherwise, there exists n ě n 0 such that r P "`n ´1 2 ˘2ν 2 , `n `1 2 ˘2ν 2 ‰ . Then we have

2 ? r ν ´1 ď 2n ď N prq ď 2n `2 ď 2 ? r ν `3,
which concludes the proof.

We finish this section by an asymptotic expension for the large eigenvalues of T . We already know that all the eigenvalues have negative imaginary parts when a ą 0 and b ‰ 0, so this asymptotics gives in particular a spectral gap for T in this case. and the conclusions follow.

Riesz basis property

For b P R the operator T 0,b is selfadjoint with compact resolvent, so there exists an orthonormal basis of eigenfunctions for T 0,b .

In this section we consider pa, bq P R ˚ˆR and we prove that there exists a Riesz basis of S made with generalized eigenfunctions of T " T a,b . Since T is not selfadjoint, we do not necessarily have a Hilbert basis of eigenvectors. However, we will prove that the generalized eigenvectors are not too far, in a suitable sense, from being orthogonal. We have no control on these vectors for n small, but when n is large we will see that they are in fact close to a family of eigenvectors of T 0,0 which forms a Riesz basis.

We recall that a family pΨ k q kPN ˚in S is a Riesz basis if the operator " 2 pN ˚q Ñ S pu k q kPN ˚Þ Ñ ř 8 k"1 u k Ψ k is well defined, bounded, bijective and boundedly invertible. Then for any U P S there is a unique sequence pu k q kPN ˚P 2 pN ˚q such that U "

ř 8 k"1 u k Ψ k and, for some C ě 1 independant of U , C ´1 8 ÿ k"1 |u k | 2 ď }U } 2 S ď C 8 ÿ k"1 |u k | 2 . (4.1)
For general results about Riesz bases we refer for instance to [START_REF] Agranovich | On series with respect to root vectors associated with forms having symmetric principal part[END_REF].

If a 2 ‰ 4b 2 then for all n ě n 0 (n 0 P N ˚given by Proposition 3.2) we set

Φ 2n`1 " A ´ez n,´a nd Φ 2n`2 " A `ez n,`, (4.2)
where A ˘" ˆµb ˙P kerpT ´µ˘q (4.3) (we omit the subscripts a, b). Then Φ 2n`1 , Φ 2n`2 P DompT q, T Φ 2n`1 " z 2 n,´Φ 2n`1 and T Φ 2n`2 " z 2 n,`Φ 2n`2 . Similarly, if a 2 " 4b 2 we set for n ě n 0

Φ 2n`1 " A 1 e zn , Φ 2n`2 " A 1 ẽzn `A2,n e zn , (4.4)
where

A 1 " ˆa 2 ´b˙a nd A 2,n " ˆηpz n q 0 ˙.
We recall that the parameter ηpz n q is defined in (3.9) (µ `" µ ´in this case). We have

A 2,n " A 2,8 `Opn ´1q, where A 2,8 " ˆi 2 0 ˙. (4.5)
In particular, choosing n 0 larger if necessary, we can assume that ηpz n q ‰ 0 for n ě n 0 . Then we have Φ 2n`1 , Φ 2n`2 P DompT q, T Φ 2n`1 " z 2 n Φ 2n`1 and T Φ 2n`2 " z 2 n Φ 2n`2 `Φ2n`1 .

In both cases, we also consider a basis pΦ k q 1ďkď2n 0 of generalized eigenfunctions for the subspace of S spaned by the generalized eigenspaces corresponding to the eigenvalues of T in Dp0, R 2 q (see Proposition 3.3).

All this defines a family pΦ k q kPN ˚of generalized eigenfunctions for T . Our purpose is to prove that this is a Riesz basis of S . Notice that we could normalize these vectors, but this is not necessary ((4.10) below is enough).

We begin with the following computation.

Lemma 4.1. (i) We have }e z } 2 L 2 p0, q Ý ÝÝÝÝÝÝ Ñ |Repzq|Ñ`8 ImpzqÑ0
2 .

(4.6) (ii) There exists c ą 0 such that for z P C with |Impzq| ď 1 we have We can first consider separately the vectors Φ k for 1 ď k ď 2n 0 , and then the vectors Φ 2n`1 and Φ 2n`2 for each n ě n 0 . It is obvious that a finite family of linearly independant vectors is a Riesz basis of the finite dimensional subspace that it spans. The interest of the following proposition is that we can take the same constant for each of these finite subfamilies of eigenfunctions.

