Guizard

A Julbe
email: ajulbe@cit.enscm.fr

D Farrusseng

J C Jalibert

C Mirodatos

C Guizard

Characteristics and performance in the oxidative dehydrogenation of propane of MFI and V-MFI zeolite membranes

Keywords: Membrane reactor, Zeolite membranes, V-MF1 membranes, Oxidative dehydrogenation of propane

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

Introduction

Among the large research effort devoted to the oxidative dehydrogenation of alkanes, the potential use of membrane reactors appears to be of particular interest for (i) controlling the oxygen feeding to limit the highly exothermic total combustion [START_REF] Dalmon | Catalytic membrane reactors[END_REF][START_REF] Pantazidis | [END_REF] and (ii) improving the contact between the reactant and the catalyst. Zeolite membranes are thermally stable and well adapted to high temperature applications [START_REF]Fundamentals of Inorganic Membrane Science and Technology[END_REF] since they can be used as separators only and/or as active contactors if they are catalytically active. Molecular sieves catalysts are also widely studied for selective oxidation reactions [START_REF] Bellusi | Advanced Zeolite Science and Applications[END_REF][START_REF] Weitkamp | Catalysis by Microporous Materials[END_REF]. Immobilizing transition metal ions in zeolites by ion exchange or by incorporation into the lattice leads to stable, isolated and well defined redox active catalytic sites. Since the industrial applications of titanium silicalite, other redox molecular sieves are developed currently. Based on the catalytic properties of vanadium oxides for selective oxidations, the insertion of vanadium in MFI and MEL structures is widely investigated [START_REF] Bellusi | Advanced Zeolite Science and Applications[END_REF][6][7][8][9][START_REF] Sayari | Synthesis of Porous Materials[END_REF][START_REF] Bellussi | Catalytic Selective Oxidation, ACS Symp. Series[END_REF]. The present study reports preliminary data obtained for the preparation and characterization of supported V-MFI membranes by in situ insertion of V species during the hydrothermal treatment. The catalytic performance of both MFI and V-MFI membrane reactors are compared for the reaction of oxidative dehydrogenation of propane (ODHP).

Experimental

Synthesis of V-MFI composite membranes

The V-MFI membranes were prepared according to the method already reported for MFI membranes 1 [START_REF] Giroir-Fendler | French Patent 94-5562[END_REF]. A sol (VOSO 4 /SiO 2 /TPAOH/H 2 O) was treated at 185 • C in contact with an ␣-alumina support (enameled tubes 15 cm long, T1-70, from US Filter-SCT). The tubes have a three layers asymmetric structure with mean pore sizes of 10 m (external layer), 0.8 m (intermediate layer) and 0.2 m (internal layer). The molar ratio Si/V in the starting sol was 40 and the pH was about 13.5. After synthesis the template was eliminated by a thermal treatment in air at 600 • C for 2 h. Vanadium-free-MFI membranes were also prepared (without VOSO 4 in the starting sol) and characterized for comparison. The hydrothermal treatment and consecutive thermal treatment resulted typically in an increase of about 5 wt.% for the support. The weight of zeolite in the studied MFI and V-MFI membranes (15 cm long) was 1.1 g.

Membrane characterizations

The vanadium species contained in a scrapped V-MFI membrane were investigated by elemental analysis, FTIR, ESR and XRD. The morphology and texture of MFI and V-MFI membranes were studied by SEM, mercury porosimetry and N 2 adsorptiondesorption. For the single gas permeance studies, membranes were sealed in a stainless steel module using graphite o-rings and outgassed overnight at 450 • C under N 2 before each test. During gas permeation measurements, one end of the membrane tube was closed and the internal pressure was varied between 1.5 and 4 bars at room temperature. The pressure in the permeate side (outside part of the tube) was the atmospheric pressure. No sweep gas was used. During the measurements as a function of temperature the transmembrane pressure was fixed to 500 mbar by a pressure regulator equipped with a pressure sensor. The membrane permeance for a given temperature was measured after stabilization of the flow rate. Single gas permeances were measured as a function of transmembrane pressure (from 0.5 to 3 bars) for N 2 and of temperature (from room temperature to 500 • C) for O 2 ,C O 2 and C 3 H 8 . The measured permeances varied in the range (1-5) × 10 -7 mol/(m 2 s Pa) with a maximum error of ±2%.

Catalytic tests

After calcination and single gas permeance studies, the above mentioned membranes were tested for ODHP in a stainless steel membrane reactor designed to operate with two separated inlet flows. Two operating configurations were tested: (i) the separate feed configuration (oxygen distributor) where propane is fed in the inner compartment and oxygen in the outer and (ii) the cofeed configuration (flow-through contactor) where both reactants are fed in the inner compartment. In the oxygen distributor configuration the exit was in the inner compartment and in the flow-through configuration, the exit was in the outer compartment. The reactant feed composition was typically 3 or 6% C 3 H 8 and X%O 2 in He, with X varying from 0 to 12% with a total flow rate varying from 36 to 52 ml/min in the temperature range 550-650 • C. A transmembrane pressure ( P = P int -P ext ) ranging from -0.1 to +0.5 bar was applied between the two compartments of the reactor. The propene yield (product of the propane conversion by the propene selectivity) was calculated from the gas analysis at the inlet and outlet parts of the reactor by gas chromatography. Permeability measurements were performed before and after reaction up to 650 • C in order to detect eventual defects or coke deposition.

Results and discussions

Characterization of the zeolite membranes

A typical atomic ratio Si/V = 160 was found for the scrapped V-MFI membrane, representing the mean surface composition. The possible incorporation of aluminum from the support during the hydrothermal synthesis would induce a variation of the vanadium insertion (quantity and nature) from the support interface to the outer part of the membrane. The presence of the vanadium precursor in the starting sol did not seem to influence the growth of MFI crystals (quantity, morphology, location in the support and crystalline phase). The characteristics of the V-MFI membrane were found similar to those of the MFI one.

Morphology and porous structure of the V-MFI/␣-Al 2 O 3 composite membrane

As observed by SEM (Fig. 1), the zeolite crystals have grown both on the inner layer of the support and inside the pores. An external layer of non-connected zeolite crystals (region a) has grown on a layer of connected zeolite crystals (region b) which is firmly stuck to the support (region c). The pores of the two internal layers of the support (0.2 and 0.8 m pore sizes) were almost totally filled with zeolite (Fig. 2). The grains of the external layer (10 m pore sizes) were only partially covered with zeolite crystals.

These observations were in good accordance with the mercury porosimetry analysis of the support before and after synthesis. As shown in Fig. 3, the large 10 m pores were still present after synthesis but the two families of pores centered on 0.2 and 0.8 m were largely modified and occluded by zeolite crystals. The N 2 adsorption-desorption isotherms of the scrapped zeolite were comparable to those obtained with a commercial MFI zeolite powder (Na-ZSM5 Uetikon).

Location and type of vanadium species in the V-MFI membrane

The location of the V ions either in the zeolite framework or randomly distributed within the pores and/or anchored to the crystal surface has been discussed in the literature on vanadium zeolite powders [START_REF] Bellusi | Advanced Zeolite Science and Applications[END_REF][7][8][9].

The cell parameters of the synthesized V-MFI crystals, calculated from the X-ray diffraction patterns [START_REF] Mr | U-FIT Program, cell parameter refinement program[END_REF], are presented in Table 1. Compared to the result obtained for MFI, a slight increase of the cell parameters c, β and consequently of the cell volume was noticed, which may indicate the insertion of vanadium species in the zeolite structure [START_REF] Hari Prasad Rao | [END_REF]15]. Furthermore FTIR spectra revealed a small absorption at 963 cm -1 characteristic of V-O-Si bonding [START_REF] Bellusi | Advanced Zeolite Science and Applications[END_REF].

Although the presence of polymeric vanadium species is not excluded, the hyperfine structure observed at room temperature in the ESR spectrum of the as-synthesized V-MFI membrane (Fig. 4a) reflected a very high dispersion of V 4+ species. The ESR signal parameters were close to those reported by different authors for V 4+ species in a distorted square pyramidal coordination [START_REF] Bellusi | Advanced Zeolite Science and Applications[END_REF]. After calcination at 600 • C, the ESR spectra disappeared (Fig. 4b) indicating the oxidation of V 4+ to V 5+ species. Reduction of the calcined sample in H 2 /Ar mixture at 500 • C restored the ESR spectra and demonstrated a reversible V 5+ /V 4+ transformation. Some isolated V 5+ species were consequently accessible to gas molecules and may act as active selective sites for ODHP as mentioned in [START_REF] Bellusi | Advanced Zeolite Science and Applications[END_REF].

Single gas permeances through the MFI and V-MFI membranes

The N 2 permeance of MFI and V-MFI membranes was measured at room temperature as a function of transmembrane pressure. The absence of viscous-flow when the transmembrane pressure was increased from 0.5 to 3 bars indicated that the membranes were free of macrodefects (>50 nm). The N 2 permeance of the MFI membrane was not modified after ODHP catalytic tests up to 650 • C, although it decreased by 10% for the V-MFI membrane. This could result from some V-MFI membrane coking during the ODHP reaction [START_REF] Julbe | Inorganic Membranes, Proc. ICIM 98[END_REF].

The O 2 ,C 3 H 8 and CO 2 permeances of MFI and V-MFI membranes were measured after the ODHP tests as a function of temperature in the range (1-3) × 10 -7 mol/(m 2 s Pa) (Fig. 5). These data were typical of MFI membranes of relatively good quality [START_REF]Fundamentals of Inorganic Membrane Science and Technology[END_REF]16]. The corresponding main transport mechanisms are tentatively schematized in Fig. 6. The O 2 and CO 2 permeances correspond to BCD curves between room temperature and 500 • C. An ABCD curve was observed for C 3 H 8 . At high temperature (400-500 • C) the measured activation energies for C 3 H 8 and O 2 transport were about 11 kJ/mol for the MFI membrane, in good agreement with the activation energies reported in [17] for gas translation diffusion at high

MFI and V-MFI membranes reactivity for ODHP

Effect of the O 2 content

The reactor was configured as a flow-through contactor, i.e. fed in the tube side (3% C 3 H 8 and 0-12% O 2 in He) with a total flow rate of 45 ml/min, at 550 • C, internal pressure 1.7 bar and transmembrane pressure 0.4 bar. The catalytic data are reported in Figs. 7 and8 for MFI and V-MFI membranes, respectively.

Whatever the O 2 concentration, higher oxygen and propane conversions were achieved with the V-MFI with respect to the MFI membranes. Furthermore, a higher propene selectivity was obtained with the V-MFI membrane for low oxygen content (<1%). However, by increasing the O 2 content from 1 to 12%, the propene selectivity decreased from 50 to 20% for the V-MFI membrane, while it remained around 50% for the MFI one. The largest propene yield (7%) was obtained with the V-MFI membrane with 1% O 2 in the feed. A higher oxygen content (6% O 2 ) was necessary to reach the same yield with the MFI membrane. In the absence of oxygen, propene yields of 4% (with a selectivity of 85%) and 2.6% (with 73% selectivity) were obtained with the V-MFI and MFI membranes, respectively.

The observed activity of the MFI membrane which should be theoretically inactive may be related to acidic sites probably created after aluminum dissolution from the support during the hydrothermal synthesis (high pH) and leading to a kind of H-ZSM-5 membrane. Within this assumption, a mechanism involving radical formation on Lewis acid sites could be considered. In this case, a propyl radical formed Fig. 6. Schematic evolution with temperature of single gas permeance through MFI membranes. ABC: activated configuration diffusion, AB: mobility of adsorbed species increases and occupancy decreases, B: the increase in mobility cannot compensate the decrease in occupancy, C: adsorption is negligible, CD: activated transport through micropores (gas translation diffusion). on an Alᮀ site could either be dehydrogenated into propene by H abstraction on a second lattice oxygen or reacting with an electrophilic oxygen radical O -, similarly activated on lattice vacancy, to be step-wise oxidized into CO x [START_REF] Kung | [END_REF]. Another possibility, favored at high temperature, would be to consider that the formed alkyl radical may desorb from the surface and react with gaseous oxygen to form propene and chain propagation radicals like OH . . In addition to these radical induced processes, a cracking reaction on these acid sites (via carbenium ions) could also account for the formation of C 1 -C 2 molecules, independently of the oxygen pressure as observed in Fig. 7.

In addition to these acid sites arising from the hydrothermal synthesis, the V-MFI membrane also contains vanadyl Lewis acid sites, able to activate propane and oxygen according to a redox process [START_REF] Bellussi | Catalytic Selective Oxidation, ACS Symp. Series[END_REF]. The presence of these sites could explain the higher oxygen and propane conversions observed on V-MFI membrane. When the oxygen concentration increases above 1%, the V sites in the V-MFI membrane would favor the formation of electrophilic oxygen adspecies leading to total oxidation products (CO and CO 2 ). This is not observed with the MFI membrane.

