Fabrice Ardhuin
email: fabriceardhuin@ifremer.fr

Mark Otero

Sophia Merrifield

Antoine Grouazel

Eric Terrill

Ice break-up controls dissipation of wind-waves across Southern Ocean sea ice

Wind-waves attenuate across the Antarctic sea ice with a narrow directional distribution • Scattering of waves by ice floes plays a negligible role in wave attenuation • Observed wave attenuation is consistent with ice break-up modulating the dissipation strength

Introduction

Recent studies have revealed that ocean waves play a significant role in the ice momentum balance [START_REF] Stopa | Strong and highly variable push of ocean waves on southern ocean sea ice[END_REF], ice extent [START_REF] Kohout | Storminduced sea-ice breakup and the implications for ice extent[END_REF], and the rapid thickening of ice when the ocean freezes [START_REF] Sutherland | Marginal ice zone thickness and extent due to wave radiation stress[END_REF]. Previous wave measurements in ice-covered waters [START_REF] Doble | Wave buoy measurements at the Antarctic sea ice edge compared with an enhanced ECMWF WAM: Progress towards global wavesin-ice modelling[END_REF][START_REF] Kohout | Storminduced sea-ice breakup and the implications for ice extent[END_REF] have shown that waves in ice attenuate over longer distances when the open water wave height increases, often due to longer corresponding wave periods [START_REF] Meylan | In situ measurements and analysis of ocean waves in the antarctic marginal ice zone[END_REF].

This result was used to define an ad hoc parameterization of wave attenuation [START_REF] Kohout | In situ observations of wave-induced sea ice breakup[END_REF] that fails to reproduce the large range of observed attenuations for any given off-ice wave height and wave period [START_REF] Stopa | Strong and highly variable push of ocean waves on southern ocean sea ice[END_REF]. This poor performance calls for a detailed physical understanding of ice-wave interaction processes. Following early works [START_REF] Wadhams | Winter observations of iceberg frequencies and sizes in the south atlantic ocean[END_REF], the attenuation of waves that propagate under the ice is generally attributed to scattering [START_REF] Squire | Ocean wave interactions with sea ice: A reappraisal[END_REF]. Scattering is a partial reflection of waves at the boundaries of ice floes, broadening the distribution of wave directions. Other processes dissipate wave energy into heat and narrow the wave direction distribution around the shortest propagation path. Measured wave directions within the ice can thus reveal the importance of scattering. For example, the narrow directional spread of waves with periods 19 to 23 s measured with directional tilt-meters in the Arctic showed that scattering could not be a significant source of attenuation for these very long waves [START_REF] Ardhuin | Ocean waves across the Arctic: attenuation due to dissipation dominates over scattering for periods longer than 19 s[END_REF]. Here we generalize these observations to more typical wave periods from 10 to 20 s using wave buoy measurements. We evaluate two different dissipation processes: under-ice friction [START_REF] Stopa | Wave climate in the Arctic 1992-2014: seasonality and trends[END_REF], and the anelastic dissipation associated with ice flexure [START_REF] Cole | Cyclic loading and creep response of aligned first-year sea ice[END_REF]]. The latter process yields large dissipation rates when the ice is flexing. However, flexing only occurs for waves shorter than about twice the ice floe diameter as floes tilt over longer waves [START_REF] Boutin | Floe size effects on wave-ice interactions: theoretical background, implementation and applications[END_REF]. This varying behavior as a function of wavelength can explain the observed sudden drop in wave attenuation when ice is broken by waves [START_REF] Collins | In situ measurements of an energetic wave event in the Arctic marginal ice zone[END_REF]. Here the floe sizes are constrained by Synthetic Aperture Radar (SAR) imagery.

Measured wave properties and sea ice conditions

Freely drifting wave buoys [START_REF] Drazen | Interpretation of in-situ ocean environ-mental measurements[END_REF] were deployed in the Southern Ocean in 2018, and advected into advancing sea ice. Fig. 1.a shows the positions of three buoys numbered 623, 624 and 625. Successive positions of the ice edge, defined as the 15% ice concentration contour, are estimates from the Advanced Microwave Scanning Radiometer 2 (AMSR2) radiometer [START_REF] Spreen | Sea ice remote sensing using AMSR-E 89-GHz[END_REF]. Buoy 624 measured a typical wave signal as shown in Fig. 1.c. High wave frequencies (above 0.3 Hz) vanish once the buoys is a few kilometers into the ice on May 26. This attenuation is similar to other measurements obtained during the formation of meter-scale rafts of ice known as pancakes [START_REF] Thomson | Overview of the arctic sea state and boundary layer physics program[END_REF]. Frequencies between 0.10 and 0.30 Hz gradually disappear as the buoy gets deeper into the ice, 80km from the ice edge on June 12. This evolution is presumably associated with the progressive welding of pancakes into much larger floes forming solid plates that can extend over kilometers. After June 12, only frequencies under 0.1 Hz are episodically recorded as the ice margin continues to extend further offshore reaching a distance of 150 km from 624 by June 24. Wave energy exceeds the instrument noise floor during seven events observed by 624; each of which is associated with a storm (Fig. 2.a). Buoy 625 kept a nearly constant distance south of 624, 120 km further into the ice field, We now turn to Sentinel-1 radar imagery for information on ice properties. These SAR images were analyzed for two types of features (Fig. 3).

First we determined the presence (blue symbols) or absence (red symbols) of leads. Leads are straight features at the boundaries of large scale floes that may correspond to an ice-free surface, which appear as dark bands, or that can re-freeze and appear bright due to the presence of frost flowers. When leads are visible, the diameters of floes exceed 1 km and waves have not been able to break the ice. In the absence of leads, it is expected that ice is broken into small floes and their differential advection yields a horizontally uniform brightness. The second type of feature is wave patterns.

Images numbered 4-6 acquired on July 5 clearly show a wave field propagating from the West, with a dominant wavelength around 400 m that is reduced to 270 m in images 11, 12 and 14 acquired on July 6. Estimated wave heights [START_REF] Ardhuin | Measuring ocean waves in sea ice using SAR imagery: A quasi-deterministic approach evaluated with Sentinel-1 and in situ data[END_REF] range from 0.7 to 2.5 m. Much fainter wave patterns are also visible in images 2, 3 and 7. All these wave features are long-crested except in image 8 where ring waves are observed in probable association with reflections off an iceberg. The radar imagery acquired around the trajectories of the buoys thus confirms the penetration, 200 km inside the ice, of long-crested waves with heights on the order of 1 m. Also, we interpret the disappearance of leads towards the ice edge on July 5 and 6 as the result of ice break up by wave action.

3 Interpretation using a wave attenuation and ice break-up mode

The magnitude and a plausible cause of wave attenuation will now be examined with a numerical wave model, based on version 6.05 of the WAVEWATCH III modelling

framework [The WAVEWATCH III R Development Group, 2019, hereinafter WW3],
that is constrained by these ice observations. We consider three model simulations that differ only in their parameterizations of wave-ice interactions:

• A simple "no attenuation" simulation in which the effect of the ice is limited to shutting off the generation of waves by wind forcing and dissipation associated with wave breaking. The difference between the measurements and the "no attenuation" run provides an estimate of the ice-induced attenuation.

• A scattering + viscous basal friction ("S1V1" for short) in which the scatter- • An anelastic dissipation based on known microscopic rheological properties of dislocations in ice cristals [START_REF] Cole | Cyclic loading and creep response of aligned first-year sea ice[END_REF][START_REF] Cole | On the physical basis for the creep of ice: the high temperature regime[END_REF], adapted by [START_REF] Boutin | Floe size effects on wave-ice interactions: theoretical background, implementation and applications[END_REF] to represent the dissipation of waves when floes with diameters larger than half the wavelength are flexing. This parameterization is combined with viscous basal friction and scattering, with the scattering coefficient reduced by a factor 5.

Apart from the differences in parameterizations, the common features of the three model simulations reported here are a spatial resolution of 0.1 degree in latitude and 0.2 degree in longitude forced at the boundaries (red line in Fig. Weather Forecasts (ECMWF) operational analyses and forecasts. Ice concentration was provided by AMSR2 with a correction designed to limit the errors caused by atmospheric water vapor [START_REF] Gloersen | Reduction of weather effects in the calculation of sea ice concentration from microwave radiances[END_REF]]: all concentrations above 70% was set to 100%. Based on climatology data, the ice thickness was set to a constant 0.77 m. Given our focus on wave periods longer than 10 s, we neglected the effect of ice on wave dispersion [START_REF] Collins | Observations of surface wave dispersion in the marginal ice zone[END_REF]. This is discussed in the Supplementary Information. The model uses 24 directions and an exponential frequency grid from 0.037 to 1 Hz.

The ice flexural strength σ c was adjusted to obtain a position of the break-up front consistent with SAR imagery, as shown in Fig. 3, giving σ c = 0.6 MPa.