}ẽ z } L 2 p0, q ď c |z| . ( 4
Proposition 4.2. There exists C ą 0 such that for all pu k q kPN ˚P 2 pN ˚q we have

C ´1 2n 0 ÿ k"1 |u k | 2 ď }U 0 } 2 S ď C 2n 0 ÿ k"1 |u k | 2 and C ´1`| u 2n`1 | 2 `|u 2n`2 | 2 ˘ď }U n } 2 S ď C `|u 2n`1 | 2 `|u 2n`2 | 2 ˘, where we have set U 0 " ř 2n 0 k"1 u k Φ k and, for n ě n 0 , U n " u 2n`1 Φ 2n`1 `u2n`2 Φ 2n`2 .
Proof. ' The vectors pΦ 1 , . . . , Φ 2n 0 q are linearly independant by definition, and we see directly that for n ě n 0 the vectors Φ 2n`1 and Φ 2n`2 are not colinear. It remains to check that we have the estimates with a constant independant of n. ' Assume that a 2 ‰ 4b 2 . By Lemma 4.1 we have

}Φ 2n`1 } 2 S " |A ´|2 C 2 › › e z n,´› › 2 L 2 p0, q Ý ÝÝÝ Ñ nÑ`8 2 |A ´|2 C 2 ,
and similarly }Φ 2n`2 } 2 S goes to 2 |A `|2 C 2 . Thus there exists C 1 ě 1 such that, for all k P N ˚,

C ´1 1 ď }Φ k } 2 S ď C 1 . (4.10)
In particular,

}U n } 2 S ď 2 |u 2n`1 | 2 }Φ 2n`1 } 2 S `2 |u 2n`2 | 2 }Φ 2n`2 } 2 S ď 2C 1 `|u 2n`1 | 2 `|u 2n`2 | 2 ˘.
(4.11)

' Since A ´and A `are not colinear in C 2 there exists ε ą 0 such that

| A ´, A ` C 2 | ď p1 ´εq |A ´|C 2 |A `|C 2 . Then | Φ 2n`1 , Φ 2n`2 S | " | A ´, A ` C 2 | ˇˇ e z n,
´, e z n,` L 2 p0, q ˇď

p1 ´εq |A ´|C 2 |A `|C 2 › › e z n,´› › L 2 p0, q › › e z n,`› › L 2 p0, q ď p1 ´εq }Φ 2n`1 } S }Φ 2n`2 } S ,
and hence

}U n } 2 S ě |u 2n`1 | 2 }Φ 2n`1 } 2 S `|u 2n`2 | 2 }Φ 2n`2 } 2 S ´2 |u 2n`1 | |u 2n`2 | | Φ 2n`1 , Φ 2n`2 S | ě ε `|u 2n`1 | 2 }Φ 2n`1 } 2 S `|u 2n`2 | 2 }Φ 2n`2 } 2 S ě εC ´1 1 `|u 2n`1 | 2 `|u 2n`2 | 2 ˘. The proof is complete if a 2 ‰ 4b 2 .
' Now assume that a 2 " 4b 2 . By Lemma 4.1 and (4.5) we have

}Φ 2n`1 } 2 S Ý ÝÝÝ Ñ nÑ`8 2 |A 1 | 2 C 2 , }Φ 2n`2 } 2 S Ý ÝÝÝ Ñ nÑ`8 2 |A 2,8 | 2 C 2 .
Since A 1 and A 2,8 are not colinear, there exists ε ą 0 such that for n large enough we have

| A 1 , A 2,8 | ď p1 ´εq |A 1 | |A 2,8 |.
We can proceed as in the previous case. Now we consider pairs of generalized eigenfunctions which are not associated to the same subfamilies of eigenfunctions.

We set S 0 " spanpΦ 1 , . . . , Φ 2n 0 q and, for n ě n 0 , S n " spanpΦ 2n`1 , Φ 2n`2 q. Proposition 4.3. There exists C ą 0 such that for n ě n 0 , m P N ˚, U n P S n and U n`m P S n`m we have

| U n , U n`m S | ď C }U n } S }U n`m } S nm .
Proof. We begin with the case a 2 ‰ 4b 2 . We apply (4.8) with z " z n,´a nd ζ " z n`m,´. With Proposition 3.4 we get

ˇˇ Φ 2n`1 , Φ 2pn`mq`1 ˇˇ" |A ´|2 ˇˇ e z n,
´, e z n`m,´ L 2 p0, q ˇˇÀ 1 nm and then

ˇˇ u 2n`1 Φ 2n`1 , u 2pn`mq`1 Φ 2pn`mq`1 ˇˇÀ }U n } S }U n`m } S nm .
We estimate similarly

| Φ 2n`1 , Φ 2m`2 |, | Φ 2n`2 , Φ 2m`1 | and | Φ 2n`2 , Φ 2m`2 |
, and the conclusion follows in this case. For the case a 2 " 4b 2 we proceed similarly, now using (4.8), (4.9) and the fact that by (4.7) we have

ˇˇ ẽzn , ẽz n`m ˇˇÀ 1 npn `mq À 1 nm .
To prove that the family pΦ k q kPN is a Riesz basis, we compare it with a family of eigenfunctions for the model operator T 0 .