The ODHP catalytic performance of the V-MFI membrane is compared in Table 2 to data reported for V-silicalite powder with SiO 2 /V 2 O 3 = 545 [START_REF] Bellussi | Catalytic Selective Oxidation, ACS Symp. Series[END_REF]. In spite of different experimental conditions, a three times lower propene selectivity is obtained with the V-MFI membrane at comparable propane conversion and temperature. The V-MFI membrane might contain a higher quantity of extra-lattice non-selective vanadium species compared to the V-MFI powder reported in [START_REF] Bellussi | Catalytic Selective Oxidation, ACS Symp. Series[END_REF] which has been treated with an ammonium acetate solution. The low selectivity obtained with the V-MFI membrane could also be related to a larger amount of aluminum than in the V-MFI zeolite studied in [START_REF] Bellussi | Catalytic Selective Oxidation, ACS Symp. Series[END_REF]. Experiments are currently performed with other supports in order to avoid the insertion of disturbing foreign species. The observation reported in [START_REF] Bellussi | Catalytic Selective Oxidation, ACS Symp. Series[END_REF] that the conversion increased and the selectivity decreased when the molar ratio SiO 2 /V 2 O 3 decreased is in good agreement with the relative performance obtained for MFI and V-MFI membranes.

Influence of contact time

The effect of contact time in the flow-through contactor configuration was studied at 550 • C with 6% C 3 H 8 and 1% O 2 in He by varying the total flow rate from 36 to 52 ml/min (Figs. 9 and 10, Table 3). As expected, with both membranes a decrease in C 3 conversion and an increase in propene selectivity was observed when the flow rate was increased, i.e. the contact time was decreased. Note that however the oxygen conversion was always 100% for the V-MFI membrane. For the MFI membrane, the maximum propene yields at 550 • C were obtained with the highest and lowest flow rate (36 or 52 ml/min) whereas for the V-MFI membrane the maximum yield corresponded to a low flow rate (40 ml/min). The effect of contact time on the V-MFI membrane performance was no more observed at 650 • C (no change in selectivity and conversion with the flow rate), indicating that at this temperature the gas phase mechanism tends to become prominent.

Influence of temperature and of reactor configuration

The ODHP data for the V-MFI membrane between 550 and 650 • C are reported in Fig. 11 and Table 4, both for the flow-through contactor and oxygen distributor configurations. The increasing contribution of the homogeneous gas phase reaction above 550 • C may explain the stability of the propene selectivity and the increasing propane conversion with temperature. In all cases the oxygen conversion was 100%. Under the studied conditions, the flow-through contactor always gave the highest propene selectivity and yield, the conversion being also slightly higher in this configuration compared to the oxygen distributor configuration.

The zeolite membrane is not permselective to the reaction gases (possible retrodiffusion of both C 3 and O 2 species) and the gas transport mechanisms in such a zeolite membrane are not affected by pressure. Due to the fact that the propane conversion was similar in both cases and that the oxygen consumption was total, the same quantity of propane is expected to diffuse through the catalytic membrane in both configurations. The difference in selectivity could then be attributed to the residence time in the membrane of the primarily formed propene which could be further oxidized into CO x . A higher residence time in the case of the oxygen distributor configuration, which is plausible due to counter flow effects, would explain the observed lower selectivity. It would be worthwhile to compare these results with those obtained with a similar membrane providing a viscous-flow transport contribution. Indeed in this case the permeance is a function of pressure and the flow rate and retrodiffusion of species might be controlled by pressure.

Conclusion

V-MFI membranes synthesized in/on a porous ␣-alumina support were found to present structural characteristics very similar to those obtained for V-MFI powders: well dispersed V 4+ species and vanadium inserted in the zeolite framework. These V-MFI membranes presented the same morphology of the zeolite phase as compared to MFI membranes.

Single gas transport through the V-MFI and MFI membranes was controlled by adsorption, configuration diffusion and gas translation diffusion, as expected for such zeolite membranes. The O 2 /C 3 H 8 permselectivity of the membranes was comparable to the Knudsen selectivity at 500 • C.

Both MFI and V-MFI membranes were found active for the ODHP at 550 • C, but behaved differently according to the O 2 partial pressure leading to slightly better propene yield for the V-MFI. Acid sites related to some aluminum insertion during the hydrothermal synthesis are likely to be responsible for the MFI membrane activity. The vanadyl sites present in the V-MFI would contribute to the improved performances of this membrane. The oxygen distributor configuration was not found to improve the V-MFI performance in the flow-through configuration, possibly due to back mixing and residence time effects. However, in spite of the very low vanadium content in this V-MFI membrane, a promising propene yield up to 8% was obtained at 550 • C with a selectivity in the range 40-50%.

Both MFI and V-MFI membranes were thermally stable for cycles between 500 and 650 • C in ODHP conditions (i.e. in the presence of water). Indeed the initial permeance and catalytic performance were restored after 200 h ODHP test. The direct insertion of a very low vanadium content during the formation of the silica rich zeolite framework lead to stable V-MFI/␣-Al 2 O 3 composite membranes. In such membranes the vanadium active sites does not seem to migrate or alter the zeolite framework in ODHP conditions.

Further improvements of the V-MFI membrane material composition (elimination of aluminum and polymeric V species) as well as a modification of the oxidative gas reactivity (N 2 O instead of O 2 ) will be attempted to improve the catalytic performance of these original and stable membranes.

Fig. 1 .

 1 Fig. 1. FESEM cross-section image of a supported MFI zeolite membrane.

Fig. 2 .

 2 Fig. 2. FESEM cross-section images of the SCT support: (a) inner layer, (b) intermediate layer, (c) external layer before (I) and after (II) zeolite synthesis.

Fig. 3 .

 3 Fig. 3. Pore size distributions of the support before and after zeolite synthesis (mercury porosimetry).

  temperature in MFI membranes. In the case of V-MFI membrane, lower activation energies were measured for O 2 (around 8 kJ/mol), indicating a lower membrane quality (larger number of microdefects). As shown in Fig.5, the C 3 H 8 and CO 2 permeance curves still show an influence of adsorption at 400 • C and it was meaningless to measure the activation energy between 400 and 500 • C.At 500 • C, the ratio of permeances (or ideal selectivity) α(O 2 /C 3 H 8 ) is 1.09 and 1.13 for MFI and V-MFI membranes, respectively. In good agreement with a gas transport mechanism by translation diffusion these values concerning gases with comparable activation energies are close to the Knudsen ratio: α Knudsen (O 2 /C 3 H 8 ) = 1.17. A separation test with an O 2 /C 3 H 8 mixture (50/50) showed that the V-MFI membrane permeance selectivity varied from 0.77 at 25 • C up to 0.96 at 200 • C. No test was performed at higher temperature because of the reaction between the two gases. It may be assumed however that at 500 • C the membrane selectivity for the mixture is identical to the one obtained for single gases (i.e. <1.17).

Fig. 4 .

 4 Fig. 4. ESR spectra at room temperature of (a) as-synthesized V-MFI membrane scrapped off the support, (b) V-MFI membrane scrapped off the support and treated in air at 600 • C.

Fig. 5 .

 5 Fig. 5. Single gas permeances as a function of temperature ( P = 500 mbar) (a) O 2 and C 3 H 8 through the MFI membrane after the catalytic tests at 650 • C, (b) CO 2 ,O 2 and C 3 H 8 through the V-MFI membrane after the catalytic tests at 650 • C; CO 2 permeance before the catalytic tests.

Fig. 7 .

 7 Fig. 7. C 3 H 6 ,C 2 H 6 ,C 2 H 4 ,C H 4 , CO, CO 2 selectivities; C 3 H 8 and O 2 conversions and C 3 H 6 yield for the MFI membrane as a function of the oxygen content in the feed (3% C 3 H 8 and 0-12% O 2 in He), for a temperature of 550 • C and flow-through contactor configuration with total flow rate: 45 ml/min, P int = 1.7 bar and P = 0.4 bar.

Fig. 8 .

 8 Fig. 8. C 3 H 6 ,C 2 H 6 ,C 2 H 4 ,CH 4 , CO, CO 2 selectivities; C 3 H 8 and O 2 conversions and propene yield for the V-MFI membrane as a function of the oxygen content in the feed (3% C 3 H 8 and 0-12% O 2 in He), for a temperature of 550 • C and flow-through contactor configuration with total flow rate: 45 ml/min, P int = 1.7 bar and P = 0.4 bar.

Fig. 9 .

 9 Fig. 9. C 3 H 6 ,C 2 H 6 ,C 2 H 4 ,C H 4 , CO, CO 2 selectivities; C 3 H 8 and O 2 conversions and propene yield for the MFI membrane as a function of the total flow rate with flow-through contactor configuration for feed composition of 6% C 3 H 8 and 1% O 2 in He at temperature 550 • C.

Fig. 11 .

 11 Fig. 11. C 3 H 8 conversion, CO x selectivity, C 3 H 6 selectivity and yield between 550 and 650 • C with a V-MFI membrane reactor in flow-through contactor (FT) or oxygen distributor (OD) configuration for feed composition: 6% C 3 H 8 and 1% O 2 in He and total flow rate: 45 ml/min.

Table 1

 1 Values of cell parameters (from XRD) for V-MFI and MFI scrapped from alumina support after the firing treatment at 600 • C a

	Cell parameter	a (Å)	b (Å)	c (Å)	β ( • )	Volume (Å 3 )
	MFI zeolite	19.877 (±0.005)	20.103 (±0.005)	13.365 (±0.003)	90.59 (±0.03)	5340 (±4)
	V-MFI zeolite	19.876 (±0.003)	20.104 (±0.003)	13.373 (±0.002)	90.63 (±0.02)	5343 (±2)

a Estimated standard deviations are indicated in brackets.

Table 2

 2 Catalytic performance of V-silicalite powders reported in[START_REF] Bellussi | Catalytic Selective Oxidation, ACS Symp. Series[END_REF] (SiO 2 /V 2 O 3 = 545 for flow reactor with 2.8% C 3 H 8 , 8.4% O 2 or N 2 Oi n He, 4.2 g of catalyst, total flow rate = 51 ml/min) and of the studied V-MFI membrane (SiO 2 /V 2 O 3 = 320 for flow-through membrane reactor with 6% C 3 H 8 , 12% O 2 in He, 1.1 g of catalyst, total flow rate = 45 ml/min)

	Material	V-silicalite powder [11]			V-MFI membrane
	oxidative gas	O 2	O 2	N 2 OO 2
	Temperature	400 ( • C)	540 ( • C)	400 ( • C)	550 • C
	C 3 H 8 conversion	8%	28%	45%	30.6%
	C 3 H 6 selectivity	84%	64%	76%	16.2%
	C 3 H 6 yield	6.7%	18%	34%	5.0%

Table 3

 3 Comparison of MFI and V-MFI membrane performance in flow-through contactor configuration at 550 • C with 6% C 3 H 8 and 1% O 2 in He, total flow rate = 36 or 40 ml/min a Membrane/flow rate (ml/min) XC 3 H 8 XO 2 SC 3 H 6 SC 2 H 6 SC 2 H 4 SCH 4 SCO SCO 2 YC 3 H 6 Transmembrane and internal pressures are indicated in the last column ( P = P int -P ext ). Fig. 10. C 3 H 6 ,C 2 H 6 ,C 2 H 4 ,C H 4 , CO, CO 2 selectivities; C 3 H 8 conversion and propene yield for the V-MFI membrane as a function of the total flow rate. The O 2 conversion was always 100% with flow-through contactor configuration for feed composition 6% C 3 H 8 and 1% O 2 in He at temperature 550 • C.

	P/P int (mbar)

a

Table 4 V

 4 -MFI membrane reactor: evolution of propane conversion, product selectivity and propene yield as a function of temperature with two reactor configurations (oxygen distributor and flow-through contactor) a Transmembrane and internal pressures are indicated in the last column ( P = P int -P ext ). Feed conditions: 6% C 3 H 8 and 1% O 2 in He. Total flow rate: 45 ml/min.