In all cases the maximum floe size diameter is estimated based on the expected local maximum flexural strength over the model time step of 600 s [START_REF] Boutin | Floe size effects on wave-ice interactions: theoretical background, implementation and applications[END_REF].

This floe size only has an impact on wave dissipation rate in the case of the anelastic simulation.

We also note that other wave-ice interaction effects have been investigated in other studies [START_REF] Squire | Ocean wave interactions with sea ice: A reappraisal[END_REF], including ice floe collisions [START_REF] Herman | Wave energy attenuation in fields of colliding ice floes part 2: A laboratory case study[END_REF] or the breaking of steep waves over ice floes [START_REF] Toffoli | Sea ice floes dissipate the energy of steep ocean waves[END_REF]. Although these processes are reasonably important right at the ice edge, we expect them to play only a minor role overall because they are typically non-linear with higher relative dissipation for higher waves, hence leading to a shorter attenuation distance for steeper ocean waves. This is not consistent with available large scale observations that show a constant or weaker attenuation for steeper waves [START_REF] Meylan | In situ measurements and analysis of ocean waves in the antarctic marginal ice zone[END_REF][START_REF] Stopa | Strong and highly variable push of ocean waves on southern ocean sea ice[END_REF], and we thus have not considered these processes here.

Modeled wave parameters are compared to the buoy measurements in Fig. 4.

A detailed analysis of the shape of wave spectra is performed in the Supporting Information, see Figures S16-S21. Starting with the directional spread in Fig. 4.a, measurements during wave events are only slightly larger than those modeled without ice attenuation. In contrast, the modeled spread is much larger than observed values when scattering is a dominant term in the attenuation (run S1V1). Reducing the spread to levels comparable to measured values requires a reduction of the scattering coefficient by at least a factor 5, as in the "Anelastic" simulation. This reduced scattering requires stronger dissipation processes to explain the measured wave height attenuation. This attenuation is estimated as the ratio between the "no attenuation" simulation and the measured height, and it varies dramatically in time and across frequencies. Some of these variations are caused by different offshore directions (Fig. shown with red contours on Fig. 3. When buoy 624 is far from the "break up front", the distance traveled by waves across unbroken ice is large and the attenuation is very strong. In contrast, when the ice is broken up closer to the buoy, the attenuation is much less with an overall wave height attenuation by ice as low as 20%.

The anelastic model scenario presented above is not a unique solution. We assumed an ice thickness h = 0.77 m and adjusted the ice flexural strength to σ c = 0.6 MPa in order to obtain a spatial extent of broken ice consistent with SAR imagery.

Because the ice break-up condition in the model is set by a maximum stress that is proportional to σ c h 2 , the same results may be obtained by increasing h by a factor a and reducing σ c by a factor a 2 . Likewise, the magnitude of the anelastic dissipation is controlled by the compliance of the relaxation of dislocations δ, and the number of dislocations per unit area [START_REF] Cole | On the physical basis for the creep of ice: the high temperature regime[END_REF]. Since the wave dissipation rate is proportional to h 3 δ, any change in h can be compensated with a change of δ to give the same wave evolution. Conversely, assuming that δ and σ c are well constrained, it is possible to estimate the ice thickness from SAR imagery using either wave heights and periods at the break-up front or the attenuation of waves in unbroken ice.

For such applications a more realistic dissipation model may be needed. Indeed, the present parameterization tends to overestimate the dissipation of frequencies above error is the neglected effect of ice in the dispersion relation [START_REF] Robinson | A modal analysis of a rectangular plate floating on an incompressible liquid[END_REF], but other dissipative processes may be involved, including flow in brine pockets during ice deformation, and viscous effects in ice deformation [START_REF] Meylan | Dispersion relations, power laws, and energy loss for waves in the marginal ice zone[END_REF][START_REF] Marchenko | Three physical mechanisms of wave energy dissipation in solid ice[END_REF][START_REF] Cole | On the physical basis for the creep of ice: the high temperature regime[END_REF].

Conclusions

As storms passed over the open ocean, we measured waves in the ice with narrow directional spreads that are not compatible with existing parameterizations of wave scattering by ice floes. Instead, dissipation associated with ice flexure is a plausible explanation for the previously observed non-linear wave attenuation [START_REF] Stopa | Strong and highly variable push of ocean waves on southern ocean sea ice[END_REF] with weaker attenuation for higher waves. This dissipation provides a transition from a non-broken and highly dissipative flexing ice field to one which is fragmented into small floes with weak attenuation, consistent with our wave and ice observations. This new paradigm also has consequences on how the waves impact the ice. In particular, a dissipation-dominated attenuation concentrates the wave-induced push to the ice around the ice break-up front. In the melting season, the waves may have a more lasting impact as large floes may not reform after each storm, allowing break-up to progress further with the next storm.

Introduction

This Supporting information contains a movie that complements Figure 1 and helps visualize buoy locations relative to the ice edge in the context of the storms observed. The supplementary figures S1-S15 provide a more detailed view of the ice and wave features observed on July 5 and 6, complementing figure 4. Figures S16 to S21 are associated with the discussion of spectral shapes and the possible identification of a linear dissipation process proportional to f n , following the type of analysis performed by [START_REF] Meylan | Dispersion relations, power laws, and energy loss for waves in the marginal ice zone[END_REF].

Movie S1

Location of buoys 623, 624, and 625 in the context of the evolving sea ice concentration field estimated from the Advanced Microwave Scanning Radiometer 2 and processed by the University of Bremen [START_REF] Spreen | Sea ice remote sensing using AMSR-E 89-GHz[END_REF]. Maximum floe size contour Dmax = 300 m, wave heights, and mean directions are taken from the the "Anelastic" model run.

Figures S1-S15

Quicklooks of SAR images labeled 1-15 in Fig. 3. These can be obtained from http://www.ifremer.fr/datavore/exp/dvor/#/s1quicklook Images are displayed in acquisition geometry with range (satellite cross-track direction) on the x-axis and azimuth (satellite along-track direction) on the y-axis. The original Level 1 product, with a resolution of 4 m, is subsampled to a 600 x 600 pixel image of 35 m resolution. The grey scale representing radar backscatter after SAR processing is automatically adjusted in each image to linearly cover the 1st to 99th percentiles of the image intensity in order to maximize contrast. The mean sea surface roughness value (µ) is indicated on each figure. The samples shown in Fig. 3 were rotated to orient North upward.

Discussion and Figures S16-S21

Whereas the main body of the paper focuses on the temporal evolution of wave properties, here we examine the spectral distribution of wave energy which provides a complementary perspective of possible dissipation processes. Indeed, [START_REF] Meylan | Dispersion relations, power laws, and energy loss for waves in the marginal ice zone[END_REF] have argued that different dissipation processes have distinct spectral signatures. This idea has been used to characterize the effect of frazil and pancake ice on spectral shape.

This approach is particularly well suited for linear processes. Considering a 1-dimensional propagation along the x axis, the wave spectrum E(x, t, f ) is attenuated by a source function Sice(x, t, f ) = a(x)f m E(x, t, f ), namely

∂E(x, t, f )/∂t + ∂ [Cg(f )E(x, t, f )] /∂x = Sice(x, t, f ). (1)
Using the method of characteristics this gives,

E(x, t, f ) = E (x0, t -Cg(f ) × (x -x0), f ) × exp f m /Cg(f ) x x 0 a(x ′ )dx , (2)
which can be generalized to any polynomial function of the frequency. Also, for a simple dispersion relation, Cg(f ) is also a power law of f and than can be combined with Sice to give,

E(x, t, f ) = E (x0, t -Cg(f ) × (x -x0), f ) × exp f n x x 0 b(x ′ )dx . (3)
Without ice effect, the spectrum would have been

Enoice(x, t, f ) = E (x0, t -Cg(f ) × (x -x0), f ) , (4)
such that the power n can be obtained by taking the log of the ratio r(x, t, f

) = log [E(x, t, f )/Enoice(x, t, f )].
From values of r at two frequencies, fi and fj, we get, n(x, t, fi, fj) = log (r(x, t, fi)/r(x, t, fj)) / log(fi/fj) (5)

For Airy waves, n = m + 1 and the wavenumber k is proportional to f 2 . For viscous friction Sice is proportional to k √ f and thus m = 2.5 and n = 3.5. For anelastic dissipation, we expect a much steeper variation with frequency, as illustrated in Figure S16, because the stress and strain are both proportional to the surface curvature, giving a k 4 contribution, that is reduced a little by the f -0.5 variation of the phase shift between stress and strain [START_REF] Cole | Cyclic loading and creep response of aligned first-year sea ice[END_REF] , leading to n = 9.5. However, this is only valid as long as the elastic contribution to the wave energy can be neglected, which is a function of the ice thickness. Indeed, the conservation of the energy flux of waves propagating from the open water under the ice leads to a smaller wave amplitude.