Assume that a 2 ‰ 4b 2 . For n P N we set Φ 0 2n`1 " A ´enν and Φ 0 2n`2 " A `enν . (4.12)

Since pA ´, A `q is a basis of C 2 , there exists C 0 ě 1 such that for V P C 2 and the unique pv ´, v `q P C 2 such that V " v ´A´`v`A`w e have

C ´1 0 `|v ´|1 `|v `|2 ˘ď }V } 2 C 2 ď C 0 `|v ´|1 `|v `|2 ˘.
Let U P S . There exist u ´, u `P L 2 p0, q unique such that for almost all y Ps0, r U pyq " u ´pyqA ´`u `pyqA `.

Since pe nν q nPN is an orthonormal basis of L 2 p0, q, there exist unique sequences pu ń q nPN and pu ǹ q nPN in 2 pN ˚q such that U pyq "

ÿ nPN u ń e nν pyqA ´`ÿ nPN u ǹ e nν pyqA `.
Moreover,

C ´1 0 ÿ nPN `ˇu ń ˇˇ2 `ˇu ǹ ˇˇ2 ˘ď }U } 2 S ď C 0 ÿ nPN `ˇu ń ˇˇ2 `ˇu ǹ ˇˇ2 ˘.
This means that the family pΦ 0 k q kPN ˚is a Riesz basis of S . Now assume that a 2 " 4b 2 . Then we get the same conclusion if we set Φ 0 2n`1 " A 1 e nν and Φ 0 2n`2 " A 2,8 e nν .

(4.13)

Proposition 4.4. The family pΦ k q kPN ˚is a Riesz basis of S .

Proof. ' We consider the map Θ :

" 2 pN ˚q Ñ S pu k q kPN Þ Ñ ř `8 k"1 u k Φ k
and we prove that Θ is well defined, continuous, bijective and has continuous inverse.

' Let pU n q nPN be a sequence in S with U n P S n for all n P N " t0u Y tn 0 , n 0 1, . . . u and ř nPN }U n } 2 S ă `8. For N ě n 0 and p P N we have

› › › › › N `p ÿ n"N U n › › › › › 2 S ´N`p ÿ n"N }U n } 2 S " N `p ÿ n"N N `p´n ÿ m"1 2Re U n , U n`m S , so by Proposition 4.3 ˇˇˇˇˇ› › › › › N `p ÿ n"N U n › › › › › 2 S ´N`p ÿ n"N }U n } 2 S ˇˇˇˇˇÀ `8 ÿ n"N }U n } S n `8 ÿ m"1 }U n`m } S m À g f f e `8 ÿ n"N 1 n 2 `8 ÿ n"N }U n } 2 S .
(4.14)

This proves in particular that the series ř nPN U n converges in S and that there exists C ą 0 independant of the sequence pU n q such that

› › › › › ÿ nPN U n › › › › › 2 S ď C ÿ nPN }U n } 2 S .
With Proposition 4.2, this proves that Θ is well defined and bounded. ' For n ě n 0 we set

Π n " ´1 2iπ ż C `n2 ν 2 , nν 2 2
˘pT ´ζq ´1 dζ P LpS q.

Then S n " kerpΠ n ´Id S q, and S j Ă kerpΠ n q for all j P N z tnu. Now assume that the sequence u " pu k q kPN is such that Θpuq " 0. For n P N we have u 2n`1 Φ 2n`1 ù2n`2 Φ 2n`2 " Π n pΘpuqq " 0, so u 2n`1 " u 2n`2 " 0. Since this holds for all n ě n 0 we have ř 2n 0 k"1 u k Φ k " 0, and then u 1 " ¨¨¨" u 2n 0 " 0. This proves that Θ is injective.

' By (4.14) there exists

N ě n 0 independant of U such that ˇˇˇˇˇ› › › › › N `p ÿ n"N U n › › › › › 2 S ´N`p ÿ n"N }U n } 2 S ˇˇˇˇˇď 1 2 `8 ÿ n"N }U n } 2 S . (4.15)
Assume by contradiction that there exists a family pU p n q pPN,nPN such that U p n P S n for all p P N and n P N ,

ř nPN }U p n } 2 S " 1 for all p P N but › › › › › ÿ nPN U p n › › › › › S Ý ÝÝÝ Ñ pÑ`8 0.
For p P N we set

V p 1 " ÿ nPN ,năN U p n , V p 2 " ÿ něN U p n .
After extracting a subsequence if necessary, we can assume that V p 1 has a limit V P spanpΦ k q kď2N in S . Since V p 1 `V p 2 goes to 0, V p 2 goes to ´V , so V also belongs to spanpΦ k q kě2N `1. By injectivity of Θ, this implies that V " 0. Then U n goes to 0 for all n ă N , and hence