	Temperature ( • C)	Conversion (X%), selectivity (S%) and yield (Y%)				Pressure (mbar)
		XC 3 H 8	SC 3 H 6	SC 2 H 6	SC 2 H 4	SCH 4	SCO	SCO 2	YC 3 H 6	P/P int
	Oxygen distributor (OD)
	555	10.2	40.6	0.4	8.62 .61 0 .1	37.8	4.1	-165/1720
	579	14.7	39.7	0.8	11.44 .58 .1	35.4	5.9	-153/1730
	604	11.7	42.6	1.2	15.46 .5	8	26.4	5	-151/1650
	628	14.7	42.5	1.8	20.19 .66 .5	19.6	6.2	-143/1690
	650	19.8	43	0	26.41 1 .34 .9	14.4	8.5	-137/1650
	Flow-through contactor (FT)
	555	13.8	52.2	0	6.41 .61 1 .7	28.2	7.2	410/1730
	575	11.1	52	0	9	2.47 .7	28.9	5.8	392/1710
	603	12.7	52.5	0.4	12.33 .97 .1	23.8	6.7	374/1690
	629	15.4	49.1	0.8	17.29 .25 .9	17.7	7.6	356/1680
	654	21.1	47.5	1.3	23.81 0 .3	5	12.1	10	339/1660

a

Acknowledgements

The authors gratefully acknowledge financial support by the European Commission (Joule program JOE3 CT95-0022) and express their sincere thanks to the LMPM-Montpellier staff: Dr. J. Sanchez for single gas permeation facilities, G. Volle for gas permeance tests, Dr. A. van der Lee for XRD studies, A. El Mansouri for N 2 adsorption-desorption and Hg porosimetry analysis, and G. Nabias and D. Cot for FESEM observations.


Theo Mckenzie
email: tmckenzie@math.harvard.edu

Mostafa Sabri
email: mmsabri@sci.cu.edu.eg

Quantum

Quantum ergodicity for periodic graphs

Keywords: 2020 Mathematics Subject Classification. Primary 58J51. Secondary 39A12 Quantum ergodicity, periodic graphs, periodic Schrödinger operators

come

Introduction

Our aim in this article is to prove quantum ergodicity for large periodic graphs. If Γ N is a sequence of finite graphs "converging" 1 to some infinite graph Γ, and if we study a Schrödinger operator H N = A N + Q N on ℓ 2 (Γ N ), then quantum ergodicity is a delocalization criterion stating that, in a weak sense, most eigenvectors of H N are equidistributed on the graph Γ N .

The terminology comes from [START_REF] De Verdière | Ergodicité et fonctions propres du laplacien[END_REF][START_REF] Šnirel | man, Ergodic properties of eigenfunctions[END_REF][START_REF] Zelditch | Uniform distribution of eigenfunctions on compact hyperbolic surfaces[END_REF], where the ergodicity of the geodesic flow on a compact manifold M of unit volume (meaning the classical particle's free motion covers the manifold uniformly) is shown to imply a quantum counterpart of ergodicity, namely, the Laplacian wavefunctions ψ λ are equally likely to be anywhere on M (more precisely |ψ λ (x)| 2 dVol(x) approaches the uniform measure dVol(x), when λ gets large).

Quantum ergodicity for large regular graphs with few cycles was first proved in [START_REF] Anantharaman | Quantum ergodicity on large regular graphs[END_REF], for the adjacency matrix H N = A N . In this case the limiting graph Γ was the (q + 1)-regular tree T q . The general case where Γ is an infinite tree which is not necessarily regular and H N = A N + Q N was later established in [START_REF] Anantharaman | Quantum ergodicity on graphs : from spectral to spatial delocalization[END_REF], assuming H Γ has absolutely continuous spectrum. This includes regimes of the Anderson model [START_REF] Anantharaman | Quantum ergodicity for the Anderson model on regular graphs[END_REF], as well as "periodic trees with periodic potentials", more precisely universal covers of finite graphs [START_REF] Anantharaman | Recent results of quantum ergodicity on graphs and further investigation[END_REF].

The previous results also required the Γ N to be expanders. It was shown in [START_REF] Mckenzie | The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph[END_REF] that counterexamples exist if expansion is dropped. Examples of regular expander graphs violating quantum ergodicity were also constructed in [START_REF] Mckenzie | The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph[END_REF]; there Γ was no longer a tree. However, it remained open whether more specific families of graphs satisfy quantum ergodicity.

In this paper we show that quantum ergodicity is in fact satisfied for a large family of non-tree graphs Γ, namely graphs which are periodic with respect to a basis of Z d . The simplest example is the adjacency matrix on Z d , but the results also apply to some classes of Schrödinger operators with periodic potentials on various lattices. These graphs do not satisfy the expansion or tree properties of previous proofs. Therefore we need new, different techniques to solve the problem in this case.

On Z d , if we consider the eigenbasis e (N ) r (k) = 1 N d/2 e 2πik•r/N for the adjacency matrix on a sub-cube Γ N of sidelength N with periodic boundary conditions, we note that |e

(N ) r (k)| 2 = 1
N d is perfectly uniformly distributed on Γ N . Similarly, for a periodic Schrödinger operator H on a periodic lattice, the Bloch theorem ensures that for any λ ∈ σ(H N ), we can find an eigenfunction Ψ λ such that |Ψ λ (x)| 2 is a periodic function (see §5.2 for a discussion and a proof of this result in our context). Here we study whether such delocalization is satisfied for any eigenbasis of the Schrödinger operator.

By virtue of their homogeneity, it is quite intuitive to expect delocalization on periodic lattices. Indeed, the spectrum is generally absolutely continuous, though flat bands (infinitely degenerate eigenvalues) can appear [START_REF] Korotyaev | Schrödinger operators on periodic discrete graphs[END_REF]. The dynamics is also ballistic [START_REF] Boutet De Monvel | Ballistic transport in periodic and random media[END_REF], meaning the waves spread at maximum speed with time. Here we show that from a spatial point of view, the behavior is quite rich :

• There is a simple family of periodic graphs which is quantum ergodic, i.e. the probability measure x∈Γ N |ψ does not favor any particular block, but the mass of ψ (N ) u may not be uniform within the block.

• In other classes of periodic graphs, quantum ergodicity fails completely ( §3. 3 and §3.4).

Examples of these three types are A on Z d , on an infinite cylinder (Fig. 4), and on the graph in Figure 3, respectively.

1.1. Main results. Let Γ be a connected, locally finite graph in some Euclidean space. We assume Γ is invariant under translations of some linearly independent vectors a 1 , . . . , a d .

If

C a = {x 1 a 1 + • • • + x d a d :
x ∈ [0, 1) d } is the basic cell, then

V f = V ∩ C a = {v 1 , . . . , v ν }
is the unit crystal, containing ν vertices. Given x = (x 1 , . . . , x d ) ∈ R d , denote x a = d i=1 x i a i . The graph Γ then consists of periodic V f blocks of size ν, in the sense that (1.1)

V = Z d a + V f ,
where Z d a = {n a : n ∈ Z d }. Any v ∈ V is thus of the form v = ⌊v⌋ a + {v} a , where ⌊v⌋ a ∈ Z d a and {v} a ∈ V f represent the integer and fractional parts of u, respectively. In the case of the simple lattice V = Z d we have a j = e j the standard basis and V f = {0}. In general one can view (1.1) as expressing the vertex set V as ν copies of the sub-lattices Z d a shifted by vertices v n ∈ V f . We consider a Schrödinger operator H = A + Q on Γ, where A is the adjacency matrix and Q satisfies

Q(v + a i ) = Q(v) for v ∈ Γ and i = 1, . . . , d. The potential Q is thus periodic with at most ν different values. Let Γ N = Γ ∩ {x 1 a 1 + • • • + x d a d : x ∈ [0, N ) d }.
Let H N be the Schrödinger operator defined analogously on Γ N . We endow Γ N with periodic boundary conditions : if ψ ∈ ℓ 2 (Γ N ), then ψ(v + N a j ) := ψ(v). Our first result is the following.

Theorem 1.1 (Case ν = 1). Let ψ (N ) u
be an orthonormal basis of ℓ 2 (Γ N ) consisting of eigenvectors of H N . Suppose the fundamental crystal has only one vertex, V f = {o}. Then for any observable a N : Γ N → C such that |a N (v)| ≤ 1 for all v and N , we have

(1.2) lim N →∞ 1 |Γ N | u∈Γ N ψ (N ) u , aψ (N ) u -a 2 = 0 , where ψ (N ) u , aψ (N ) u = v∈Γ N |ψ (N ) u (v)| 2 a(v) and a = 1 |Γ N | v∈Γ N a(v) is the uniform average.
This means that in a weak sense, we have |ψ

(N ) u (v)| 2 ≈ 1 |Γ N | when N is large enough. That is, the eigenvectors ψ (N ) u
are uniformly distributed. This theorem applies to the adjacency matrix on Z d and the triangular lattice, see §4.1 for more graphs.

To our knowledge, Theorem 1.1 is the first positive result establishing quantum ergodicity for a general family of graphs Γ having cycles.

This statement is generally false for higher ν, quantum ergodicity can be completely violated for ν = 2 without further assumptions, see §3.4.1. For general ν, we make an assumption on the Floquet eigenvalues and relax the conclusion. Let b 1 , . . . , b d be the dual basis, satisfying a i • b j = 2πδ i,j . We similarly denote θ b = d i=1 θ i b i . Then we have :

Theorem 1.2 (General case). Let ψ (N ) u
be an orthonormal basis of ℓ 2 (Γ N ) consisting of eigenvectors of H N . Let H(θ b ) be the ν × ν matrix arising in the Floquet decomposition, with eigenvalues E 1 (θ b ), . . . , E ν (θ b ). Suppose that for any s, w ∈ {1, . . . , ν}, we have

(1.3) lim N →∞ sup m∈L d N m =0 #{r ∈ L d N : E s ( r b +m b N ) -E w ( r b N ) = 0} N d = 0 ,
where

L N = {0, 1, . . . , N -1}. Then, (i) For any observable a N : Γ N → C such that |a N (v)| ≤ 1
for all v and N , we have

(1.4) lim N →∞ 1 |Γ N | u∈Γ N ψ (N ) u , aψ (N ) u -ψ (N ) u , Op N (a)ψ (N ) u 2 = 0 ,
where Op N (a) is an explicit operator (see (2.13)). (ii) If a is locally constant, in the sense that it takes a constant value on each periodic block, then

(1.5) ψ (N ) u , Op N (a)ψ (N ) u = a := 1 |Γ N | v∈Γ N a(v).
Specifically, this is true if ν = 1.

This theorem applies, for example, to the honeycomb lattice and periodic Schrödinger operators on Z, see §4.2 and §4. [START_REF] Anantharaman | Quantum ergodicity for the Anderson model on regular graphs[END_REF].

Assumption (1.3) says in particular that the Floquet eigenvalues should not have a short period. Specifically {θ ∈ [0, 1)

d : E s (θ b + α b ) = E s (θ b )}
should be of zero measure for any nonzero α. For example, for d = 1, we should not have

E s (θ b ) = cos 4πθ, as then for α = 1 2 , we get E s (θ b +α b ) = E s (θ b ) for all θ.
The assumption also implies there is no point spectrum, because flat bands require E s to be constant for some s.

Theorem 1.1 only applies to the adjacency matrix on regular graphs of even degree (as follows from the assumption ν = 1, see §3.1). The following proposition uses Theorem 1.2 to provide concrete applications to non-regular graphs endowed with a periodic potential. Proposition 1.3 (Cartesian products). Suppose that Γ is a Z d a -periodic graph with ν = 1, and let G F be any finite graph, endowed with some potential Q. Then the Cartesian product Γ G F is a periodic graph with fundamental crystal V f = G F . Moreover, assumption (1.3) is satisfied in this case, so conclusion (1.4) is true.

If for Γ G F , the orthonormal basis is of the form ψ n,j = φ n ⊗ w j , where (φ n ) is an orthonormal eigenbasis for H Γ N , and (w j ) is an orthonormal eigenbasis for H G F , then

(1.6) ψ n,j , Op N (a)ψ n,j = vq∈G F a(• + v q ) |w j (v q )| 2 ,
where a(•

+ v q ) = 1 N d k∈L d N a(k a + v q ).
For general ν and general observables, the quantity ψ

(N ) u , Op N (a)ψ (N ) u
in (1.4) can be complicated and it can depend on the basis ψ (N ) u as in (1.6). Such base-dependence never appeared in the tree models of [START_REF] Anantharaman | Quantum ergodicity on regular graphs[END_REF][START_REF] Anantharaman | Quantum ergodicity on graphs : from spectral to spatial delocalization[END_REF]. There, the theorems established that ψ

(N ) λ j , aψ (N ) λ j ≈ v∈Γ N a(v) Im g λ j N (ṽ,ṽ) v∈Γ N Im g λ j N (ṽ,ṽ)
, where g z N is the Green's function of the universal cover of Γ N .

In particular, it is certainly independent of ψ (N )

λ j .
Here we have a different phenomenon which can be regarded as partial quantum ergodicity. For a concrete example in which ψ

(N ) u , Op N (a)ψ (N ) u
is basis-dependent, see §4.5. Such partial quantum ergodicity can be violated even in dimension one : Proposition 1.4. There exist Z-periodic graphs which violate (1.4).