In order to be fully consistent (which was not the case in the main body of the paper, for the sake of simplicity), the ice thickness should also be taken into account in the wave dispersion relation. For this we may use the dispersion relation by [START_REF] Robinson | A modal analysis of a rectangular plate floating on an incompressible liquid[END_REF]. This gives a transition of the wavenumber from k ∝ f 2 at low frequencies to k ∝ f 0.5 at high frequencies, and Cg ∝ f -1 to Cg ∝ f 0.5 . That effect of dispersion depends on the ice thickness, and for the valie hi = 0.77 m used in the paper, it is significant for wave frequencies above 0.1 Hz.

With all this in mind, we can now look at frequency spectra from the buoy measurements and model results. The average spectra at the off-ice buoy 623 and in-ice buoy 624 are shown in Figure S17.

From the monthly-averaged spectra, it is clear that below 0.08 Hz the model underestimates the off-ice wave energy. Above 0.08 Hz, the shape of the modeled spectra is too steep for the anelastic dissipation, and with about the right shape but too high by a factor 10 or more for the viscous dissipation. As shown in Figure S18, part of the discrepancy in the anelastic model could be due to the use of the Airy dispersion relation, with the [START_REF] Robinson | A modal analysis of a rectangular plate floating on an incompressible liquid[END_REF] dispersion working better especially if we use an ice thicknesses larger than the 0.77 m used all the model results shown here. Possibly the frequency smoothing that is used in the buoy data processing also contributes to that difference. Indeed, the default data processing has each spectral component obtained with a 3-point frequency (Hann filter) smoothing with weights [0.25 0.5 0.25]. That processing step tends to reduce the spectral slope at both 0.08 Hz (near the typical spectral peak) and at 0.097 Hz (where the spectrum approaches the noise level). Also, we note that when averaged over 1 day, the spectral estimate uses 128 degrees of freedom, hence a +/-24% error at the 95% confidence level, which may be used to estimate an error on the fitted power law.

We may now take a look at the values of n estimated from the data. In practice we have computed used the discrete frequencies f13 = 0.0815, f14 = 0.0854, f15 = 0.0894, f16 = 0.0933, f17 = 0.0972 to compute n14, 15, n15,16, n16,17 and n14,17, using daily-averaged spectra. The same n parameters were computed from the modeled spectra after their interpolation on the observed frequency grid. The first result is that, even in the model runs, there is a wide variability of n. Also, on most of the days the energy level is below the noise floor and n cannot be estimated, so that any estimate of n will be biased towards the strong energy event.

: X -13 Estimates are only available when the spectrum is larger than 2 times the mean noise level.

Daily average estimates of n range from 0.2 to 11. The scatter is even larger for the anelastic model run, and the scattering-viscous model run (S1V1) has a large range of n values, showing that the simple linear attenuation of unidirectional waves does not represent well the more complex situation with waves from all directions. In order to understand a bit better this variability we now look at spectra for individual days, overlaying analytical examples of attenuations with powers n = 9.5, n = 4 and n = 3.

It is particularly interesting to contrast the spectral shape on June 17 at the start of a wave event, with a very steep high frequency decay consistent with 5 < n < 11, and the next day with n ≈ 4 (Figure S20). In the chosen frequency range (above 0.08 Hz) the model performs well off the ice at buoys 623.

On the wave event from 27 to 28 June, n estimated from the buoy data changes from 2 to 6 before the energy level disappears below the noise level on the 29th whereas a constant attenuation rate would have given a strong signal. Steep spectra can also be found such as on July 1st (bottom line, figure S21), but n cannot be estimated because only E(f14) is well above the noise.

Figure 1 .

 1 Figure 1. (a) Positions of the ice edge, defined as the 15% ice concentration contour, every 4 days from June 12 to July 10, 2018, and positions of three mini-buoys, number 623 (triangles in open water), 624 (circles in the ice) 625 (crosses, further in the ice). (b) Geographical context and domain of the numerical wave model used in section 3. The black rectangle corresponds to the region shown in (a), the colors show simulated significant wave heights from 0 to 10 m on July 4th at 12:00 UTC. (c) Wave energy spectra at buoy 624. (d) Time series of measured and modeled wave height at the open water buoy 623 for period ranges 5 to 20 s, and 13.2 to 15.9 s. See also supplementary video S1.

Figure 2 .

 2 Figure 2. Wave properties measured by buoy 624 as a function of time and frequency. (a) Energy spectrum, with dotted boxes around the short and long period ranges used in Fig. 4, (b) Mean directions, shown when the energy is above -10 dB, avoiding random noise for low energies. (c) directional spread.

Figure 3 .

 3 Figure 3. Daily maps of ice concentrations (from AMSR2 -ASI, as provided by University of Bremen), buoy positions and locations of Sentinel 1 Wave Mode SAR imagery classified by the presence of leads and waves, through a storm event from (a) July 4th to (d) July 7th. Each SAR image is represented by a blue symbol if leads are detected, red if no lead is detected, triangles if waves are visible, squares otherwise. For some of the images, numbered 1 to 15, sub-sets of the SAR images are also shown. Each subset is 5 km by 5 km and full images are shown in Fig. S1-S15. Finally, the contour of maximum floe diameter Dmax = 300 m, as estimated from the model run using anelastic dissipation, is shown with a dashed line at 6 AM and solid line at 6 PM marking the extent of the regions where ice is broken by waves in the model.

  ing coefficient is determined by the ice thickness and maximum floe size following[START_REF] Williams | Wave-ice interactions in the marginal ice zone. part 1: Theoretical foundations[END_REF] with the addition of back-scattering (so that the scattering term conserves energy) and viscous dissipation taken from Liu and Mollo-Christensen[1988] with the viscosity taken to be the molecular viscosity of sea water at the freezing point. Our scattering term conserves wave energy which is isotropically redistributed as discussed in[START_REF] Boutin | Floe size effects on wave-ice interactions: theoretical background, implementation and applications[END_REF]. Scattering strength is based on the normal reflection of waves travelling from the open ocean under a semi-infinite ice floe with a straight boundary[START_REF] Williams | Wave-ice interactions in the marginal ice zone. part 1: Theoretical foundations[END_REF].

  1.b) by a 0.5 degree global model. Wind forcing is provided by the European Center for Medium Range

  2.b). Still, for similar wave directions, the observed attenuation varies from 80% to 20% between June 20 and June 30 (purple arrows in Fig.4.c). The gradual variation of wave attenuation in the S1V1 simulation is markedly different from observations which exhibit a sharp cut-off after each wave event such as on July 2 or 8 when no wave energy is measured at buoy 624. The anelastic simulation reproduces that effect which, in the model, is due to displacement of the "break up front"

Figure 4 .

 4 Figure 4. Time series of measured and modeled (a) directional spread and (b-e) wave heights for the ranges of wave periods shown in Fig. 2.a. The model results are shown for the 3 different parameterizations of wave-ice interactions. Purple arrows in (c) point to markedly different attenuation rates, 80% on June 20 and 20% on June 30.

Figure S1 .

 S1 Figure S1. SAR image acquired 2018/07/05 at 06:01:08

Figure S13 .

 S13 Figure S13. SAR image acquired by Sentinel 1B on 2018/07/06 at 14:43:03.

Figure S14 .

 S14 Figure S14. SAR image acquired by Sentinel 1B on 2018/07/06 at 14:44:18.

Figure S16 .

 S16 Figure S16. Amplitude decay (imaginary wavenumber) as a function of wave frequency for the anelastic model of Cole et al. (1998) when using a Robinson-Palmer (blue) or Airy (red) dispersion relation.

Figure S17 .Figure S18 .

 S17S18 Figure S17. Mean spectra at buoys 623 and 624, from June 12 to July 12. A f -2 noise floor was added to the model to make it comparable to the measurements.

Figure S19 .

 S19 Figure S19. Dissipation power law parameters n14,15, n15,16, n16,17 and n14,17 estimated from daily-averaged (a) measured spectra, (b) modeled spectra with anelastic dissipation (c) modeled spectra with viscous dissipation. Estimates are only available when the spectrum is larger than 2 times the mean noise level.

Figure S20 .

 S20 Figure S20. Measured and modelled spectra on June 17 (top) and June 18 (bottom), at buoys 624 (left) and 623 (right).

Figure S21 .

 S21 Figure S21. Measured and modelled spectra on (from top to bottom), June 27 , June 28, June 29, and July 1st, at buoys 624 (left) and 623 (right).

Copyright 2020 by the American Geophysical Union. 0094-8276/20/$5.00

Acknowledgments

We thank the New Zealand Navy for buoy deployments, and the EU/ESA Copernicus program for the acquisition of SAR imagery. The analysis was supported by ONR. Buoy data has been uploaded to the UC San Diego Library Digital Collections (https://doi.org/10.6075/J0V40SM5), Level-1 SAR wave mode data was provided by the European Space Agency and their processing is supported by Sentinel-1 A Mission Performance Center (Contract No. 4000107360/12/I-LG), and can be viewed at http://www.ifremer.fr/datavore/exp/dvor/#/s1quicklook. Finally model simulations are available at http://tiny.cc/wavesinice. Comments from anonymous reviewers and discussions with David Cole are greatefully aknowledged.