`8 ÿ n"N }U n } 2 S Ý ÝÝÝ Ñ pÑ`8 1.
Since V p 2 Ñ 0, this gives a contradiction with (4.15). Thus there exists C ą 0 independant of the sequence pU n q such that

› › › › › ÿ nPN U n › › › › › 2 S ě C ´1 ÿ nPN }U n } 2 S .
With Proposition 4.2 we deduce that for u P 2 pN ˚q we have }Θpuq} 2 S Á }u} 2 2 pN ˚q . ' To prove that Θ is surjective we follow the proof of [Kat80, Theorem V.2.20], except that the reference basis is not orthogonal. We denote by Θ 0 the map defined as Θ with the family pΦ k q kPN ˚replaced by pΦ 0 k q kPN ˚defined by (4.12) or (4.13). Since we already know that the family pΦ 0 k q kPN ˚is a Riesz basis, Θ 0 is boundedly invertible. Then ΘΘ ´1 0 ´Id S P LpS q is the map ΘΘ ´1 0 ´Id S :

8 ÿ k"1 u k Φ 0 k Þ Ñ 8 ÿ k"1 u k pΦ k ´Φ0 k q.
For u " pu k q kPN ˚P 2 pN ˚q and N P N we have

› › › › › 8 ÿ k"N `1 u k pΦ k ´Φ0 k q › › › › › 2 S ď }u} 2 2 pN ˚q 8 ÿ k"N `1 › › Φ k ´Φ0 k › › 2 S .
By Proposition 3.4, (4.5) and (4.7) we have

› › Φ k ´Φ0 k › › 2 S À
1 k 2 , so ΘΘ ´1 0 ´Id is the limit in LpS q of a family of finite rank operators, and hence it is compact. This implies that ΘΘ ´1 0 is a Fredholm operator. Since we already know that it is injective, it is surjective. Then Θ is surjective and the proof is complete.

Resolvent for the Schrödinger operator on the wave guide

In this section we prove Theorem 1.3 for a ą 0 and b P R ˚. For this we deduce spectral properties of P " P a,b on H from those of T " T a,b on S . The intermediate result (Proposition 5.1 below) is valid for any a, b P R with pa, bq ‰ p0, 0q (in general we could proceed similarly, repeating the eigenvalues according to their geometric multiplicities, but the model case a " b " 0 is already clear).

We denote by pλ k q kPN ˚the sequence of (distinct) eigenvalues of T . These eigenvalues have finite algebraic multiplicities and we know that for k large enough the multiplicity of λ k is 1 if a 2 ‰ 4b 2 and 2 if a 2 " 4b 2 . In particular, if we denote by m k P N ˚the multiplicity of the eigenvalue λ k , we know that the sequence pm k q kPN is bounded. We denote by m its maximum.

For u P L 2 pR d´1 q and V P S we set u b V : px, yq P Ω Þ Ñ upxqV pyq. Then, if we denote by L the usual selfadjoint realization of the Laplacian on R d´1 , we have

P " L b Id L 2 pωq `Id L 2 pR d´1 q b T .
Since the spectrum of L is the half-line r0, `8r, it is known (see for instance Section XIII.9 in [START_REF] Reed | Method of Modern Mathematical Physics, volume IV, Analysis of Operators[END_REF]) that the spectrum of P is given by Σ " tλ k `r, k P N ˚, r P r0, `8ru .

We can recover this fact directly in our context. Let k P N ˚and r ě 0. We consider an eigenfunction Ψ k P S corresponding to the eigenvalue λ k of T , and a sequence pu n q nPN in H 2 pR d´1 q such that }u n } L 2 pR d´1 q " 1 for all n P N and pL ´rqu n Ñ 0 in L 2 pR d´1 q. Then for all n P N the function u n b Ψ k belongs to DompPq and in H we have

`P ´pλ k `rq ˘pu n b Ψ k q " `pL ´rqu n ˘b Ψ k Ý ÝÝÝ Ñ nÑ`8 0.
This proves that λ k `r P σpPq, and hence Σ Ă σpPq. The converse inclusion will be a part of Proposition 5.1 below.

Let k P N ˚. By Proposition 3.1, λ k is an eigenvalue of T of geometric multiplicity 1. There exist Ψ k,1 , . . . , Ψ k,m k such that pT ´λk qΨ k,1 " 0 and pT ´λk qΨ k,j " Ψ k,j´1 for j P t2, . . . , m k u. By Proposition 4.4, we can choose these vectors in such a way that pΨ k,j q kPN ˚,1ďjďm k is a Riesz basis of S .