We give examples in §3.3 and §3.4. These graphs have point spectrum and (1.3) is not satisfied. It is natural to ask if the assumption is always satisfied when the spectrum of H on Γ is purely absolutely continuous. We construct a counterexample in §3.4 : Proposition 1.5. There exist periodic graphs with purely absolutely continuous spectrum violating assumption (1.3).

However, perhaps graphs with a more regular shape satisfy this property. The most immediate open example is the integer lattice.

Open problem. Is assumption (1.3) satisfied for Schrödinger operators on Z d with a periodic potential ?

If the answer is yes, then Theorem 1.2 gives quantum ergodicity for these classical operators. We only solve the one-dimensional case (positively) in §4.4.

Remark 1.6. Instead of considering the whole spectrum in Theorem 1.2, we can instead suppose that (1.3) is satisfied in some interval I, then the conclusion (1.4) now holds if we average over λ (N ) u ∈ I instead of u ∈ Γ N . This is similar to what is done in [START_REF] Anantharaman | Quantum ergodicity on graphs : from spectral to spatial delocalization[END_REF] for the high girth regime. In other words, if part of the spectrum is well-behaved, then the corresponding eigenfunctions are quantum ergodic. This is helpful for example for graphs having flat bands but satisfying (1.3) away from the degenerate eigenvalue. Then our theorem applies to these regions. For the technical details, see Remark 2.3.

Remark 1.7 (Convergence rate). The proof shows that the variance on the LHS of (1.4) is essentially bounded from above by the fraction in (1.3). For ν = 1, we bound the latter by C N in §3.1, so the speed of convergence is at least 1 N in Theorem 1.1, which is significantly faster than the logarithmic rate 1 log N of the tree case [1, 2, 3]. 1.2. Stronger statements. The following two paragraphs illustrate that one cannot obtain much stronger results than the ones we provide. 1.2.1. Quantum Unique Ergodicity. In [START_REF] Anantharaman | Quantum ergodicity on large regular graphs[END_REF], it was suggested to check whether (1.7) lim

N →∞ sup 1≤j≤|Γ N | | ψ (N ) j , a N ψ (N ) j -a N | = 0
as an indication of quantum unique ergodicity (QUE). This would mean that we can avoid the Cesàro average in (1.2). This criterion is too strong however, at least in our context, in fact it is already violated for the adjacency matrix on Z d . See §5.1.

1.2.2. Eigenvector correlators. In [START_REF] Anantharaman | Quantum ergodicity on regular graphs[END_REF], instead of taking observables a N (n) which are functions on Γ N , a quantum ergodicity theorem was proved more generally for band matrix observables, that is, K N (n, m), where

K N (n, m) = 0 if d(n, m) > R. It was shown (in Cesàro sense) that ψ (N ) j , K N ψ (N ) j ≈ K N λ j , where K N λ = 1 |Γ N | n,m K N (n, m)Φ λ (d(x, y)
) and Φ λ is the spherical function of the tree; it has an explicit form in terms of Chebyshev polynomials. Since ψ

(N ) j , K N ψ (N ) j = n,m K N (n, m)ψ (N ) j (n)ψ (N ) j (m), this shows that the eigenfunction correlator ψ (N ) j (n)ψ (N ) j (m) ≈ 1 |Γ N | Φ λ (d(n, m
)), a universal quantity; this generalizes the statement that |ψ

(N ) j (n)| 2 ≈ 1 |Γ N |
. This stronger statement fails in our case; ψ

(N ) j (n)ψ (N )
j (m) is not universal, it depends on the basis, even for A Z d . See §5.1.

Still, our proof can be generalized to matrix observables

K N . If ν = 1, we show that 1 N d j∈L d N ψ (N ) j , Kψ (N ) j -K ψ (N) j 2 → 0 , where K ψ = 1 N d n∈L d N |τ |≤R K(n a , n a + τ a ) ψ, ψ(• + τ a )
, and R is the width of the band matrix. So in a weak sense, ψ

(N ) j (n a )ψ (N ) j (n a + τ a ) ≈ 1 N d ψ (N ) j , ψ (N )
j (• + τ a ) . 1.3. Structure of the paper. We prove the general Theorem 1.2 in Section 2. In §3.1, we prove that (1.3) is satisfied for ν = 1, thereby proving Theorem 1.1. We then discuss Cartesian products in §3.2 and prove Proposition 1.3. In §3.3 and §3.4, we discuss graph decorations, tensor products and strong products of graphs, giving examples of graphs violating quantum ergodicity. In Section 4, we give more specific examples satisfying quantum ergodicity. Finally in Section 5, we discuss complementary results such as quantum unique ergodicity, eigenvector correlators, the Bloch theorem, as well as a general method that could be useful for checking (1.3).

Proof of the general criterion

Here we prove Theorem 1.2. The argument is very different than the proof for trees [START_REF] Anantharaman | Quantum ergodicity on regular graphs[END_REF][START_REF] Anantharaman | Quantum ergodicity on large regular graphs[END_REF][START_REF] Anantharaman | Quantum ergodicity on graphs : from spectral to spatial delocalization[END_REF]. We will use some ideas from [START_REF] Klimek | Ergodic Properties of the Quantum Geodesic Flow on Tori[END_REF] where ergodic averages for the continuous Laplacian -∆ on the torus R d /Z d are studied, in the high frequency limit.

2.1. Step 1. Since e -itH N ψ (N ) u = e -itλ (N) u ψ (N ) u , ψ (N ) u , e itH N ae -itH N ψ (N ) u = ψ (N ) u , aψ (N ) u
and we have

(2.1) ψ (N ) u , aψ (N ) u = ψ (N ) u , 1 T T 0 e itH N ae -itH N dtψ (N ) u .
In the spirit of Egorov's theorem, we show the sandwich e itH N ae -itH N can be expressed as a kind of phase space operator.

Let

L d N = [[0, N -1]] d and define U : ℓ 2 (Γ N ) → ⊕ j∈L d N ℓ 2 (V f ) by (2.2) (U ψ) j (v n ) = 1 N d/2 k∈L d N e -ij b N •ka ψ(v n + k a ) .
We see as in [6, §3.2] that U is unitary and

(2.3) U H N U -1 = ⊕ j∈L d N H j b N ,
where

H(θ b ) acts on ℓ 2 (V f ) by (2.4) H(θ b )f (v n ) = u∼vn e iθ b •⌊u⌋a f ({u} a ) + Q(v n )f (v n ) ,
the sum is over the nearest neighbors u of v n in the whole graph Γ (not just V f ) and using (1.1), we expanded any u ∈ V by u = ⌊u⌋ a + {u} a , with ⌊u⌋ a ∈ Z d a and {u} a ∈ V f . We have

U -1 (g j ) j∈L d N = ψ, where ψ(k a + v n ) = 1 N d/2 r∈L d N g r (v n )e ika• r b N . To see this, note that ( 1 N d/2 e -ika•j b /N ) j∈L d N is an orthonormal basis of ℓ 2 (L d N ) (use the relation k a • j b = 2πk • j). So for such ψ we have (U ψ) j (v n ) = 1 N d k,r∈L d N e -ij b N •ka g r (v n )e ika• r b N = k∈L d N 1 N d/2 e -2πik N •• , g • (v n ) ℓ 2 (L d N ) ( 1 N d/2 e -ij b N •ka ) = g j (v n ). Note that H(θ b ) is a ν × ν matrix with eigenvalues E 1 (θ b ), . . . , E ν (θ b ). Let P s (θ b ) be the corresponding eigenprojections. Let e (N ) r (k) := 1 N d/2 e 2πik•r/N . Given F ∈ ℓ 2 (L 2d N × V 2 f ), we now let (2.5) Op N (F )ψ(k a + v n ) := r∈L d N ν ℓ=1 (U ψ) r (v ℓ )F (k, r; v n , v ℓ )e (N ) r (k) , The "quantization" (2.5) is such that if F (k, r; v n , v ℓ ) = F (k a +v n )δ vn,v ℓ , then Op N (F )ψ = F ψ.
The presence of δ vn,v ℓ may seem unusual; indeed it would not be here if we were dealing with just the adjacency matrix on Z d . The presence of δ vn,v ℓ is related to the fact that the Floquet transform (2.2) is only a partial transform in the sense that it keeps v n fixed. Define

(2.6) F T (k, r; v n , v ℓ ) := m∈L d N ν q,s,w=1 1
T T 0 e it[Es( r b +m b N )-Ew( r b N )] dt × P s r b + m b N (v n , v q )a (N ) m (v q )P w r b N (v q , v ℓ )e (N ) m (k) ,
where a

(N ) m (v q ) := e im b ••a N N d/2 , a(v q + • a ) ℓ 2 (L d N ) are the Fourier coefficients of a. Lemma 2.1. We have 1 T T 0 e itH N ae -itH N dt = Op N (F T ).
Although the actual definitions are somewhat long, the meaning is straightforward: this sandwich can be expressed in phase space. F T "smoothes" the function over different eigenvalues of the phase space operator, and Op N gives the averaging under which this occurs.

Proof. First, we expand ψ in order to relate it to the form of Op N (F T ).

ψ(k a + v n ) = (U -1 U ψ)(k a + v n ) = r∈L d N (U ψ) r (v n )e (N ) r (k) .
Recalling (2.3), we obtain

(H N ψ)(k a + v n ) = r∈L d N H r b N (U ψ) r (v n )e (N ) r (k) .
Knowing this, we can now examine the operator e itH N ae -itH N and expand over the various (v n , v q ). This yields

(e itH N ae -itH N ψ)(k a + v n ) = r∈L d N ν q=1 e itH( r b N ) (v n , v q )(U ae -itH N ψ) r (v q )e (N ) r (k) . Expanding a(v q + n a ) = 1 N d/2 m∈L d N a (N ) m (v q )e im b •na N , we have (U ae -itH N ψ) r (v q ) = 1 N d n∈L d N m∈L d N e -ir b N •na a (N ) m (v q )e im b •na N (e -itH N ψ)(v q + n a ) = 1 N d/2 m∈L d N a (N ) m (v q )(U e -itH N ψ) r-m (v q ) .
Here, rm is understood in (Z/N Z) d . More precisely, if r im i is negative for some i, it is replaced by N + r im i . But this last term can be further reduced as

(U e -itH N ψ) r-m (v q ) = [e -itH( r b -m b N ) (U ψ) r-m ](v q ) = ν ℓ=1 e -itH( r b -m b N ) (v q , v ℓ )(U ψ) r-m (v ℓ ) . Moreover, we can write H(θ b ) = ν s=1 E s (θ b )P s (θ b ) through its eigendecomposition. Sim- ilarly, e ±itH(θ b ) = ν s=1 e ±itEs(θ b ) P s (θ b ).
Applying this to our original operator gives us

(e itH N ae -itH N ψ)(k a + v n ) = 1 N d/2 r,m∈L d N ν q,ℓ,s,w=1 e itEs( r b N ) P s r b N (v n , v q ) × a (N ) m (v q )e -itEw( r b -m b N ) P w r b -m b N (v q , v ℓ )(U ψ) r-m (v ℓ )e (N ) r (k) = 1 N d/2 r,m∈L d N ν q,ℓ,s,w=1 e it[Es( r b +m b N )-Ew( r b N )] P s r b + m b N (v n , v q ) × a (N ) m (v q )P w r b N (v q , v ℓ )(U ψ) r (v ℓ )e (N ) r+m (k) with r + m again understood in (Z/N Z) d . Since 1 N d/2 e (N ) r+m (k) = e (N ) r (k)e (N ) m (k), we get 1 T T 0 e itH ae -itH dtψ(k a + v n ) = r∈L d N ν ℓ=1 (U ψ) r (v ℓ )F T (k, r; v n , v ℓ )e (N ) r (k) , with F T in (2.6). Therefore, according to (2.5), 1 T T 0 e itH N ae -itH N dt = Op N (F T ). 2.2. Step 2. Now we observe that if m ∈ L d N and s, w ∈ {1, . . . , ν} are such that E s ( r b +m b N ) -E w ( r b N ) = 0, then the corresponding term in F T vanishes as T → ∞. So define (2.7) b(k, r, v n , v ℓ ) = m∈L d N ν q,s,w=1 1 Sr (m, s, w)P s r b + m b N (v n , v q ) × a (N ) m (v q )P w r b N (v q , v ℓ )e (N ) m (k) , where S r = {(m, s, w) : E s ( r b +m b N ) -E w ( r b N ) = 0}. Lemma 2.2. We have convergence in norm, 2 lim T →∞ Op N (F T ) -Op N (b) 2 HS = 0.
Proof. To show this, we use a special basis of ℓ 2 (V N ), namely φ