Supporting Information for "Ice break-up controls dissipation of wind-waves across Southern Ocean sea ice"


Douglas N Arnold
email: arnold@umn.edu

Guy David
email: guy.david@math.u-psud.fr

Marcel Filoche
email: cel.filoche@polytechnique.edu

AND David Jerison ¶
email: jeri-son@math.mit.edu

Svitlana Mayboroda
email: svit-lana@math.umn.edu

Simons

Dj

COMPUTING SPECTRA WITHOUT SOLVING EIGENVALUE PROBLEMS *

Keywords: localization, spectrum, eigenvalue, eigenfunction, Schrödinger operator AMS subject classifications. 65N25, 81-08, 82B44

The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation of the spectrum of a Schrödinger operator with a disordered potential. Unlike plane waves or Bloch waves that arise as Schrödinger eigenfunctions for periodic and other ordered potentials, for many forms of disordered potentials the eigenfunctions remain essentially localized in a very small subset of the initial domain. A celebrated example is Anderson localization, for which, in a continuous version, the potential is a piecewise constant function on a uniform grid whose values are sampled independently from a uniform random distribution. We present here a new method for approximating the eigenvalues and the subregions which support such localized eigenfunctions. This approach is based on the recent theoretical tools of the localization landscape and effective potential. The approach is deterministic in the sense that the approximations are calculated based on the examination of a particular realization of a random potential, and predict quantities that depend sensitively on the particular realization, rather than furnishing statistical or probabilistic results about the spectrum associated to a family of potentials with a certain distribution. These methods, which have only been partially justified theoretically, enable the calculation of the locations and shapes of the approximate supports of the eigenfunctions, the approximate values of many of the eigenvalues, and of the eigenvalue counting function and density of states, all at the cost of solving a single source problem for the same elliptic operator. We study the effectiveness and limitations of the approach through extensive computations in one and two dimensions, using a variety of piecewise constant potentials with values sampled from various different correlated or uncorrelated random distributions.

1. Introduction. Eigenfunctions of elliptic operators are often widely dispersed throughout the domain. For example, the eigenfunctions of the Laplacian on a rectangle are tensor products of trigonometric functions, while on a disk they vary trigonometrically in the angular variable and as Bessel functions in the radial argument. By contrast, in some situations eigenfunctions of an elliptic operator localize, in the sense that they are practically zero in much of the domain (after normalizing the L 2 or L ∞ norm, say, to 1), like the two functions pictured on the right of Figure 1 (which will be explained shortly). Localization may be brought about by different mechanisms including irregular coefficients of the elliptic operator, certain complexities of the geometry of the domain such as thin necks or fractal boundaries, confining potential wells, and disordered potentials. A celebrated example is Anderson localization [START_REF] Anderson | Absence of diffusion in certain random lattices[END_REF], which refers to localization of the eigenfunctions of the Schrödinger operator on R n induced by a potential with random values. The eigenfunctions of the Anderson system model the quantum states of an electron in a disordered alloy, and localization can even trigger a transition of the system from metallic to insulating behavior. Over the past 60 years, analogous phenomena have been observed in many other fields, and found numerous applications to the design of optical [START_REF] Riboli | Anderson localization of near-visible light in two dimensions[END_REF], acoustic [START_REF] Sapoval | Acoustical properties of irregular and fractal cavities[END_REF][START_REF] Félix | Localization and increased damping in irregular acoustical cavities[END_REF], electromagnetic [START_REF] Laurent | Localized modes in a finite-size open disordered microwave cavity[END_REF][START_REF] Sapienza | Cavity quantum electrodynamics with Anderson-localized modes[END_REF], and photonic devices [START_REF] Filoche | Localization landscape theory of disorder in semiconductors[END_REF][START_REF] Li | Localization landscape theory of disorder in semiconductors. iii. application to carrier transport and recombination in light emitting diodes[END_REF].

The localized functions shown on the right of Figure 1 are the first and second eigenfunctions of the Schrödinger operator H = -∆ + V on the square [0, 80] × [0, 80] with periodic boundary conditions. The potential V is of Anderson type: it is a piecewise constant potential obtained by dividing the domain into 80 2 unit subsquares and assigning to each a constant value chosen randomly from the interval [0, 20]. Although we will consider numerous disordered potentials generated by random distributions in this paper, we emphasize that our approach is deterministic. We aim to efficiently predict aspects of the spectrum that depend on a particular realization of the potential, rather than to give statistical or probabilistic results about the spectrum associated to a family of potentials with a certain distribution. For example, the location of the eigenfunctions shown in Figure 1-where they achieve their maxima, what is the shape of their effective supports-depends sensitively on the precise configuration of the disordered potential. The determination of characteristics of the spectrum such as these is an example of the issues addressed here.

In this paper we shall focus on the Schrödinger operator H = -∆ + V . The domain Ω will always be either an interval in one dimension or a square in two, the boundary conditions will be periodic, and the potential V will be piecewise constant with respect to a uniform mesh of Ω having positive values chosen from some probability distribution, either independently or with correlation. Much of the work can be extended, e.g., to more general domains, potentials, PDEs, and boundary conditions. In particular, we remark that the choice of periodic boundary conditions is for simplicity, and that similar localization occurs with Neumann or Dirichlet boundary conditions. We generally choose unit-sized subsquares for the constant regions of the potential, but this is merely a convenient normalization. For example, instead of considering an 80 × 80 square broken into unit subsquares as the domain in Figure 1, we could have chosen instead to take the unit square as the domain, with subsquares of side length 1/80 for the potential. Had we scaled the potential to take values in the range from 0 to 128, 000, we would have obtained the same localization (128, 000 being 20 × 80 2 ).

There is a large literature concerning the localization of eigenfunctions, approaching the phenomenon from various viewpoints: spectral theory, probability, and quantum mechanics. But there is nothing like a complete explanatory and predictive theory which can quantitatively and deterministically answer such basic questions as:

• How are the eigenvalues and eigenfunctions determined by a particular potential? • Given a potential, do the eigenfunctions localize, and, if so, how many?

• What are the size, shape, and location of the approximate supports and how do the eigenfunctions decay away from them? • What are the associated eigenvalues? The present work aims at providing answers to these questions, based on a theory of localization recently conceived and under active development [START_REF] Filoche | Universal mechanism for Anderson and weak localization[END_REF][START_REF] Filoche | The landscape of Anderson localization in a disordered medium[END_REF][START_REF] Arnold | Effective confining potential of quantum states in disordered media[END_REF][START_REF] Lefebvre | One single static measurement predicts wave localization in complex structures[END_REF][START_REF] Arnold | Localization of eigenfunctions via an effective potential[END_REF].

In the next section of the paper, we briefly survey some classical tools used to understand localization. Then, in Section 3, we introduce two more recent tools: the localization landscape function and its associated effective potential, introduced in [START_REF] Filoche | Universal mechanism for Anderson and weak localization[END_REF][START_REF] Arnold | Effective confining potential of quantum states in disordered media[END_REF]. These are easily defined. The landscape function u is the solution to Hu = 1 subject to periodic boundary conditions, and the effective potential W is its reciprocal. Our approach is guided by estimates and relations between these objects and the spectrum of the Schrödinger operator from [START_REF] Filoche | Universal mechanism for Anderson and weak localization[END_REF], [START_REF] Arnold | Effective confining potential of quantum states in disordered media[END_REF], and the recent theoretical work of [START_REF] Arnold | Localization of eigenfunctions via an effective potential[END_REF]. In Section 4 we show how the structure of wells and barriers of the effective potential can be incorporated into numerical algorithms to predict the locations and approximate supports of localized eigenfunctions. Then, in Section 5, we show how the values of the minima of the effective potential can be used to predict the corresponding eigenvalues and density of states. Throughout, the performance of our algorithms is demonstrated for various types of 1D and 2D random piecewise constant potentials (uniform, Bernoulli, Gaussian, uncorrelated and correlated).

Note that the computation of the effective potential involves the solution of a single source problem, and so is far less demanding than the computation of a significant portion of the spectrum by traditional methods. However, a remarkable conclusion of our results is that, for the localized problems studied here, a great deal of information about the spectrum can be extracted easily from the effective potential.

Classical confinement.