Let F P H » L 2 pR d´1 , S q. For almost all x P R d´1 we have F px, ¨q P S , so there exist f k,j pxq, k P N ˚, 1 ď j ď m k , such that, in S , F px, ¨q "

ÿ kPN ˚mk ÿ j"1 f k,j pxqΨ k,j .
(5.1) Moreover, by the Riesz basis property, there exists C ě 1 independant of F such that, for almost all x P R d´1 ,

C ´1 ÿ kPN ˚mk ÿ j"1 |f k,j pxq| 2 ď }F px, ¨q} 2 S ď C ÿ kPN ˚mk ÿ j"1 |f k,j pxq| 2 .
This will prove that ζ P ρpPq. Let ζ 0 P ρpPq. We can check that for F P H we have RpζqRpζ 0 qF " Rpζ 0 qRpζqF . On the other hand we have pP ´ζ0 q `RpζqF ´Rpζ 0 qF ˘" pζ ´ζ0 qRpζqF so, by (5.5), R satisfies the resolvent identity RpζqF ´Rpζ 0 qF " pζ ´ζ0 qRpζ 0 qRpζqF " pζ ´ζ0 qRpζqRpζ 0 qF.

Applied with F " pP ´ζ0 q ´1U this gives RpζqpP ´ζqU " RpζqF ´pζ ´ζ0 qRpζqU " U.

This proves (5.6) and completes the proof.

6. More about the low frequency transverse eigenvalues

In Section 3 we did not say much about the low frequency eigenvalues of the transverse operator T " T a,b . We were quite accurate with the large eigenvalues, and then we said that the part of the spectrum in Dp0, R 2 q (see Proposition 3.3) consists of a finite number of eigenvalues with finite multiplicities, and negative imaginary parts if a ą 0 and b ‰ 0. That was enough to prove that there is a spectral gap for P " P a,b and hence the local energy decay for (1.1). In this final section we provide more information about these low frequency eigenvalues.

We consider a ě 0 and b P R such that pa, bq ‰ p0, 0q. Taking the adjoint, we get similar results for the case a ă 0 (for the proofs the roles of φ ´and φ `are reversed).

We recall from Proposition 3.1 that the eigenvalues of T are geometrically simple. In the following proposition we discuss their algebraic simplicity. Proposition 6.1. Let a ě 0 and b P R such that pa, bq ‰ p0, 0q.

(i) If a 2 " 4b 2 then the eigenvalues of T are not simple.

(ii) Assume that a 2 ‰ 4b 4 . Let σ P t˘u and let z P C ˚be a zero of φ σ . Then z 2 is an algebraically simple eigenvalue of T if and only if z is a simple zero of φ σ . (iii) All the zeros of φ `are simple. If a 2 ą 4b 2 then all the zeros of φ ´are simple.

There exists a countably infinite subset Θ `of pa, bq P R ˚ˆR|a 2 ă 4b 2 ( such that φ ´has a multiple zero if and only if pa, bq P Θ `.

Proof. ' If a 2 " 4b 2 then pM ´µq 2 " 0 so, by (3.8), kerppT ´z2 q 2 q is at least of dimension 2 for any z P Z. ' Now we assume that a 2 ‰ 4b 2 . We consider z P Z and σ P t˘u such that φ σ pzq " 0. We have ker `pM ´µσ q 2 ˘" kerpM ´µσ q so from (3.8) we see that ker `pT ´z2 q 2 ˘" kerpT ´z2 q if and only if η σ pzq ‰ 0. On the other hand we have φ 1

˘pzq " e 2iz ´1 `2i pz ´µσ qe 2iz . Then we can check that η σ pzq " 0 if and only if φ 1 σ pzq " 0. This gives the second statement. ' If φ `pzq " 0 and a ą 0 we have Re `µ``i pz 2 ´µ2 `q˘" Repµ `q ´ Impz 2 q ` Impµ 2 `q ą 0, (6.1) so η `pzq ‰ 0 and the zeros of φ `are simple. If a 2 ą 4b 2 then µ ´is real positive and we similarly see that the zeros of φ ´are simple. If a " 0 we have z P R so µ ``i pz 2 ´µ2 `q " i |b| `i z 2 `i b 2 ‰ 0 and, again, the zeros of φ `are simple. ' Assume that there exists z P Z such that φ ´pzq " φ 1 ´pzq " 0. We assume for instance that Repzq ě 0 and Impzq ď 0. We have e 2iz `1 ‰ 0, µ ´" e 2iz ´1 e 2iz `1z (6.2) and sinp2z q `2z " 0. (6.3) This last equality implies in particular that z is not real or purely imaginary. We write 2z " ξ `iκ with ξ ą 0 and κ ă 0. Then (6.3) gives # coshpκq sinpξq `ξ " 0, sinhpκq cospξq `κ " 0. (6.4)