(N ) r,v ℓ = e (N ) r ⊗ δ v ℓ . More precisely, φ (N ) r,v ℓ (k a + v q ) = e (N ) r (k)δ v ℓ (v q ) = e 2πir•k/N N d/2 δ v ℓ (v q ). We note that (2.2) implies (U φ r,v ℓ ) j (v q ) = e (N ) j , e (N ) r ℓ 2 (L d N ) δ v ℓ (v q ) = δ j,r δ v ℓ (v q )
. By definition (2.5), this implies Op N (F )φ

(N ) r,v ℓ (k a + v n ) = F (k, r, v n , v ℓ )e (N ) r (k). Note that F (•, r, ⋆, v ℓ )e (N ) n (•) 2 ℓ 2 (Γ N ) = 1 N d F (•, r, ⋆, v ℓ ) 2 ℓ 2 (Γ N ) , where • runs over k ∈ L d N and ⋆ runs over v n ∈ V f . Therefore, (2.8) Op N (F ) 2 HS = r∈L d N ν ℓ=1 Op N (F )φ (N ) r,v ℓ 2 = 1 N d r∈L d N ν ℓ=1 F (•, r, ⋆, v ℓ ) 2
To prove the lemma, we should thus examine the corresponding norm of the symbols,

F T (•, r, ⋆, v ℓ ) -b(•, r, ⋆, v ℓ ) 2 = m∈L d N ν q,s,w=1 1 S c r (m, s, w) e iT [Es( r b +m b N )-Ew( r b N )] -1 T [E s ( r b +m b N ) -E w ( r b N )] × P s r b + m b N (⋆, v q )a (N ) m (v q )P w r b N (v q , v ℓ )e (N ) m (•) 2 .
We deduce that

Op N (F T )-Op N (b) 2 HS = 1 T 2 N d r,m∈L d N ν ℓ=1 ν q,s,w=1 1 S c r (m, s, w) e iT [Es( r b +m b N )-Ew( r b N )] -1 E s ( r b +m b N ) -E w ( r b N ) × P s r b + m b N (⋆, v q )a (N ) m (v q )P w r b N (v q , v ℓ ) 2 C ν ≤ C N,a T 2 ,
where C N,a is finite for any N and is independent of T . Taking T → ∞ yields that Op N (F T ) → Op N (b) in HS norm.

2.3.

Step 3. We are thus reduced to studying Op N (b) with b given in (2.7). Note that ν p=1 P p (θ) = id, so ν p=1 P p (θ)(v i , v j ) = δ v i ,v j . Therefore, if we remove the 1 Sr term, (2.7) becomes (2.9)

m∈L d N a (N ) m (v n )e (N ) m (k)δ vn,v ℓ = a(k a + v n )δ vn,v ℓ
and the corresponding Op N applied to ψ simply gives a(k a +v n )ψ(k a +v n ). Hence, Op N (b)ψ is just aψ but with many suppressed Floquet modes. Let a be the part of b corresponding to m = 0. Let ã = a -Op N (a) and c = ba. Then collecting the previous steps, we have

u∈Γ N | ψ (N ) u , ãψ (N ) u | 2 = u∈Γ N lim T →∞ | ψ (N ) u , Op N (F T -a)ψ (N ) u | 2 ≤ u∈Γ N lim T →∞ 2( Op N (c)ψ (N ) u 2 + Op N (F T -b)ψ (N ) u 2 ) = 2 Op N (c) 2 HS .
2 It is worthwhile to note that in the case of trees [1, 2, 3], we usually evolve the dynamical system in time T , essentially up to the girth of the graph, take the size of the graph N → ∞, then finally take T → ∞. Here we first consider the equilibrium limit in T , then take N → ∞ in the end of the proof.

Proof of (1.4). It now suffices to show that lim

N →∞ 1 |Γ N | Op N (c) 2 HS = 0. Using (2.8), we have 1 |Γ N | Op N (c) 2 HS = 1 νN 2d r∈L d N ν ℓ=1 c(•, r, ⋆, v ℓ ) 2 ℓ 2 (V N ) . We thus consider 1 N 2d r∈L d N ν ℓ=1 m =0 ν q,s,w=1 1 Sr (m, s, w)P s r b + m b N (⋆, v q )a (N ) m (v q )P w r b N (v q , v ℓ )e (N ) m (•) 2 = 1 N 2d r∈L d N ν ℓ=1 m =0 ν n=1 ν q,s,w=1 1 Sr (m, s, w)P s r b + m b N (v n , v q )a (N ) m (v q )P w r b N (v q , v ℓ ) 2
Denote P s := P s ( r b +m b N ), P w := P w ( r b N ) and expand the square modulus to give

(2.10) 1 N 2d r∈L d N ν ℓ=1 m =0 ν n=1 ν q,s,w,q ′ ,s ′ ,w ′ =1 1 Sr (m, s, w)P s (v n , v q )a (N ) m (v q )P w (v q , v ℓ ) × 1 Sr (m, s ′ , w ′ )P s ′ (v n , v q ′ )a (N ) m (v q ′ )P w ′ (v q ′ , v ℓ ) . But ν n=1 P s (v n , v q )P s ′ (v n , v q ′ ) = ν n=1 (P s δ vq )(v n )(P s ′ δ v q ′ )(v n ) = P s ′ δ v q ′ , P s δ vq . Similarly, ν ℓ=1 P w (v q , v ℓ )P w ′ (v q ′ , v ℓ ) = ν ℓ=1 P w δ vq (v ℓ )P w ′ δ v q ′ (v ℓ ) = P w δ vq , P w ′ δ v q ′ . If E s = E s ′ or E w = E w ′ , these scalar products vanish. So (2.10) is concentrated on the s ′ , w ′ for which E s ′ = E s and E w ′ = E w , in which case 1 Sr (m, s, w) = 1 Sr (m, s ′ , w ′ ) and we obtain 1 N 2d r∈L d N m =0 ν q,s,w,q ′ ,s ′ ,w ′ =1 1 Sr (m, s, w) P s ′ δ v q ′ , P s δ vq a (N ) m (v q ) P w δ vq , P w ′ δ v q ′ a (N ) m (v q ′ ) = 1 N 2d m∈L d N m =0 ν q,q ′ =1 a (N ) m (v q ′ )a (N ) m (v q ) r∈L d N ν s,w,s ′ ,w ′ =1
1 Am (r, s, w) P s ′ δ v q ′ , P s δ vq P w δ vq , P

w ′ δ v q ′ ,
where

A m = {(r, s, w) : E s ( r b +m b N ) -E w ( r b N ) = 0}
and we used that (m, s, w) ∈ S r ⇐⇒ (r, s, w) ∈ A m . By hypothesis (1.3), we know that (2.11) lim

N →∞ sup m∈L d N m =0 |A m | N d = 0. Then since | P s ′ δ v q ′ , P s δ vq | ≤ 1, it follows that the above is o N (1) 1 N d m ν q,q ′ =1 a (N ) m (v q ′ )a (N ) m (v q ) = o N (1) 1 N d ν q,q ′ =1 a(• a + v q ′ ), a(• a + v q ) ℓ 2 (L d N ) = o N (1)
using |a(n a + v q )| ≤ 1. This completes the proof of (1.4).

2.4.

Step 4. Let us now explore the main term a. Recall that it corresponds to m = 0 in (2.7). Having (0, s, w) ∈ S r means that E s ( r b N ) = E w ( r b N ). This is automatically true for w = s. Thus,

a = ν q,s=1 a (N ) 0 (v q )e (N ) 0 (k)P s r b N (v n , v q ) P s r b N (v q , v ℓ ) + w =s Es=Ew P w r b N (v q , v ℓ ) = ν q=1 a(• + v q ) ν ′ s=1 P Es r b N (v n , v q )P Es r b N (v q , v ℓ ) , (2.12)
where a(• Proof of (1.5). By the definition of Op N , we can write out

+ v q ) = 1 N d n∈L d N a(n a + v q ), ν ′ ≤ ν
ψ, Op N (a)ψ = k∈L d N vn∈V f ψ(k a + v n )[Op N (a)ψ](k a + v n ) = vn∈V f r∈L d N ν ℓ=1 (U ψ) r (v ℓ )a(r, v n , v ℓ ) k∈L d N ψ(k a + v n )e (N )
r (k) .

But k ψ(k a + v n )e (N )
r (k) = (U ψ) r (v n ). Thus, ψ, Op N (a)ψ = vn∈V f r∈L d N ν ℓ=1 (U ψ) r (v ℓ )(U ψ) r (v n )a(r, v n , v ℓ ) = ν q=1 r∈L d N ν ℓ=1 ν ′ s=1 P Es (v q , v ℓ )(U ψ) r (v ℓ ) ν n=1 P Es (v q , v n )(U ψ) r (v n ) a(• + v q ) = ν q=1 a(• + v q ) r∈L d N ν ′ s=1 [P Es (U ψ) r ](v q )[P Es (U ψ) r ](v q ) .
where P s = P s ( r b N ). We have shown that

(2.13) ψ, Op N (a)ψ = ν q=1 a(• + v q ) r∈L d N ν ′ s=1 P Es r b N (U ψ) r (v q ) 2 .
In the special case where a(• + v q ) = a(• + v 1 ) for q = 1, . . . , ν, the above reduces to

a(• + v 1 ) r∈L d N ν ′ s=1 P Es (U ψ) r 2 C ν = a(• + v 1 ) r∈L d N (U ψ) r 2 C ν = a(• + v 1 ) ψ 2 .
In particular, ψ = ψ

(N ) u
gives the uniform average a(•

+ v 1 ) = 1 N d n∈L d N a(n a + v 1 ) = 1 |Γ N | v∈Γ N a(v).
Remark 2.3. The main theorem holds more generally if instead of summing over the whole spectrum in (1.4), we sum over eigenvalues in some interval I, in which case we only need (1.3) to hold on I. To see this, we slightly modify the proof as follows : in (2.1), we insert a spectral projection χ I (H N ), so the operator is now 1 T T 0 e itH N ae -itH N χ I (H N ) dt. In (2.6), we replace the sum over all w by the sum over E w ( r b N ) ∈ I. In fact, by adding the spectral projection through the proof of Lemma 2.1, we now get (U e -itH χ

I (H)ψ) r-m = e -itH( r b -m b N ) χ I (H( r b -m b N ))(U ψ) r-m .
Consequently, the limiting symbol b now also sums over E w ( r b N ) ∈ I instead. The proofs carry over mutatis mutandis. In the end, the symbol a in (2.12) now sums over E s ( r b N ) ∈ I. This gives the illusory impression that the weighted average changes, which of course makes no sense as the term ψ

(N ) u , aψ (N ) u
should approach a fixed quantity whether the Cesàro mean is over the whole spectrum or not. However the quantity ψ

(N ) u , Op N (a)ψ (N ) u
is indeed the same as before. In fact, if we know that λ (N ) u ∈ I, we may again insert a projector so that (U ψ

(N ) u ) r in (2.13) becomes (U χ I (H N )ψ (N ) u ) r = χ I (H( r b N ))(U ψ (N )
u ) r , so the sum over all E s in (2.13) reduces to the sum over E s ( r b N ) ∈ I, which is what we obtained when averaging over I.

Special classes of graphs

In this section we discuss the validity of assumption (1.3) for various classes of graphs. We start with graphs having ν = 1, proving Theorem 1.1. We then discuss Cartesian products, proving Proposition 1.3, and conclude with graph decorations, tensor and strong products, proving Propositions 1.4 and 1.5 along the way. for some n (1) , . . . , n (D) ∈ {0, 1, . . . } d \ {0}. We only have one eigenvalue here given by E(θ b ) = H(θ b ). So we should show that for any fixed m = 0, the equation

(3.1) E r b + m b N -E r b N = 2 D p=1 cos 2π (r + m) • n (p) N -cos 2π r • n (p) N = 0 has o(N d ) solutions in r ∈ L d N .
By the sum to product formula, we are lead to consider the zeroes of

(3.2) f m r N := D p=1 sin π m • n (p) N sin π (2r + m) • n (p) N .
For this, we consider the projection of the surface

A m = {r ∈ L d N : f m ( r N ) = 0} onto a vector φ ∈ L d
N to be specified. More precisely, given j ∈ L d N , we write j = r + yφ, for r ∈ φ ⊥ and y = φ,j φ 2 . Note that y ∈ [0,

N -1] since 0 ≤ φ i j i ≤ (N -1) φ i ≤ (N -1) φ 2 i for φ ∈ L d N .
We will show that for fixed r ∈ φ ⊥ , there are at most M points y such that f m ( j N ) = 0, with M independent of N . By varying r ∈ φ ⊥ , it follows that

|A m | ≤ M |φ ⊥ | ≤ M N d-1 = o(N d ) as required.
We therefore consider the function

g m,r (x) = f m r N + xφ = 0 for x ∈ [0, 1). Denote (3.3) α p = sin π m • n (p) N , β p = π (2r + m) • n (p) N , γ p = 2φ • n (p) .
Then

g m,r (x) = D p=1 α p sin(β p + πγ p x) = 1 2i D p=1
α p (e iβp e iπγpxe -iβp e -iπγpx ) .