A simple and well-understood example of eigenfunction localization for the Schrödinger equation occurs with a classically confining potential. Such a potential is decisively different and simpler than the highly disordered potentials we consider, but we shall draw a connection in the forthcoming discussion, and for that reason we now briefly review localization by confinement. The basic example is the finite square well potential in one dimension, for which the analytic solution is derived in first courses on quantum mechanics [19, vol. 1, p. 78]. This problem is posed on the whole real line with the potential V equal to some positive number ν for |x| > 1 and vanishing otherwise. The fundamental eigenfunction is then

(1) ψ(x) = cos( √ λx), |x| ≤ 1, c ν exp(- √ ν -λ|x|), |x| > 1,
where the eigenvalue λ is uniquely determined as the solution to the equation cos √ λ = λ/ν in the interval (0, π 2 /4) and c ν = λ/ν exp( √ ν -λ). Since λ < ν, the solution decays exponentially as |x| → ∞, which captures localization in this context. A similar calculation can be made in higher dimensions, for example for spherical wells [19, vol. 1, p. 359-361], in which case the well height ν must be sufficiently large to ensure that there exists an eigenvalue smaller than ν.

A fundamental result for the localization of eigenfunctions of the Schrödinger equation with a general confining potential is Agmon's theory [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators[END_REF], [11, § 3.3], [START_REF] Hislop | Introduction to Spectral Theory[END_REF]Ch. 3], which demonstrates a similar exponential decay for a much larger class of potentials. In this case the domain is all of R n and the Schrödinger operator is understood as an unbounded operator on L 2 . One requires that the potential be sufficiently regular and bounded below and that there is an eigenfunction ψ with eigenvalue λ such that V > λ outside a bounded set. In other words, outside a compact potential well where V dips below the energy level λ, it remains above it (thus creating confinement). Agmon defined an inner product on the tangent vectors at a point

x ∈ R n by (2) ξ, η x = [V (x) -λ] + ξ • η,
where the subscript + denotes the positive part. This defines a Riemannian metric, except that it degenerates to zero at points x where V (x) ≤ λ. Its geodesics define a (degenerate) distance dist V λ (x, y) between points x, y ∈ R n , and, in particular, we may define ρ(x) = dist V λ (x, 0) to be the distance from the origin to x computed using the Agmon metric. Agmon's theorem states, with some mild restrictions on the regularity on V , that for any > 0,

(3)

R n |e (1-)ρ(x) ψ| 2 dx < ∞.
This result describes exponential decay of the eigenfunction in an L 2 sense in regions where ρ(x) grows, which expresses localization in this context.

For the random potentials which we investigate in this paper, the Agmon distance is highly degenerate, and, consequently, an estimate like (3) is not generally useful. Consider, as a clear example, the Bernoulli potential shown in Figure 2, in which the values 0 and 4 are assigned randomly and independently to each of the 80 × 80 subsquares with probabilities 70% and 30%, respectively. As shown in color on the right of Figure 2, the region where the potential is zero has a massive connected component which nearly exhausts it. For any positive λ, the Agmon distance between any two points in this connected component is zero, and hence an estimate like (3) tells us nothing. Nonetheless, as exemplified by the first two eigenfunctions shown in the figure, the eigenfunctions do localize, a phenomenon for which we must seek a different justification.

3. The landscape function and the effective potential. An important step forward was made in [START_REF] Filoche | Universal mechanism for Anderson and weak localization[END_REF] with the introduction of the landscape function, which is simply the solution to the PDE Hu = 1 together with, for us, periodic boundary conditions. Note that, as long as the potential V is a bounded nonnegative function, nonzero on a set of positive measure, then u is a strictly positive periodic C 1 function (indeed, it belongs to W 2 p for any p < ∞). The following estimate, taken from [START_REF] Filoche | Universal mechanism for Anderson and weak localization[END_REF], relates the landscape function to the eigenvalues.

Proposition 3.1. If ψ : Ω → R is an eigenfunction of H with eigenvalue λ, then (4) ψ(x) ≤ λu(x) ψ L ∞ , x ∈ Ω.
If we normalize the eigenfunction ψ so that ψ L ∞ = 1/λ, then the theorem asserts that ψ ≤ u pointwise, a fact illustrated in Figure 3, which shows the onedimensional case where the potential has 256 values randomly chosen uniformly iid from [0, 4]. The argument was made in [START_REF] Filoche | Universal mechanism for Anderson and weak localization[END_REF] that if the landscape function u nearly vanishes on the boundary of a subregion Ω 0 of the domain Ω, then (4) implies that any eigenfunction ψ must nearly vanish there as well, and so ψ| Ω0 is nearly a Dirichlet eigenfunction for Ω 0 (with the same eigenvalue λ). Similarly ψ restricts to a near Dirichlet eigenfunction on the subdomain complementary to Ω 0 . Except for the unlikely case in which these two subdomains share the eigenvalue λ, this suggests that ψ must nearly vanish in one of them, and so be nearly localized to the other. This viewpoint gives an initial insight into the situation illustrated in Figure 3, where it is seen that the eigenfunctions are essentially localized to the subdomains between two consecutive local minima of u. However, it must be remarked, in the case shown in Figure 3 and many other typical cases, the landscape function merely dips, but in no sense vanishes, on the boundary of localization regions, and so a new viewpoint is needed in order to satisfactorily explain the localization which is observed.

Fig. 3. The potential on the left gives rise to the landscape function, plotted in blue on the right. The first four eigenfunctions are also plotted, scaled so that their maximum value is the reciprocal of their eigenvalue, illustrating the inequality of (4).

Such a new viewpoint was developed in the paper [START_REF] Arnold | Effective confining potential of quantum states in disordered media[END_REF], where the emphasis was placed on the effective potential W , defined as the reciprocal 1/u of the landscape function. A key property of W and the explanation for its name, is that it is the potential of an elliptic operator which is conjugate to the Schrödinger operator H and so has the same spectrum. The following essential identity was derived in [START_REF] Arnold | Effective confining potential of quantum states in disordered media[END_REF]. Proposition 3.2. Suppose that the potential V ∈ L ∞ (Ω) is nonnegative and positive on a set of positive measure. Let u > 0 be the landscape function and W = 1/u the effective potential. Define L :

H 1 (Ω) → H -1 (Ω) by (5) Lφ = - 1 u 2 div(u 2 grad φ). Then (-∆ + V )(uφ) = u (L + W )φ, φ ∈ H 1 (Ω).
In this result, H 1 denotes the periodic Sobolev space on Ω and H -1 its dual. The equation holds in H -1 , making sense because u ∈ C 1 .

Corollary 3.3. Let V , u, and W be as in the theorem and λ ∈ R.

Then ψ ∈ H 1 satisfies (-∆ + V )ψ = λψ if and only if φ := ψ/u satisfies (L + W )φ = λφ.
Thus the eigenvalues of the operator L + W (with periodicity) are the same as those of the original Schrödinger operator, and the eigenfunctions are closely related. However, the corresponding potentials W and V are very different. The effective potential W is often much more regular than the physical potential V . More importantly, it has a clear structure of wells and walls. As we shall see, these induce a sort of localization by confinement, which is not evident from the physical potential. For example, for the Bernoulli potential shown in Figure 2, the effective potential is shown in Figure 4. Note that the effective potential contains many wells: small regions where its value is low, but surrounded, or nearly surrounded, by crestlines where its values are relatively high. If we think of a gradient flow starting at a generic point in the domain and ending at a local minimum, thereby associating the point to one of the local minima of W , the crestlines, displayed in green in Figure 4 are the boundaries of the basins of attraction of the local minima. There are several algorithms to compute the precise location of these crestlines. The one used in this case is the watershed transform as described in [START_REF] Beare | The watershed transform in ITK-discussion and new developments[END_REF].

In recent work [START_REF] Arnold | Localization of eigenfunctions via an effective potential[END_REF], we have established a rigorous connection between the well and wall structure of W and the exponential decay of eigenfunctions. We define the W -distance dist W λ , to be the Agmon distance, as defined after (2) in Section 2, but with the potential V replaced by the effective potential W . Then we show, roughly speaking, that whenever a well of the effective potential exists with sufficient separation of the well depth from the height of the surrounding barriers, then eigenfunctions ψ of the operator H with eigenvalue λ are localized to {W ≤ λ} in the sense that they decay exponentially in the W -distance associated to the eigenvalue. More precisely, in [START_REF] Arnold | Localization of eigenfunctions via an effective potential[END_REF], we prove:

Theorem 3.4. Suppose (ψ, λ) is an eigenpair of the Schrödinger operator H = -∆ + V . Let W be the effective potential and dist W λ (x, y) the associated W -distance. Let δ > 0, S = { x ∈ Ω | W (x) ≤ λ + δ },
a sublevel set of W , and h(x) the W -distance from x to S. Then there exists a constant C depending only on V L ∞ and δ (but not the domain Ω) such that

Ω e h(x) ψ 2 dx ≤ C Ω ψ 2 dx.
This result can be found, stated in considerably more generality and in sharper form in [START_REF] Arnold | Localization of eigenfunctions via an effective potential[END_REF]Corollary 3.5]. The dependence of the constant is given there explicitly. It grows at most linearly in V L ∞ . The same paper also proves exponential decay for the gradient of the eigenfunctions. On the other hand these theoretical results do not capture fully the accuracy with which W predicts the behavior of the eigenfunctions. Our numerical results show that the eigenfunctions typically occupy a single connected component of the set W ≤ λ and decay exponentially across each green crestline of Figure 4, whereas the theorems do not rule out that a resonance occurs resulting in eigenfunctions that have significant mass in several different components of W ≤ λ.