In particular cospξq ă 0 and sinpξq ă 0. Then ' Let k P N. The left-hand side of (6.6) has a positive derivative in I k " ‰ p2k `1qπ, p2k `3 2 qπ ‰ , it goes to ´8 at p2k `1qπ and is positive at p2k `3 2 qπ. We denote by ξ k the unique solution of (6.6) in I k . Then we define κ k by (6.5) and we set

z k " ξ k `iκ k 2 . (6.7)
We finally define µ k by (6.2). Notice that if Repµ k q ď 0 then there exist α ď 0 and β P R such that z k P Z α,β and µ k " µ `,α,β or µ k " µ ´,α,β . Since Impz 2 k q ă 0 this gives a contradiction and proves that Repµ k q ą 0. Then we set

a k " 2Repµ k q ą 0 and b k " 1 2 b a 2 k `Impµ k q 2 ą 0.
' If a ą 0 and b P R ˚are such that a 2 ă 4b 2 and T a,b has an eigenvalue with algebraic multiplicity greater than 1, then this eigenvalue is necessarily z 2 k for some k P N, we have µ ´,a,b " µ k and, finally, a " a k and |b| " b k . Conversely, T a k ,b k and T a k ,´b k have an eigenvalue z 2 k of algebraic multiplicity greater than 1 for all k P N. This proves that T a,b has a non-simple eigenvalue if and only if pa, bq belongs to Ť kPN tpa k , b k q, pa k , ´bk qu and concludes the proof. For a single equation with damping at the boundary, is is proved in [Roy15] that for each n P N there is exaclty one square root of an eigenvalue with real part in snν, pn `1qνr. This came from the continuity of the spectrum with respect to the absorption index a and the fact that the square roots of the eigenvalues do not have their real parts in νN. This ensured in particular that each eigenvalue of the transverse operator is simple. Here we prove an analogous result, but the situation is not as simple as it was for a single equation. In this case z is unique and is given by

z " nν ´inν c 4b 2 a 2 ´1. (6.9)
This is a simple zero of φ

´.

Proof. ' We easily see that nν belongs to Z if and only if b " 0, and that in this case it is a simple zero of φ ´and not a zero of φ `.

' Assume that z P Z satisfies Repzq " nν ą 0 and Impzq ‰ 0. We necessarily have a ą 0 and Impzq ă 0 so e 2iz " e ´2 Impzq Ps1, `8r.

We set

κ " e 2iz `1 e 2iz ´1 Ps1, `8r. By (3.13) we have z " κµ `or z " κµ ´. If a 2 ě 4b 2 then µ `and µ ´are real, which gives a contradiction. Thus we have a 2 ă 4b 2 . Since Impµ `q ą 0 we cannot have z " κµ `. Therefore z " κµ ´. Since Repµ ´q " a 2 we necessarily have κ " 2nν a , (6.10) which gives (6.9) and implies 2nν ą a. Then we can write

e 2nπ b 4b 2 a 2 ´1 ´2nν `a 2nν
´a " e 2iz ´κ `1 κ ´1 " 0. (6.11)

Conversely, if (6.8) holds then with z defined by (6.9) we have z " κµ ´with κ given by (6.10), so the equality (6.11) now gives φ ´pzq " 0, and hence z P Z.

Finally, for z given by (6.9) we have Im `φ1 ´pzq ˘" Im `e2iz ´1 `2i µ ´pκ ´1qe 2iz ˘" 2 pκ ´1qe 2iz Repµ ´q ‰ 0, so φ 1 ´pzq ‰ 0 and z is a simple zero of φ ´ For n P N we set C n " tz P C : nν ă Repzq ă pn `1qνu. We recall that if a 2 ‰ 4b 2 then the functions φ ´and φ `have no common zeros. Proposition 6.3. Let a ě 0 and b P R. Let n P N.

(i) If a ą 0 and b " 0 then φ ´pnνq " 0 (this is a simple zero), φ ´has no zero in C n and φ `has a unique zero in C n (which is also simple). (ii) If a 2 ą 4b 2 ą 0 then φ ˘has a unique zero ζ n,˘i n C n (and this zero is simple). (iii) If a 2 " 4b 2 ą 0 then φ `" φ ´has a unique zero ζ n in C n , and this zero is simple (therefore it is a zero of order 2 for the product φ ´φ`) .

(iv) In particular, for s ą 0 small enough we have ζ n,˘p sq P C n . Moreover, by Proposition 6.1 applied to T s " T sa,sb , the eigenvalues ζ n,˘p sq 2 of T s are simple (we can also abserve that n 2 ν 2 has multiplicity 2 for T 0 and splits into two distinct eigenvalues z n,˘p sq 2 of T s , so each of these two eigenvalues is necessarily simple for s ą 0 small). We proceed similarly around pn `1qν, and we see that the zero pn `1qν of φ ˘,0 moves to C n`1 for s ą 0 small. Finally, for s ą 0 small the functions φ ˘,s have exactly one simple zero in C n . We recall that by Proposition 3.1 applied to T s , the zeros of φ `,s and φ ´,s cannot meet. Then the functions φ ˘,s have exactly one simple zero in C n for any s Ps0, 1s . All this holds in particular for s " 1 and the first statement is proved for n ‰ 0.