Setting z = e iπx , this reduces to

gm,r (z) = D p=1 (ρ p z γp + ρ ′ p z -γp )
for some ρ p , ρ ′ p ∈ C. By definition (3.3), γ p is an integer. We thus seek the solutions of gm,r (z) on the unit circle. We have gm,r (z) = 0 iff D p=1 (ρ p z γ⋆+γp + ρ ′ p z γ⋆-γp ) = 0, where γ ⋆ = max p γ p . This is a polynomial in z. By the fundamental theorem of algebra, if this polynomial is nontrivial, it has at most M = 2 max p γ p roots. In turn, we have at most M solutions x j for g m,r (x) = 0, and the proof of (2.11) is complete (recall the discussion after (3.2)).

So it remains to check the polynomial z γ * g m,r (z) = D p=1 (ρ p z γ⋆+γp + ρ ′ p z γ⋆-γp ) is nontrivial. For this, we check that 1. At least one ρ p is nonzero. 2. We can choose φ such that γ p = γ p ′ for p = p ′ . This way, no two terms in the sum have the same power, so no cancellation can occur. Note that no γ p is zero, so no cancellation can occur from ρ p ′ = -ρ p .

Proof of 1. Since m = 0, we have m j = 0 for some j. Note that o + a j ∈ Γ by translation invariance. Since Γ is connected, some integer combination

o+ D p=1 k p n (p)
a of the neighbors of o is o + a j , where k p ∈ Z is the number of adjacencies of type n (p) traversed on the geodesic from o to o + a j . It follows that

(3.4) sin π m N • D p=1 k p n (p) = sin m b 2N • D p=1 k p n (p) a = sin m b 2N • a j = sin πm j N = 0 .
If we had sin(π m•n (p) N ) = 0 for all p, we would have m•n (p)

N

∈ Z for all p and thus m N • D p=1 k p n (p) ∈ Z, contradicting (3.4). Thus, α p = 0 for at least one p. This completes the proof. Proof of 2. We need φ to avoid the subspaces

V p,p ′ = {v : v •n (p) = v •n (p ′ ) } for all D(D -1) pairs of p = p ′ . Each of these is d -1 dimensional, since the n (p) are distinct.
It is not difficult to see that such a φ exists. However, we give a quite explicit construction below, which in turn gives an explicit bound on M .

Suppose we give a list of ℓ D = (d -1)D(D -1) + 1 vectors in L d N such that any d of them forms a basis. Then each of the subspaces V p,p ′ can only contain at most d -1 of our vectors, therefore there must be some vector not contained in any of the subspaces and we are done.

A possible list is given by the row vectors

       1 1 1 • • • 1 1 2 2 2 • • • 2 d-1 1 3 3 2 • • • 3 d-1 . . . . . . . . . . . . 1 ℓ D ℓ 2 D • • • ℓ d-1 D        .
Indeed, any subset of d of these vectors, say the ones from the x 1 , . . . , x d rows, forms a Vandermonde matrix with determinant i<j (x ix j ), which is nonzero, meaning any set of d vectors is linearly independent. This finishes the proof.

We may obtain an upper bound over M = 2 max p γ p . In fact, the worst case is if the last vector in the list is the first φ that avoids all V p,p ′ . In this case,

γ p = 2φ • n (p) ≤ 2dℓ d-1 D q, where q = max i,p n (p) i , so M ≤ 4dℓ d-1 D q.
3.2. The case of Cartesian products. The Cartesian product Γ G of Γ and G is the graph with vertex set

V (Γ) × V (G), in which (u, v) ∼ (u ′ , v ′ ) if either (i) (u = u ′ and v ∼ v ′ ), (ii) or (u ∼ u ′ and v = v ′ ).
For example, to construct Z P 2 , where P 2 is the 2-path, replace each vertex of Z with a 2-path, and connect edges between matching vertices. The result is an infinite ladder. Similarly, for Z C p , where C p is a p-cycle, replace each vertex of Z with a p-cycle, and connect edges between matching vertices (Figure 4). The graph is 4-regular, naturally embedded in R 3 , and is clearly Z-periodic with fundamental crystal V f = C p . We may endow C p with a potential Q and copy it in each layer. Then

H(θ b )f (u, v) = 2 cos 2πθf (u, v) + f (u, v + 1) + f (u, v -1) + Q v f (u, v). In other words, H(θ b ) = A Z (θ b ) ⊗ I + I ⊗ H G F .
The eigenvalues are thus {2 cos 2πθ + µ j }, where {µ j } p j=1 are the eigenvalues of the Schrödinger operator of the p-cycle. These observations are general :

Lemma 3.1. If Γ is a periodic graph with ν = 1 and G F is a finite graph endowed a potential Q, then Γ G F is a periodic graph with fundamental crystal V f = G F and (3.5) H Γ G F (θ b ) = H Γ (θ b ) ⊗ I + I ⊗ H G F . Proof. Replace each u ∈ Γ by a copy of G F . The result has vertex set V (Γ) × V (G F ). According to rules (i)-(ii), we should have A Γ G F = I ⊗ A G F + A Γ ⊗ I.
This means that if we arrange the vertices of Γ G F as successive G F -layers, then a given (u, v) is connected on the one hand to the neighbors (u, v ′ ) in the same layer (rule (i)) and to the neighbors (u ′ , v) outside (rule (ii)). This means that the edges are precisely the old edges of G F in each layer, as well as bridges between successive layer between the matching vertices. Recalling (2.4), we see that the θ-dependence only arises in the bridges from (u, v) to another layer (the neighbors within G F have ⌊u⌋ a = 0). The bridges occur precisely at the bridges from u to its neighbors in Γ. We conclude that

A Γ G F (θ b ) = A Γ (θ b ) ⊗ I + I ⊗ A G F .
If we finally endow G F a potential and copy it across the layers, then (Qf

)(u, v) = Q v f (u, v), so we obtain (3.5) (note that A Γ (θ b ) = H Γ (θ b ) as ν = 1). Proof of Proposition 1.3. Since ν = 1 for Γ, H Γ (θ b ) has just one eigenvalue E Γ (θ b ). So the spectrum of H Γ G F is the set {E Γ (θ b ) + µ j }
, where µ j are the eigenvalues of H G F on the finite graph G F .

Given nonzero m, we should thus control the quantity

E s r b + m b N -E w r b N = E Γ r b + m b N -E Γ r b N + µ s -µ w .
Here,

E Γ (θ b ) = 2 D p=1 cos(2πθ • n (p)
) is precisely the quantity we controlled in §3.1. Following the arguments, we see that the same proof continues to hold here. In fact, gm,r (z) only has an additional term µ sµ w , and the proof continues to hold, as no γ p is zero so this term cannot induce cancellations in the polynomial z γ * g m,r (z). Thus, the quantity in (1.3) is ≤ M N -1 → 0 as required, with the same M ≤ 4dℓ d-1 D q of the case ν = 1. This shows that the assumption of Theorem 1.2 is satisfied for Γ G F .

By (3.5), the eigenvectors of

H Γ G F (θ b ) are just the eigenvectors of H G F (recall that H Γ (θ b
) is just a scalar 1×1 matrix). They are thus independent of θ b , and so are the eigenprojectors P s (θ b ). This makes (2.13) a bit simpler here. If moreover we choose ψ = ψ (N ) u to consist of a tensor basis, that is, ψ (N ) u = φ n ⊗ w j , where (φ n ) is an orthonormal eigenbasis of H Γ on Γ and (w j ) is an orthonormal eigenbasis of H G F , then the expression simplifies further. In fact, recalling (2.2), we have (U ψ) r (v q ) = 1

N d/2 k e -2πir•k/N φ n (k a )w j (v q ) = φ n (r)w j (v q ), where φ n (r) is the Fourier coefficient of φ n in the basis e (N ) k of ℓ 2 (L d N )
. Hence, (P s (U ψ) r )(v q ) = φ n (r)(P s w j )(v q ). Thus, (2.13) simplifies to

ν q=1 a(• + v q ) r ν ′ s=1 | φ n (r)| 2 |(P Es w j )(v q )| 2 = ν q=1 a(• + v q ) ν ′ s=1 |(P Es w j )(v q )| 2 ,
where we used that φ n 2 = 1. But w j is an eigenvector, so P Es w j = w j if E s = E j and P Es w j = 0 otherwise. This completes the proof.

Graph decorations.

Another way to create a new graph from given infinite and finite graphs Γ and G F is to simply attach a copy of G F at each vertex of Γ. More precisely, we identify a special vertex o F ∈ G F to each v ∈ Γ. This process is called graph decoration. A very simple example is given in Figure 2. The resulting graph is sometimes denoted by Γ ⊳ G F (which reflects the procedure). In contrast to Cartesian products, this process can be problematic for delocalization. For example, as shown in Figure 2, this can create compactly supported eigenfunctions. The corresponding eigenvalue is a flat band, i.e. an infinitely degenerate eigenvalue. The example in Fig. 2 has the Floquet eigenvalues {-1, 2 cos 2πθ+1± √ 4 cos 2 2πθ-4 cos 2πθ+9 2

}. This generates the spectrum of H = A consisting of two bands which do not intersect. This spectrum is not very nice as the eigenvalue -1 is embedded in the left band, as can be seen by taking θ = 1 4 . It may be interesting to observe that in general, if Γ is a periodic graph having ν = 1, then Γ, Γ G F and Γ ⊳ G F are all "loop graphs" in the sense of Korotyaev and Saburova [11]. This class of graphs was singled out in [START_REF] Korotyaev | Schrödinger operators on periodic discrete graphs[END_REF] for being more amenable to spectral analysis. We see that not all graphs in this class are quantum ergodic.

Proof of Proposition 1.4. For the graph in Fig. 2, we have |Γ N | = 3N , and on Γ N , we may construct N localized eigenfunctions f j , one on each triangle, each supported on only two vertices. Let N be even and take the locally constant observable a which is identically 1 on triangles attached to even vertices, and identically zero on triangles attached to odd vertices. Then a = 1 2 . On the other hand, if we normalize the eigenfunctions f j so that their values are ( 1

√ 2 , -1 √ 2 , 0, 0, . . . , 0), then f 2j , af 2j = v a(v)|f 2j (v)| 2 = 1, while f 2j+1 , af 2j+1 = 0 for each j. Hence, 1 |Γ N | u∈Γ N | ψ (N ) u , aψ (N ) u -a | 2 ≥ 1 3N N j=1 | f j , af j -a | 2 = 1 3N N 2 1 - 1 2 2 + 0 - 1 2 2 = 1 12 .
3.4. More product operations. Besides the Cartesian product, some known operations to construct new graphs from old are the tensor product and the strong product of graphs.

3.4.1. Strong products. The strong product G ⊠ H has vertex set V (G) × V (H), with (u, v) ∼ (u ′ , v ′ ) iff (u = u ′ and v ∼ v ′ ) or (u ∼ u ′ and v = v ′ ) or (u ∼ u ′ and v ∼ v ′ ).
We thus add more edges to the Cartesian product. This operation is not as well behaved as the Cartesian one. For example, consider Z ⊠ P 2 , where P 2 is a 2-path. The result (Figure 3) is an infinite sequence of boxes ⊠. Still, this product sometimes behaves well. For example, Z ⊠ Z gives the king's graph, which is quantum ergodic since it is periodic with ν = 1.

Tensor products. Next, the tensor product

G× H has vertex set V (G)× V (H), with (u, v) ∼ (u ′ , v ′ ) iff u ∼ u ′ and v ∼ v ′ . Equivalently, A G×H = A G ⊗ A H .
The edges of this product are precisely the ones we added to the Cartesian product when discussing strong products.

The product of two connected graphs is not necessarily connected. For example, the tensor product of two path graphs of length 2 {a, b} and {v, w} gives the union of the two graphs {(a, v), (b, w)} and {(a, w), (b, v)}. To consider a product graph of the form Γ × G F for quantum ergodicity, where Γ is a quantum ergodic graph and G F is some finite graph, we first need Γ × G F to be connected. It turns out this is satisfied if and only if either Γ or G F contains an odd cycle, see [START_REF] Weichsel | The Kronecker product of graphs[END_REF].