4. Eigenfunction prediction. We may apply the theory described in the previous section to predict the location and extent of the supports of localized eigenfunctions. For this we proceed in four steps.

1. Compute the landscape function u from the PDE Hu = 1, and define the effective potential W = 1/u. 2. To approximate a desired number of localized eigenfunctions, identify the same number of local minima of W , selected in order of increasing minimal value. The location of these minima will be our prediction of the place where the eigenfunctions localize, in the sense that the maximum of the localized eigenfunction will occur nearby there. 3. To each of the selected local minimum we associate an energy level E given by the minimum value of W in the well times a constant greater than 1. The constant is chosen so that E is close to fundamental eigenvalue of the well, or perhaps somewhat larger. (We show in the next section how this may be achieved.) 4. From the corresponding sublevel set, consisting of all x for which W (x) ≤ E, we compute the connected component which contains the selected local minimum. This is the region we predict to be occupied by the eigenfunction. Figure 5 shows the outcome of applying this approach to the 1D Schrödinger equation with the potential shown in Figure 3. The landscape function was computed using Lagrange cubic finite elements with a uniform mesh of 2, 560 subintervals (10 element per constant piece of the potential). The finite element solution was evaluated at 15, 360 equally-spaced points (6 per element), with the reciprocals giving the values of the effective potential. The local maxima and local minima of the effective potential were then identified by comparing the value at each point to that of its two immediate neighbors. For comparison, the true eigenvalues and eigenfunctions were computed by using the same finite element discretization and solving the resulting sparse matrix real symmetric generalized eigenvalue problem using a Krylov-Schur solver. The finite element discretizations were implemented using the FEniCS software environment [START_REF] Logg | Automated Solution of Differential Euqations by the Finite Element Method[END_REF], calling the SLEPc library [START_REF] Hernandez | SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems[END_REF] for the eigenvalue solves. Note that the locations of the four local minima of the effective potential, indicated by small circles in both plots of Figure 5, very nearly coincide with the locations of the maxima of the four corresponding eigenfunctions. Moreover the correspondence respects the ordering of the eigenvalues, in the sense that the ith eigenfunction corresponds to the ith well for i = 1, 2, 3, 4. To predict the extent of the localized eigenfunctions, we use as outer boundaries the level curves of the effective potential at an energy level E set to be 1.875 times the depth of the wells. This is 150% of the value we justify in the next section as an approximation of the eigenvalues. Of course, this choice is somewhat arbitrary, since the effective support of a localized function is not an absolute notion, but must be defined with respect to some tolerance. We could as well have chosen a somewhat larger level to get wider regions incorporating more of the tail of the eigenfunctions, or have chosen a somewhat smaller level to get narrower regions. 3. The first, second, third, and fourth deepest local minima are marked and labeled. The small yellow circles signify the positions of these minima. We expect the corresponding eigenfunctions to be centered near the location of the minima, with extent related to the surrounding basin of attraction. This prediction is plotted in green, and the actual first four eigenfunctions superimposed over the predictions on the right.

Next, we vary this example by increasing the amplitude of the potential by a factor of 64, so that it takes values between 0 and 256, but is otherwise identical to the potential shown on the left of Figure 3. The results analogous to Figure 5 for this potential are shown in Figure 6. Note that, despite the fact that the potentials are proportional in the two cases, the effective potentials look quite different and the eigenfunctions localize in entirely different places. The eigenvalue maxima occur at 17.85, 41.47, 90.75, and 72.92 for the smaller potential and at 90.57, 73.42, 204.5, and 110.5 for the second. These locations again are captured very accurately by the minima of the effective potential, and in the correct order. A crucial difference between the two examples is that the eigenvalues for the problem with the larger potential are much more tightly localized, as predicted by the thinner wells of its effective potential. In the examples depicted in Figures 5 and6, we looked at the first four eigenvalues and found that the eigenvalues are very accurately located by the local minima of the effective potential. We now look at what happens for a larger number of local minima. Obviously, at some point the eigenfunction locations cannot be predicted by the local minima, since there are infinitely many eigenfunctions and only finitely many local minima. Figure 7, which is similar to the plot on the right-hand side of Figure 6, and, in particular, uses the same potential, shows the locations of the first 16 local minima of W (as yellow dots), plotted over the first 16 eigenfunctions. We see that for all 16, the location of the local minimum of W predicts very accurately the location of the corresponding eigenfunction. Figure 8 and Table 1 explore the situation further, comparing the location of the nth local minimum of W , plotted on the x-axis, to that of the maximum of the nth eigenfunction, plotted on the y-axis, for n = 1, . . . , 20. Since these nearly coincide for n ≤ 16, the first 16 points lie very nearly on the line y = x. From then on, however, the points deviate from the line because the ordering of the local minima does not perfectly match the ordering of the most closely associated eigenfunctions. Specifically, as can be seen from Table 1, the 17th local minimum of W occurs at the location of the 19th eigenfunction, and the 18th occurs at the location of the 20th eigenfunction.

We now consider the two-dimensional case where the potential is the random 80 × 80 Bernoulli potential shown in Figure 2, for which the effective potential is shown in Figure 4. To compute the landscape function we again used the finite element method with Lagrange cubic finite elements on a uniform mesh. The mesh was obtained by dividing each of the unit squares into 10×10 subsquares, each of which was further divided into two triangles, resulting in 1,280,000 triangles altogether. We then evaluated the solution at a uniform grid of 400 × 400 points and found the local minima of the effective potential by comparing each of these values to the values at the eight nearest neighbors (horizontally, vertically, and diagonally). In Figure 9 the first plot shows the first four local minima of the effective potential. For each, a corresponding sublevel set of the effective potential is shown. The four minima and sublevel sets are our predictors for the locations of the eigenfunctions. The energy level E of the sublevel sets was taken as 1.56 times the well depth, just slightly larger than the prediction for the eigenvalue, namely 1.5 times the well depth, which we propose in the next section. Recall that the choice of E is somewhat arbitrary. This choice gives a good visual match with the apparent support of the eigenfunctions. The second plot in Figure 9 The data plotted in Figure 8.

the L ∞ norm. Since they are localized one can easily distinguish the location of each within the sum, which is very close to that predicted. The third plot is a superposition of the first two, to facilitate comparison.

In the three cases just considered, there is a clear correspondence between the first four eigenfunctions and the four deepest wells of the effective potential, with each of eigenfunctions, when ordered as usual by increasing eigenvalue, centered at the corresponding local minimum of the effective potential, ordered by depth of the minimum. However, this ideal situation does not always pertain. When two eigenvalues, or two of the minima, are nearly equal, their ordering may not be respected by the correspondence. Another situation which may arise is that the basin surrounding one of the minima may include another. In that case the second minima does not lead to a separate eigenfunction. Both of these issues arise in the the case of the uniformly random potential of Figure 1. Figure 10 shows the first five local minima of the effective potential, and their basins. The numbers provided show the ordering by the depth of the wells. Note that the third and fourth minima nearly coincide in location, and they only contribute one well, even though they are, technically, two distinct minima. The actual minimum values and eigenvalues are given in Table 2. It reveals that the third and fourth minima are not only close in location, but nearly coincide in value as well. Moreover the difference between their value and values of the preceding and following minima is rather small. These close values account for the fact that the correspondence between the eigenfunctions and minima clearly visible in Figure 10 does not respect the precise ordering. Nonetheless, the structure of the effective potential clearly provides a lot of information on the localization structure of the eigenfunctions. To account for such near coincidences we could seek to develop an algorithm to identify clusters of minima with nearly equal values and relate them to clusters of nearly equal eigenvalues. However we shall not pursue this direction here.