We proceed similarly for n " 0. 0 is a zero of φ ˘,0 of multiplicity 2, which splits into two opposite simple zeros for s ą 0 small (once squared, they correspond to the same eigenvalue of T s ). One of these two zeros is in C 0 , and we conclude similarly. ' When a ą 0 and b " 0 we know that nν is a zero of φ ´,s for all s P r0, 1s and the zero of φ `,s near nν behaves as in the previous case. ' We continue with the case a 2 " 4b 2 . We set φ s " φ ˘,s . As above, if n ‰ 0 then nν is a simple zero of φ 0 , and for s ą 0 small there is a unique zero ζ n of φ s near nν, and (6.13) holds for ζ n psq with µ ˘" a 2 . Then for s ą 0 small the function φ s has a unique zero in C n . Since this is a simple zero there is no splitting, we have a unique simple zero in C n for all s Ps0, 1s. Then, by continuity of the spectrum of T s with respect to s, the corresponding eigenvalue ζ n psq 2 of T s has algebraic multiplicity 2, as is the case for the eigenvalue n 2 ν 2 of T 0 . As above we deal similarly with the case n " 0. ' We finish with the case a 2 ă 4b 2 . The zeros of φ s,`b ehave exactly as in the first case: the asymptotic expansion (6.13) still holds and, since µ `has a positive imaginary part, we still have ζ n P C n for s ą 0 small. The same applies to the zeros of φ ´,s if a " 0 and b ‰ 0. ' Things are different for the zeros of φ s,´i n general, since by Proposition 6.2 they can go through the vertical lines Repzq P νN. Assume that a ą 0. For s P r0, 1s we set apsq " sa, bpsq " saε b 2 d 1 `1 4θ 2 π 2 ln ˆ2θν `sa 2θν ´sa ˙2,

where ε b P t˘1u is the sign of b and θ is defined by (6.12). We also set µpsq " µ ´,apsq,bpsq and, for z P C, φs pzq " φ ´,apsq,bpsq pzq. We have µpsq " sa 2 ´is 2 a 2 4θ 2 πν `Ops 3 q.

We set Ts " T apsq,bpsq . Then we can proceed as above. If n ‰ 0, nν is a simple zero of φ0 . By the implicit function Theorem there exist s 0 Ps0, 1s, a neighborhood V of nν and an analytic function ζn : r0, s 0 s Ñ V such that for s P r0, s 0 s and ζ P V we have φs pζq " 0 if and only if ζ " ζn psq. Moreover, ζn psq " nν ´isa 2nπ `s2 a 2 pθ 2 ´n2 q 4π 2 νn 3 θ 2 `O sÑ0 ps 3 q.

We see that for s ą 0 small enough we have ζn psq P C n if θ ą n and ζn psq P C n´1 if θ ă n. If θ " n, we necessarily have Rep ζn psqq " nν by Proposition 6.2. We proceed similarly around pn ´1qν and pn `1qν, and we obtain that for s ą 0 small the numbers of zeros of φs with real part in spn ´1qν, nνr, tnνu and snν, pn `1qνr depend on the value of θ as follows.

Repζq P spn ´1qν, nνr tnνu snν, pn `1qνr

θ ď n ´1 1 0 1 n ´1 ă θ ă n 2 0 1 θ " n 1 1 1 n ă θ ă n `1 1 0 2 θ ě n `1 1 0 1 (6.14)
By continuity of the zeros of φs , we can extend this observation for all s P r0, 1s. If θ ‰ n, then the zeros of φs cannot go through the line Repζq " nν. If θ " n, there is always one zero of φs on this line. Since this zero is always simple, the zeros in C n´1 or C n cannot meet it. Since the same argument is valid for the lines Repζq " pn ´1qν and Repζq " pn `1qν, we obtain that (6.14) holds for any s Ps0, 1s. The only difference is that when we have two zeros (for instance in C n if θ Psnν, pn `1qνr) then for s small we know that we have two simple zeros (one close to nν and the other close to pn `1qν), while for large s they could meet and produce a zero of multiplicity 2 (and this happens for some values of pa, bq by Proposition 6.1). The discussion of the case n " 0 is the same as above (in this case we necessarily have θ ą n).

Proposition 2. 2 .

 2 Let a, b P R. (i) The operator P a,b is maximal sectorial. (ii) If a ě 0 then P a,b is maximal dissipative. (iii) The adjoint of P a,b is P ´a,b .

e

  z pxq " e izx `e2iz e ´izx (3.2) and ẽz pxq " 1 2iz p ´xq `eizx ´e2iz e ´izx ˘. (3.3) We have ´e2 z ´z2 e z " 0 and ´ẽ 2 z ´z2 ẽz " e z . (3.4) Notice that 0 is an eigenvalue of T if and only if b " 0. In the following proposition we give a characterization of non-zero eigenvalues. Proposition 3.1. Let a, b P R and z P C ˚.