Assume now that we are given a periodic Γ with ν = 1, for simplicity. Just like Cartesian products, the tensor structure of the adjacency matrix translates well into the Floquet fibers. To see this, it is best to first picture the product operation. Geometrically, we simply consider the G F -layers structure of Cartesian products, but then we remove all edges and add instead the following ones : a given (u, v) in a G F layer is connected to all vertices (u ′ , v ′ ), where u ′ is in a different G F layer and v ′ ∼ v in G F . Note that V f = G F contains no edges. Instead, if we "project" the edges going from a neighboring G F layer to the starting one, we obtain the finite graph G F that we started with. We may also endow G F with some potential Q which is copied across the layers.

By definition (2.4), we have

H(θ b )f (u, v) = u ′ ∼u,v ′ ∼v e iθ b •⌊u ′ ⌋a f (u, v ′ ) + Q v f (u, v) = H Γ (θ b ) ⊗ H G F f (u, v), where we used here that {(u ′ , v ′ )} a = (u, v ′ ) and ⌊(u ′ , v ′ )⌋ a = ⌊u ′ ⌋ a by construction. This shows that H Γ×G F (θ b ) = H Γ (θ b ) ⊗ H G F . Consequently, (3.6) σ(H Γ×G F (θ b )) = {µ j E Γ (θ b )} ν j=1
, where µ j are the eigenvalues of H G F . Note that if µ j = 0 for some j, then this creates a flat band {0} for H Γ×G F , i.e. an infinitely degenerate eigenvalue.

We now consider the special case of Z × G F . So E Γ (θ b ) = 2 cos 2πθ.

Proof of Proposition 1.5. To construct a counterexample, we take G F such that (i) G F is not bipartite, (ii) 0 / ∈ σ(A G F ), (iii) there exists s such that µ s and -µ s belong to σ(A G F ).

Point (i) is necessary to make Z × G F connected, (ii) is necessary to avoid a point spectrum {0}, and (iii) is what will contradict (1.3).

We take G F = C 3 P 4 as an example, where C 3 and P 4 are the 3-cycle and the 4-path, respectively. We have σ(A C 3 ) = {2, -1, -1} and σ(A

P 4 ) = { 1+ √ 5 2 , √ 5-1 2 , 1- √ 5 2 , -1- √ 5 2 }. So σ(A C 3 P 4 ) is the sum 5 + √ 5 2 , √ 5 + 3 2 , 5 - √ 5 2 , 3 - √ 5 2 , √ 5 -1 2 , √ 5 -3 2 , -1 - √ 5 2 , -3 - √ 5 2 , √ 5 -1 2 , √ 5 -3 2 , -1 - √ 5 2 , -3 - √ 5 2
.

We see all properties are satisfied, take e.g. µ s = √ 5+3 2 . By (3.6), σ(A Z×G F (θ b )) is just {2µ j cos 2πθ}, where µ j runs over the above list of eigenvalues. It follows that σ(A Z×G F ) is purely absolutely continuous (as each Floquet eigenvalue is analytic and nonconstant, see [START_REF] Reed | Methods of Modern Mathematical Physics[END_REF]Th. XIII.86]). The graph Z × G F is also connected, since [[-n, n]] × G F is connected for any n by [START_REF] Weichsel | The Kronecker product of graphs[END_REF].

If

µ s = √ 5+3 2
and µ w = - √ 5-3 2

, we find that

E s (θ b + α b ) -E w (θ b ) = µ s (2 cos(2π(θ + α)) + 2 cos 2πθ) = 4µ s cos π(2θ + α) cos πα .
This is zero if α = 1 2 , for all θ. This suffices to contradict (1.3). In fact, taking m = N 2 ∈ L N for even N , the fraction in (1.3) is equal to 1 and does not vanish.

Concrete examples

4.1. Graphs with scalar fibers. For the adjacency matrix H = A on Z d or the triangular lattice (sometimes called hexagonal, see [START_REF] Korotyaev | Schrödinger operators on periodic discrete graphs[END_REF]Fig. 3]) where each vertex has 6 neighbors, or the king's graph (sometimes called EHM lattice), we have ν = 1 so Theorem 1.1 applies.

The family of periodic graphs having ν = 1 is quite rich. For example, one can consider Z and add edges up to some fixed distance k from each vertex. More precisely,

(Af )(n) = f (n -k) + f (n -k + 1) + • • • + f (n + k -1) + f (n + k) .
Then V f = {0}, a 1 = e 1 and H(θ b ) = 2 cos 2πθ + 2 cos 4πθ + • • • + 2 cos 2πkθ. Similar variations can be performed on Z d .

We remark that the connectedness of Γ is important. For example, if we consider Z with (Af )(n) = f (n-2)+f (n+2), then the graph is disconnected (there are two copies of Z, for the even and odd vertices, respectively). Here, V f = {0}, a 1 = e 1 and H(θ b ) = 2 cos 4πθ, which does not obey (1.3), since for α = The eigenvectors

w ± (θ b ) = 1 √ 2 (1, ±e -iφ(θ b ) ) ⊺ , where φ(θ b ) is the argument of ξ(θ b ). So P ± (θ b )f (v 1 ) = f (v 1 )±e iφ(θ b ) f (v 2 ) 2 , P ± (θ b )f (v 2 ) = f (v 1 )±e iφ(θ b ) f (v 2 ) 2 (±e -iφ(θ) b ). It follows that |P + f (v 1 )| 2 + |P -f (v 1 )| 2 = |f (v 1 )+e iφ(θ b ) f (v 2 )| 2 +|f (v 1 )-e iφ(θ b ) f (v 2 )| 2 4 = |f (v 1 )| 2 +|f (v 2 )| 2 2 = f 2 2 = |P + f (v 2 )| 2 + |P -f (v 2 )| 2 .
We showed that for the honeycomb lattice, (2.13) reduces to

2 q=1 r∈L d N (U ψ) r 2 C ν 2 a(• + v q ) = a(• + v 1 ) + a(• + v 2 ) 2 ψ 2
which is the uniform average. 4.3. Ladder graph. Consider the ladder graph Z P 2 in Figure 1. As a Cartesian product, we already know that Proposition 1.3 holds true, but we show here that we always get the uniform average in this example.

We have and

H(θ b )f (v 1 ) = e 2πiθ f (v 1 ) + e -2πiθ f (v 1 ) + f (v 2 ) and H(θ b )f (v 2 ) = e 2πiθ f (v 2 ) + e -2πiθ f (v 2 )+ f (v 1 ). Thus, H(θ b ) = 2 cos
P ± f (v 2 ) = -f (v 1 )±f (v 2 ) 2
. As in the honeycomb lattice, we deduce that (2.13) reduces to a(•+v 1 ) + a(•+v 2 ) 2 ψ 2 . If we endow P 2 with a potential Q • , Q • , then we get a ladder with a potential coming in two parallel sheets, the first sheet only containing Q • , the second only Q • . The construction can be generalized to Z P k to create an infinite k-strip. Proposition 1.3 continues to apply, but the average may not be uniform. For this choice of λ we thus have p(λ; z) = p(λ; zζ) = 0. So z + z -1 = zζ + (zζ) -1 . This yields a quadratic expression for z. Hence, for any fixed α = 0, there are at most two θ such that E s (θ b + α b ) = E w (θ b ). This implies (1.3).

Q i . We have V f = {1, . . . , ν}, a 1 = νe 1 and b 1 = 2π ν e 1 . Now H(θ b )f (1) = Q 1 f 1 + f 2 + e -2πiθ f (ν), H(θ b )f (i) = Q i f i + f i-1 + f i+1 for 1 < i < ν and H(θ b )f (ν) = Q ν f ν + f ν-1 + e 2πiθ f 1 . We thus have H(θ b ) =        Q 1 1 0 • • • e -2πiθ 1 Q 2 1 0 . . . 1 e 2πiθ 0 1 Q ν        . Let z = e 2πiθ .
Let us examine (2.13) for Z with a 2-periodic potential Q

• , Q • . Here we have H(θ b ) = Q • 1 + e -2πiθ 1 + e 2πiθ Q • . The eigenvalues solve (Q • -λ)(Q • -λ) -(2 + 2 cos 2πθ) = 0, so E ± (θ b ) = Q 1 +Q 2 ±c 2 , with w ± = ( Q•-Q•±c 2(1+e 2πiθ ) , 1), where c = (Q • -Q • ) 2 + 16 cos 2 πθ.
After some tedious calculations, we conclude that (2.13) takes the form

(4.2) ψ, Op N (a)ψ = 2 q=1 a(• + v q ) N -1 r=0 [|P + r b N (U ψ) r (v q )| 2 + |P - r b N (U ψ) r (v q )| 2 ] = a(•) N -1 r=0 8 cos 2 πr N + (Q • -Q • ) 2 16 cos 2 πr N + (Q • -Q • ) 2 |(U ψ) r (0)| 2 + 8 cos 2 πr N 16 cos 2 πr N + (Q • -Q • ) 2 |(U ψ) r (1)| 2 - 2(Q • -Q • ) 16 cos 2 πr N + (Q • -Q • ) 2 Re(1 + e -2πir N )(U ψ) r (0)(U ψ) r (1)
+ a(•+1)

N -1 r=0 8 cos 2 πr N 16 cos 2 πr N + (Q • -Q • ) 2 |(U ψ) r (0)| 2 + 8 cos 2 πr N + (Q • -Q • ) 2 16 cos 2 πr N + (Q • -Q • ) 2 |(U ψ) r (1)| 2 + 2(Q • -Q • ) 16 cos 2 πr N + (Q • -Q • ) 2 Re(1 + e -2πir N )(U ψ) r (0)(U ψ) r (
w 1 = 1 2 (1, 1, -1, -1) , w 2 = 1 √ 2 (0, 1, 0, -1) , w 3 = 1 √ 2 (1, 0, -1, 0) , w 4 = (1, 1, 1 , 1)
and

κ j = 1 2 (1, ω j , ω 2j , ω 3j )
for A C 4 , where ω = e πi/2 and j = 0, . . . , 3. By Proposition 1.3, we know that the orthonormal eigenbases of Γ approach some weighted averages.

On the other hand, φ

(N ) ℓ , Kφ (N ) ℓ = cos( 2πℓ 1 N )+cos( 2πℓ 2 N ), so 1 N 2 ℓ∈L 2 N | φ (N ) ℓ , Kφ (N ) ℓ | 2 → [0,1] 2 cos 2 (2πx 1 ) + cos 2 (2πx 2 ) + 2 cos(2πx 1 ) cos(2πx 2 ) dx = 1.
This implies there can be no quantity K N λ (N) j independent of the basis such that

1 N 2 j | ψ (N ) j , K N ψ (N ) j -K N λ (N) j | 2 → 0.
5.1.3. Matrix generalization. We finally sketch how to generalize quantum ergodicity to matrix observables K. For simplicity we only discuss the case ν = 1. We may assume V f = {0} up to translating coordinates. Here, H(θ b ) = E(θ b ).

For Step 1, we note that

(e itH N Ke -itH N ψ)(k a ) = r∈L d N e itE( r b N ) (U Ke -itH N ψ) r e (N ) r (k) .
Here, (U

Ke -itH N ψ) r = 1 N d/2 n∈L d N e -ir b N •na (Ke -itH N ψ)(n a ). If R is the band width, then (Ke -itH N ψ)(n a ) = |τ |≤R K(n a , n a +τ a )(e -itH N ψ)(n a +τ a ). Denote K τ (n a ) := K(n a , n a + τ a ). Next, expand K τ (n a ) = 1 N d/2 m∈L d N K τ m e im b •na N , where K τ m = e (N ) m , K τ (• a ) ℓ 2 (L d N ) . Then we obtain (U Ke -itH N ψ) r = 1 N d n,m∈L d N |τ |≤R e -i(r b -m b )•na N K τ m (e itH N ψ)(n a + τ a ) = 1 N d/2 m∈L d N |τ |≤R K τ m e i(r b -m b )•τa N (U e -itH N ψ) r-m .
From here, we proceed as before, replacing a ,

where K τ = 1 N d n∈L d N K(n, n + τ ). Hence, ψ, Op N (K)ψ = k∈L d N ψ(k a ) r∈L d N (U ψ) r |τ |≤R K τ e ir b •τa N e (N ) r (k) = |τ |≤R K τ k∈L d N ψ(k a )ψ(k a + τ a ) = |τ |≤R K τ ψ, ψ(• + τ a ) .
This is the same expression we stated in §1.2.2. Interestingly, by examining the proof, we see that R can be taken to increase with N , like R N δ with δ < 1 2d .