Thus far we have examined random piecewise constant potentials with values taken independently and identically distributed according to some probability distribution (uniform or Bernoulli). In the final example of this section, we consider a potential for which the values are correlated rather than independent. To generate values for the potential, we use circulant embedding to convert uncorrelated Gaussian N (0, 1) random vector samples to correlated Gaussians [START_REF] Kroese | Spatial process simulation[END_REF]. We take a 1-dimensional example, in which the potential is piecewise constant with n unit length pieces, with n even. Define q i = q n-i = σ exp(-di), 0 ≤ i ≤ n/2, where d is a positive constant, The values of the effective potential at its first five minima, and the first five eigenvalues for the uniformly random potential of Figure 1. Cf. Figure 10. and let Q be the diagonal matrix with entries q 0 , q 1 , • • • q n-1 . A sample vector for the values of V is obtained by squaring the components of the vector F -1 QF z where z is a vector of length n with components sampled independently from a normalized Gaussian distribution, and F is the discrete Fourier transform on C n . This type of a random potential is typically created by optical speckles in a Bose-Einstein condensate. See, e.g., [START_REF] Modugno | Collective dynamics and expansion of a bose-einstein condensate in a random potential[END_REF], [START_REF] Falco | Density of states in an optical speckle potential[END_REF]. It is quite challenging to derive rigorous probabilistic results when the correlation is not negligibly small, especially in higher dimensions. The landscape theory, however, continues to apply. We consider an example with n = 1, 024, σ = 1.0, d = 0.01, shown in Figure 11 along with the corresponding ef-Fig. 10. For the potential of Figure 1, the correspondence between eigenfunctions and wells of the effective potential does not respect the ordering of the well minima. Moreover, the third and fourth minima effectively define a single well.

fective potential. Note that, although the potential and effective potential look quite different from the previous examples, the effective potential still has clearly defined wells which allow us to apply our theory. In the final plot in Figure 11 we use the effective potential as before to predict the location and extent of the first seven eigenfunctions. As in the uncorrelated cases, the well minima very nearly coincide with the peak of the eigenfunctions. The first three minima, in order, correspond to the first three eigenfunctions, but after that the order is not the same. The discrepancy in ordering is not very significant, however, since both the minimum values and the eigenvalues are very close to one another, as indicated in Table 3. Table 3 The values of the effective potential at its first seven minima, and the first seven eigenvalues for the correlated potential of Figure 11.

In this section we have shown how we can deduce the approximate locations and approximate supports of eigenfunctions just by processing the effective potential, without solving eigenvalue problems. We remark that this information could be refined to give an approximation of the precise shape of the eigenfunction. To do so, one could solve for the eigenfunction with a standard PDE eigensolver, but with the domain taken as a regular domain just slightly larger than the approximate support, and with Dirichlet boundary conditions. Because of the localization, this computational domain will be much smaller than the original domain and this computation much less expensive than a global eigenvalue solve. The development and study of such algorithms, however, goes beyond the scope of this paper, and is left for future work. Fig. 11. A correlated potential and the corresponding effective potential with its seven lowest minima marked. On the top left is the original correlated potential, on the top right the corresponding effective potential. On the bottom are the first seven eigenfunctions with the color giving the order as indicated by the inset. A rectangle is superimposed on each eigenfunction. The heights of the rectangles and eigenfunctions are normalized to unity, while each rectangle's width indicates the extent of the localized eigenfunction as predicted by the effective potential (using a sublevel set at an energy level equal to 1.875 times the depth of the wells as was done for Figure 5). The small yellow circles give the position of the local minima and the numbers over them, the order of the local minima. Notice that the order differs from the order of the corresponding eigenfunctions after the first three.

5. Eigenvalue prediction. We now turn to the question of predicting eigenvalues of the Schrödinger operator H from the effective potential. As a simple illustration of the utility of the effective potential for eigenvalue estimation, we start by recalling the basic lower bound on the fundamental eigenvalue in terms of the potential V , and show how it can be improved by using the effective potential.

For an eigenfunction ψ of H with eigenvalue λ, normalized to have L 2 norm 1, we have [START_REF] Falco | Density of states in an optical speckle potential[END_REF] λ = (Hψ, ψ) = grad ψ 2 + (V ψ, ψ)

which represents the decomposition into kinetic and potential energy. Dropping the kinetic energy term and replacing V by its infimum gives a lower bound on the eigen-values:

(7) λ ≥ inf V.
Now we use the fundamental identity of Proposition 3.2 to decompose the eigenvalue in terms of the effective potential:

(Hψ, ψ) = (u 2 Lφ, φ) + (W ψ, ψ),
where φ = ψ/u. In view of the form of L (5), the first term on the right hand side is positive, so dropping it and replacing W by its infimum gives another lower bound: The infimum of V is, however, very near zero: 0.00009, and so the bound ( 7) is nearly worthless. (In this realization inf V happens to be particularly small, but the expected value of 1/65 = 0.015 is again of little use.) By contrast, inf W = 1.22, which is a useful lower bound. In fact, the fundamental eigenvalue is equal to about 1.3 inf W . We shall see below that this factor of roughly 1.25 or 1.3 applies for a wide range of random potentials in one dimension.

(8) λ ≥ inf W.
Fig. 12. A random potential (in shaded gray), the corresponding effective potential (the solid blue line), and the fundamental eigenvalue (the horizontal red line).

In the remainder of this section, we shall consider two approaches to eigenvalue prediction, one based solely on the local minima values of the effective potential, and the other based on a variant of Weyl's law utilizing the effective potential. Although we shall explain the thinking behind these approaches, it has to be noted that neither has yet been justified rigorously.

5.1. Eigenvalues from minima of the effective potential. In this section we discuss the approximation of the eigenvalues of the Schrödinger operator H using the effective potential W . We shall be interested in both the approximation of individual eigenvalues, and in the distribution of the eigenvalues. The latter is captured by the density of states (DOS). Defined precisely, the DOS is the distribution on R obtained by summing the delta functions centered at each eigenvalue. For visualization, it is often converted to a piecewise constant function with respect to a partition of the real line into intervals of some length > 0, with the value over any interval being the integral of the DOS over the interval (so a plot of this function is a histogram of the eigenvalues using bins of width ). The integrated density of states (IDOS) is the integral of the DOS from -∞ to a real value E. The resulting function N : R → N is simply the eigenvalue counting function, with N (E) defined as the number of eigenvalues ≤ E.

To motivate our first approach to eigenvalue prediction, consider again the potential shown in Figure 1, a piecewise constant function with 80 × 80 pieces and constant values chosen randomly and independently from [0, 20]. In the first plot of Figure 13 we have computed the values of the effective potential W at its local minima and compared the first 100 of these, in increasing order, to the first 100 eigenvalues of W . Just below we plot the quotients of each of these eigenvalues divided by the corresponding local minimum value of W . Observe that the quotient is quite constant, taking on a value of roughly 1.5. We shall endeavor to explain this value below, but first we observe that this ratio of roughly 1.5 between the mth eigenvalue and the mth minimum value of the effective potential holds over a wide range of random potentials in two dimensions. In the remainder of the first two rows of Figure 13 we show the same results also for the Bernoulli potential of Figure 2 and the correlated potential shown in Figure 14. (The correlated potential was constructed with circulant embedding as discussed above for one dimension, except that we of course used the two-dimensional discrete Fourier transform, and we took as aperture function σχ(d|t|) with σ = 4 and d = 0.05, where χ is the characteristic function of the unit interval, instead of σ exp(-d|t|) as we took previously.) The three pairs of plots in the final two rows of the figure show the case of a uniformly random 40 × 40 potential with values taken between 0 and 4, 16, and 64, respectively. The quotient is quite close to being constant with value 1.5, particularly in the last two cases. In the first case, with the lowest disorder, the ratio drifts away from 1.5 to about 1.75 after approximately 50 eigenvalues.

Of course, the minima of the effective potential can only predict a limited number of eigenvalues. Indeed, W has only a finite number of local minima, while there are infinitely many eigenvalues. In Figure 15 we revisit the fifth case shown in Figure 13, with a uniformly random 40 × 40 potential with values in [0,[START_REF] Lefebvre | One single static measurement predicts wave localization in complex structures[END_REF]. For this realization of the potential, W has exactly 252 local minima. These are plotted alongside the first 300 eigenvalues in the left-hand side of Figure 15, and their ratios with the corresponding eigenvalues are plotted on the left. We see that, in this case, the ratio of 1.5 remains quite accurate for more than 150 of the 252 minima.

In Table 4 we display the mean and standard deviation of these ratios computed for the first 10, 50, and 100 eigenvalues for each of the six potentials of Figure 13. The main observation is that, across all this range, the ratio stays quite close to 1.5.

Table 5 is similar, but shows the results for a variety of potentials in one dimension. Again, we see that the ratio of the eigenvalue to the corresponding minimum values of the effective potential is roughly constant, as it was in two dimensions. However the constant value we find in one dimension is about 1.25 or 1.3 rather than the value of 1.5 we saw in two dimensions.

Finally, in Figures 16 we plot along the line λ = 1.25W min , and in the second along the line λ = 1.5W min .

From this and other evidence, we conclude that in many cases

(9) λ ≈ (1 + n 4 ) W min .
Here λ is one of the lower eigenvalues of H, for which the corresponding eigenfunction is localized to a subdomain Ω 0 , W min is the minimum value of the effective potential on that subdomain, and n is the number of dimensions (thus far 1 or 2). The constant 1 + n/4, i.e., 1.25 in 1D and 1.5 in 2D, is a rough approximation in accord with our observations and which we now further justify heuristically. It is remarkable that this constant, though dimension-dependent, is independent of the specific realization of the potential and of the parameters of its probability distribution, the size of the domain, etc.