Proposition 6. 2 .

 2 Let a ě 0 and b P R such that pa, bq ‰ p0, 0q. Let n P N ˚.(i) If φ `pzq " 0 then Repzq ‰ nν. (ii) If b " 0 (and a ą 0) we have φ ´pnνq " 0 (this is a simple zero) and φ ´has no other zero of real part equal to nν. (iii) If b ‰ 0 then φ ´has a zero of real part nν if and only if 0 ă a 2 ă 4b 2 , a ă 2nν and

  Proposition 3.4. Let a, b P R. We have

	z n,˘" nν	´iµ nπ `O nÑ`8	pn ´2q,
	and hence
	z 2 n,˘" n 2 ν 2 ´2iµ ˘ `O nÑ`8	pn ´1q.
	This proves that
	r n,˘"	´iµ nπ	`Opn ´2q,

Proof. Let r n,˘: " z n,˘´n ν. By Proposition 3.2, r n,˘g oes to 0 as n Ñ `8. Then we compute 0 " φ ˘pz n,˘q " pnν `rn,˘´µ˘q e 2ir n,˘ ´pnν `rn,˘`µ˘q " 2ir n,˘n π ´2µ ˘`Opr n,˘q `Opnr 2 n,˘q .

  If a 2 ă 4b 2 then the function φ `has a unique zero ζ n,`i n C n (and this zero is simple). There exists a unique θ P We begin with the case a 2 ą 4b 2 ą 0. By Proposition 3.2, Z sa,sb is included in a horizontal strip of C which does not depend on s P r0, 1s. Moreover, by Proposition 6.2, Z sa,sb does not intersect the vertical lines Repζq " nν and Repζq " pn `1qν for s Ps0, 1s. Then, by the Rouché Theorem, the number of zeros of φ ˘,s " φ ˘,sa,sb in C n is a continuous and hence constant function of s Ps0, 1s.Assume that n ‰ 0. We know that nν is a simple zero of φ 0 (see (3.16)). By the implicit functions theorem there exist s 0 Ps0, 1s, a neighborhood V of nν in C and analytic functions ζ n,˘: r0, s 0 s Ñ U such that for s P r0, s 0 s and ζ P V we have φ ˘,s pζq " 0 if and only if ζ " ζ n,˘p sq. We can compute

			‰ a 2ν ,	`8‰ such that
	4b 2 " a 2 `a2 4θ 2 π 2 ln	ˆ2θν 2θν	`a ´a˙2	,	(6.12)
	and then the number of zeros of φ ´in C n (counted with multiplicities) is
	#
	2 if n ă θ ă n `1,
	1 otherwise.
	Proof. ' ζ n,˘p sq " nν	´iµ ˘s nπ	`µ2 ˘ s 2 n 3 π 3 `O sÑ0	ps 3 q.	(6.13)

After integration over x P R d´1 we get

}f k,j } 2 L 2 pR d´1 q .

(5.2)

With this notation we set, for ζ P CzΣ,

p´1q p `pL ´pζ ´λk qq ´1´p f k,j`p ˘b Ψ k,j .

(5.3)

(5.4)

Proof. Let F P H . We use notation (5.1). ' For k P N ˚, j P t1, . . . , m k u and p P t0, . . . , m k ´ju we have by the spectral theorem

With (5.2) we deduce that for N, N 1 P N ˚we have

so RpζqF converges in H and, by (5.2) again,

This proves that Rpζq defines a bounded operator on H and (5.4) is satisfied.

Then U N P DompPq, it goes to RpζqF in H and

Since P is closed, this proves that RpζqF P DompPq and pP ´ζqRpζqF " F . In particular, for ζ 0 P ρpPq we have pP ´ζ0 q ´1 " Rpζ 0 q.

(5.5)

' It remains to prove that for ζ P CzΣ and U P DompPq we have RpζqpP ´ζqU " U.

(5.6) (R. Ayechi) LAMMDA -Université de Sousse -ESST Hammam Sousse -Rue Lamine El Abbessi, Hammam Sousse, 4011, Tunisia.

Email address: radhiaayechi@essths.u-sousse.tn (I. Boukhris) LAMMDA -Université de Sousse -ESST Hammam Sousse -Rue Lamine El Abbessi, Hammam Sousse, 4011, Tunisia.

Email address: ilhemboukhris@essths.u-sousse.tn (J. Royer) Institut de Mathématiques de Toulouse -UMR 5219 -Université Toulouse 3, CNRS -UPS, F-31062 Toulouse Cedex 9, France.

Email address: julien.royer@math.univ-toulouse.fr