5.2. Bloch's theorem. We prove here a version of the Bloch theorem for discrete periodic operators. This result is well-known in the continuum, but doesn't seem to have been explored in our setting. We also comment on the corresponding eigenfunction average. Note that on Γ N , we have

Ψ λ 2 = k∈L d N ν n=1 |f (v n )| 2 = N d f 2 C ν . If Ψ λ = 1 Ψ λ Ψ λ , then Ψ λ , a Ψ λ = 1 N d f 2 C ν k∈L d N ν n=1 |f (v n )| 2 a(k a + v n ) = ν n=1 a(• + v n ) |f (vn)| 2 f 2 C ν
. This average is in general not uniform unless a is locally constant. This is in accord with Theorem 1.2.

Remark 5.2. Note that these Bloch functions exist even in case of flat bands. For example, in Figure 3, instead of considering the localized functions (. . . , 0, 0, 1 -1 , 0, 0, . . . ), one can consider the Bloch function e 2πik•θ 1 -1 , where k ∈ Z is the position. We see that this delocalized function is also an eigenvector with the same eigenvalue -1.

This shows the limitations of this theorem; while there always exist an eigenfunction with periodic modulus (hence well distributed over the crystal and delocalized), there can also exist a lot of localized eigenfunctions for the same energy, which is the phenomenon that quantum ergodicity investigates.

5.3.

A general method to prove (1.3). We discuss an argument here for general ν, which may be helpful to find more examples satisfying (1.3) This method can be compared to [1, Section 5], in which the spectral properties of the commutator operator L(B) = AB -BA are used to show quantum ergodicity. Here, instead of the commutator we use another operator to show the chaotic nature of H.

If d = 1, then by virtue of [9, Th. 1.10, p.371], we know that λ = 0 is either an eigenvalue of B α (θ b ) for all θ ∈ [0, 1) d , or an eigenvalue for only finitely many θ 1 , . . . , θ p . In the following we indicate why in the second scenario, p can be chosen independently of α, which implies (1.3). It is not clear however how to avoid the first scenario.

The matrix B α (θ b ) has size ν 2 × ν 2 . We first observe that

H(θ b )(v n , v ℓ ) = u∼vn {u}a=v ℓ e iθ b •⌊u⌋a + Q(v n )δ vn,v ℓ .
For a fixed lattice Γ, the neighbors u of vertices in V f satisfy ⌊u⌋ a = d i=1 k u i a i for some k u i ∈ Z. For example, if Γ = Z d , then k u i ∈ {±1, 0}. Denote z j = e 2πiθ j . The matrix elements of B ζ (z) = H(zζ) ⊗ Id -Id ⊗ H(z) are of the form H(z)(v n , v ℓ ), H(zζ)(v n , v ℓ ) and H(z)(v n , v ℓ )-H(zζ)(v m , v q ). If δ = max vn∈V f ,u∼vn,i≤d

|k u i |,
then each of these entries is a polynomial in z j , z -1 j of degree at most δ (in each variable). Since the determinant is a polynomial of its entries, it follows that det(B ζ (z)) is a polynomial of degree at most ν 2 δ in each z ±1 j . To control the zeroes, we can consider the polynomial z δ 1 • • • z δ d det(B ζ (z)), of degree at most 2δν 2 in each coordinate. In particular, for d = 1, we get a polynomial z δ det(B ζ (z)). If it is nontrivial, then it has at most C = 2δν 2 roots, which proves the result. The question that remains is whether it can be trivial, i.e. whether we can exclude that E s (θ b + α b ) = E w (θ b ) for all θ, in specific examples.

For general d, analogously to §3.1, one may need to show the polynomial is nontrivial in a given direction in space.

u

  (x)| 2 δ x is close to the uniform measure 1 |Γ N | x∈Γ N δ x , for most u. See Theorem 1.1. Here (ψ (N ) u ) u is an orthonormal eigenbasis. • In another class of periodic Schrödinger operators, we have partial quantum ergodicity, in the sense that we no longer have |ψ (N ) u (x)| 2 ≈ 1 |Γ N | , but the sum of |ψ

u

  (x)| 2 over any periodic block is approximately the same (Theorem 1.2, Proposition 1.3). This means that ψ (N ) u

  is the number of distinct eigenvalues of H(θ b ) and P Es (θ b ) = Ew=Es P w (θ b ) is the orthogonal projection onto the eigenspace corresponding to E s (θ b ). In general, ν ′ is independent of θ b , i.e. the multiplicity of E s (θ b ) is fixed, expect perhaps for finitely many exceptional θ b (see e.g. [9, Chapter II.1.1]).

3. 1 .

 1 Scalar fibers. The end of §2.4 shows that if ν = 1, then ψ (N ) u , Op N (a)ψ (N ) u = a . To prove Theorem 1.1, it remains to establish assumption (1.3) in this context. Here of course w = s. If ν = 1, then the graph is 2D-regular for some D ∈ N. In fact, V f = {o} for some o ∈ C a , and Γ = Z d a + {o}. If u ∼ o, then u = ⌊u⌋ a + o. By translation invariance we have un a ∼ on a . Applying this to n a = ⌊u⌋ a gives o ∼ o -⌊u⌋ a . We may thus arrange the neighbors of o into N + o ∪ N - o , where N + o = {o + n a } and N - o = {on a }, for some D nonzero integers n a = d i=1 n i a i with n i ∈ {0, 1, . . . }. Since the rest of the graph is just a periodic copy of the star around o, we see it is 2D-regular. If ν = 1, then the potential Q must be constant. We assume without loss of generality that Q = 0. Proof of Theorem 1.1. The ν × ν matrix H(θ b ) is now just a scalar given by H(θ b ) = u∼o e iθ b •⌊u⌋a = 2 D p=1 cos(2πθ • n (p) )

Figure 1 .

 1 Figure 1. The ladder graph, Z P 2 .

-1 1 0 2 .

 12 Figure Decorating Z with triangles. The values of an eigenfunction are shown (it is then extended by zero).

1 - 1 Figure 3 .

 113 Z ⊠ P 2 . An eigenfunction localized on two vertices is shown.Unlike the ladder, this graph has some point spectrum. In fact, the Floquet matrix here isH(θ b ) {-1, 1 + 4 cos 2πθ}. Quantumergodicity is violated (use the eigenfunction shown in Figure3and argue as in §3.3).

1 2 ,

 2 we have E(θ b + α b ) = E(θ b ) for all θ.

4. 2 .

 2 Honeycomb lattice. Consider the honeycomb lattice ([11, Fig. 7], a.k.a graphene or hexagonal lattice) where each vertex has 3 neighbors. Hereν = 2, H(θ b ) = 0 ξ(θ b ) ξ(θ b ) 0 , where ξ(θ b ) = 1 + e -iθ b •a 1 + e -iθ b •a 2 for the crystal basis a 1 = a(1, 0), a 2 = a 2 (1, √3), a > 0. This gives the eigenvalues ±|ξ(θ b )| = ± 3 + 2 cos 2πθ 1 + 2 cos 2πθ 2 + 2 cos 2π(θ 1θ 2 ). Assumption (1.3) is clearly satisfied if w = s as the bands only meet at 0 (for (θ 1 , θ 2 ) = (23 , 1 3 )). On the other hand, we can control the event that |ξ(θ b +α b )| = |ξ(θ b )| by squaring, deducing as a special consequence of the arguments in §3.1 that (1.3) is satisfied. This shows that Theorem 1.2 holds true. Let us investigate (2.13).

1 √ 2 ( 1 ,

 121 are E ± (θ b ) = 2 cos 2πθ ± 1. Clearly (1, 1) and (1, -1) are eigenvectors. So the eigenprojectors are P ± (θ b )f = w ± , f w ± , with w ± = ±1), independently of θ. Thus,P ± f (v 1 ) = f (v 1 )±f (v 2 ) 2

4. 4 .

 4 1d periodic potential. Consider Z endowed with a periodic potential taking ν values

  Expanding the determinant of the characteristic polynomial p(λ; z) in detail, we see that [8, Lemma 3.1] (4.1) p(λ; z) = ∆(λ)zz -1 for some polynomial ∆(λ, Q). This splitting into pure λ and z parts is specific to one dimension. Now fix α = 0, let ζ = e 2πiα and suppose that E s (θ b + α b ) = E w (θ b ) for some s, w. Then λ = E s (θ b + α b ) solves (4.1). On the other hand, λ = E s (θ b + α b ) is also a root of the characteristic polynomial of H(θ b + α b ), which is p(λ; zζ) = ∆(λ)zζ -(zζ) -1 .

  [START_REF] Anantharaman | Quantum ergodicity on regular graphs[END_REF] .Note that if a(•) = a(• + 1) , this indeed reduces to a(•) ψ 2 . 4.5. Cylinders. Consider the Cartesian product Γ = Z C 4 , where C 4 is the 4-cycle.

Figure 4 .

 4 Figure 4. The cylinder, Z C 4 .

m

  (v q ) by |τ |≤R K τ m e ir b •τa N . There are of course many simplifications because ν = 1. In the end, a is replaced by

Theorem 5 . 1 .

 51 Let H be a periodic Schrödinger operator over the infinite periodic graph Γ, and suppose λ ∈ σ(H). Then we may find Ψ λ on Γ such that HΨ λ = λΨ λ andΨ λ (k a + v n ) = e iθ b •ka f (v n ), for some θ ∈ [0, 1) d and f on V f . Similarly, if λ ∈ σ(H N ), we may find Ψ λ on Γ N such that H N Ψ λ = λΨ λ and Ψ λ (k a + v n ) = e i j b •ka N f (v n ), for some j ∈ L d N and f on V f . Proof. H is unitarily equivalent to C b H(θ b ) dρ ⋆ , so σ(H) = ∪ ν n=1 σ n , where σ n = E n (C b ) = [E - n , E + n ], see[START_REF] Boutet De Monvel | Ballistic transport in periodic and random media[END_REF][START_REF] Korotyaev | Schrödinger operators on periodic discrete graphs[END_REF]. Hence, λ ∈ σ(H) implies λ = E r (θ b ) for some r and θ b ∈ C b . Let ψ θ b r be the corresponding eigenvector on V f and define Ψλ (k a + v n ) := e iθ b •ka ψ θ b r (v n ). Then HΨ λ (k a + v n ) = u∼ka+vn Ψ λ (u) + Q(v n )Ψ λ (k a + v n ) = w∼vn Ψ λ (w + k a ) + Q(v n )Ψ λ (k a + v n ) = w∼vn Ψ λ (k a + ⌊w⌋ a + {w} a ) + Q(v n )Ψ λ (k a + v n ) = e iθ b •ka w∼vn e iθ b •⌊w⌋a ψ θ b r ({w} a ) + Q(v n )ψ θ b r (v n ) = e iθ b •ka (H(θ b )ψ θ b r )(v n ) = e iθ b •ka E r (θ b )ψ θ b r (v n ) = λΨ λ (k a + v n ) . The case of Γ N is the same since H N ≡ ⊕ j∈L d N H( j b N ).

  Fix 0 = α ∈ [0, 1) d . Consider the Kronecker sum B α (θ b ) = H(θ b + α b ) ⊗ Id -Id ⊗ H(θ b ) . Then σ(B α (θ b )) = σ(H(θ b + α b ))σ(H(θ b )). In particular, E s (θ b + α b ) -E w (θ b ) is an eigenvalue of B α (θ b ).We are thus trying to control the set of θ for which B α (θ b ) has zero as an eigenvalue, i.e. det(B α (θ b )) = 0.

If we choose the eigenbasis ψ n,j = φ n ⊗ w j , then by (1.6),

On the other hand, if ψ n,j = φ n ⊗ κ j , then

This example shows that ψ

in general depends on the choice of the basis, even for simple regular graphs, and it may or may not be the uniform average. In fact, this gives the uniform average for the basis ψ n,j , but not for ψ n,j , take for example the observable

where we parametrized the vertices of the cylinder Z C 4 by u = i + v q , where i ∈ Z is the layer's level and

) are the vertices within it.

The problem with ψ n,j is that it is concentrated on half the cylinder for j = 2, 3, while ψ n,j is spread on the whole. The semi-delocalization of ψ n,j is not detected by locally constant observables.

Complementary results

QUE and eigenvector correlators.

5.1.1. Quantum unique ergodicity. We first investigate QUE for A Z and A Z 2 .

For Γ = Z, taking L N with periodic conditions amounts to considering N -cycles. On C 4N , consider the observable a N = (1, 0, 1, 0, . . . , 1, 0) and the eigenvector v (N ) = 1 √ 2N (0, 1, 0, -1, . . . , 0, 1, 0, -1) with eigenvalue 0, where the string (0, 1, 0, -1) is repeated N times. Then v (N ) , a N v (N ) = 0 while a N = 1 2 , so (1.7) is violated. On Z 2 , the whole sequence may converge to a nonzero limit. If e (N )

This gives an orthonormal eigenbasis with |φ