We now give some heuristic support of the eigenvalue approximation [START_REF] Filoche | The landscape of Anderson localization in a disordered medium[END_REF]. Our argument will be rather crude, and we do not claim it fully explains the numerical evidence presented above. Let ψ denote one of the eigenfunctions associated to a smaller eigenvalue. We assume ψ to be localized, i.e., essentially supported in a small subdomain Ω 0 , for which it is the fundamental Dirichlet eigenfunction. We also assume that on the subdomain Ω 0 the landscape function u is well approximated by a constant multiple of the fundamental eigenfunction ψ of the subdomain. This is roughly supported by experimental results such as shown in Figure 3. Another supporting argument comes from the expansion of the constant 1 on Ω 0 in terms of the Dirichlet eigenfunctions of the domain, retaining only the first term c 0 ψ, and dropping the terms coming from the eigenfunctions which change sign. Then u ≈ (c 0 /λ)ψ, indeed a multiple of ψ. We may use these two assumptions, together with the definition Hu = 1 of the landscape function, to approximate the Rayleigh quotient:

λ = Ω ψHψ dx Ω ψ 2 dx ≈ Ω0 ψHψ dx Ω0 ψ 2 dx ≈ Ω0 uHu dx Ω0 u 2 dx = Ω0 u dx Ω0 u 2 dx
.

Next we assume that on Ω 0 the landscape function u (or ψ which we are supposing is a constant multiple of u there) can be approximated by the simplest sort of positive bump-like function, the positive part of a concave quadratic function. After rotating and translating the coordinate system, this means that

u ≈ u max [1 - (x i /a i ) 2 ] on Ω 0 ≈ { x ∈ R n | (x i /a i ) 2 ≤ 1 },
for some positive constants a i . Thus our approximation to the eigenvalue is

λ ≈ Ω0 u dx Ω0 u 2 dx = c u max , c = Ω0 [1 -(x i /a i ) 2 ] dx Ω0 [1 -(x i /a i ) 2 ] 2 dx
.

Finally, we compute c using the change of variables xi = x i /a i to convert the integrals in the numerator and denominator into integrals over the unit ball B which can be computed with polar coordinates. This gives c = 1 + n/4 (independent of the values of the a i and so of the size and shape of the ellipsoid). Since 1/u max = W min , this indeed gives the approximation [START_REF] Filoche | The landscape of Anderson localization in a disordered medium[END_REF]. The results we have shown in Figure 13 and Tables 4 and5 demonstrate that the approximation (9) can be used to estimate 100 eigenvalues with errors of a few percent. Note that it is nonetheless very cheap to apply (9), the cost being that of solving a single source problem and the extraction of some maxima, much less than the cost of computing many eigenvalues. As another example of the utility of ( 9) we now use it to approximate the density of states in the interval [0, 1] of the 1D Schrödinger operator for which the piecewise constant potential has 2 19 = 524, 288 pieces. (Specifically, we compute on the interval [0, 2 19 ] and assign the random values to unit subintervals uniformly iid in [0, 4].) We display the DOS as a histogram with 100 bins. The top left plot in Figure 17 shows the actual density of states, requiring the computation of all 7,122 of the eigenvalues which belong to the interval [0, 1]. The finite element mesh we use to approximate the Schrödinger operator in this case has 10 × 2 19 elements, and we use piecewise cubic finite elements, so that the problem has about 15.7 million degrees of freedom. The calculation of 7,000 eigenvalues is thus a very large computation. It required about 40 CPU hours on a workstation with an Intel Core i7-4930K processor, using sprectral slicing and the Krylov-Schur method of SLEPc. However, an accurate approximation of the density of states can be obtained quickly using the effective potential without resorting to the computation of any eigenvalues. This approximation is shown in the top right plot of Figure 17. It is a histogram of all those values of 1.25W min which belong to the same interval [0, 1] (8,800 in all). The computation of these values is much less demanding. It required slightly over 5 minutes of CPU time of the same workstation, i.e., was about 480 times faster. The two histograms are compared in the bottom plot.

5.2.

Eigenvalues from a variant of Weyl's law. The approximation (9) can be used to predict, at most, one eigenvalue per local minimum of W . Intuitively, it approximates a localized eigenfunction by the fundamental mode of the well around the local minimum. Now we present an alternative approach, which again relies on the effective potential W , but which gives some sort of prediction for all the eigenvalues. For large eigenvalues, it provides information similar to Weyl's law. Recall that Weyl's law for the Schrödinger equation is an asymptotic formula for the eigenvalue counting function:

(10) N (E) ∼ N V (E) := (2π) -n vol{ (x, ζ) ∈ Ω × R n | V (x) + |ζ| 2 ≤ E } as E → ∞,
where the volume term is the 2n-dimensional measure of the indicated subset of the phase space Ω × R n . Assuming smoothness and growth conditions for the potential, Weyl's law holds asymptotically as E tends to +∞ and so for a large number of eigenvalues [START_REF] Zworski | Semiclassical Analysis[END_REF]Theorem 6.8]. Inverting the counting function, we can thus view Weyl's law as furnishing an approximation of the nth eigenvalue, which is asymptotically valid for n large. Weyl's law is generally not expected to be accurate for a small number of eigenvalues. However, experimentally we have found that, for the sorts of random potentials considered in this paper, a variant of Weyl's law invoking the effective potential W gives very good results right down to the first few eigenvalues, while remaining asymptotically correct. The variant, which we shall refer to as the effective Weyl's law, is obtained by simply replacing the potential V in (10) by the effective potential W : For this potential, there is no localization. Nonetheless, we see that, in each case, Weyl's law becomes a good approximation only after 100 or so eigenvalues and, in every case, it incorrectly predicts many eigenvalues in the interval from 0 to the least eigenvalue. By contrast, the effective Weyl's law provides a very good approximation of the counting function for many eigenvalues, starting from the first. It is revealing to compare the two potentials which take on only Boolean values (the second and the fourth). Because the classical Weyl's law is unaffected by rearrangement of the potential, it gives the same prediction for the counting function in both cases. But the actual counting functions differ very significantly, a fact which is well captured by the effective Weyl's law. (Similar results were published in [START_REF] Arnold | Effective confining potential of quantum states in disordered media[END_REF].) Finally, in Figure 19 we show similar results for a single 2D potential, namely the uniformly random 80 × 80 potential of Figure 1. The first plot shows the first 100 eigenvalues, while the second zooms in on the first 10 eigenvalues. The predictive power of the effective Weyl's law does not seem to be as great as in one dimension, but again it displays a great improvement over the classical Weyl's law for small eigenvalues.

N W (E) = (2π) -n vol{ (x, ζ) ∈ Ω × R n | W (x) + |ζ| 2 ≤ E }.

Conclusion.

We have demonstrated numerically that the effective potential, defined as the reciprocal of the localization landscape function, accurately captures a great deal of information about the localization properties of a random potential, and shown how to employ it to predict eigenvalues and eigenfunctions. These predictions are attained by a solving a single PDE source problem, without the direct solution of any eigenvalue problems, and so at a very low computational price. The wells of the effective potential reveal the main localization subdomains, and the values of its minima are found to be very good predictors of the corresponding fundamental eigenvalues. We have tested this approach on piecewise constant potentials with several types of random distributions, uniformly random and Bernoulli, and with a certain correlated distribution, both in one and two dimensions. We have further used the effective potential to predict the density of states and obtain good precision even for small eigenvalues, something which is not attained by the classical Weyl law asymptotics. In highly demanding computations where the Schrödinger equation has to be solved for a large number eigenfunctions and eigenvalues (as for instance in semiconductor physics), the resulting computational efficiency makes it now possible to reproduce numerically, to analyze, and to understand the behavior of quantum disordered materials.

Fig. 1 .

 1 Fig. 1. A piecewise constant potential with randomly chosen iid values, and the first two eigenfunctions of the corresponding Schrödinger operator.

Fig. 2 .

 2 Fig. 2. For this Bernoulli potential there are no wells surrounded by thick walls. Nonetheless, the eigenfunctions localize.

Fig. 4 .

 4 Fig. 4. The effective potential associated to the Bernoulli potential of Figure 2. It is shown with its crestlines which partition the domain into a few hundred basins of attraction surrounding wells.

Fig. 5 .

 5 Fig.5. On the left is the effective potential corresponding to the piecewise constant potential with 256 uniformly iid randomly selected values shown on the left of Figure3. The first, second, third, and fourth deepest local minima are marked and labeled. The small yellow circles signify the positions of these minima. We expect the corresponding eigenfunctions to be centered near the location of the minima, with extent related to the surrounding basin of attraction. This prediction is plotted in green, and the actual first four eigenfunctions superimposed over the predictions on the right.

Fig. 6 .

 6 Fig. 6. The effective potential for the potential equal to 64 times that shown on the left of Figure 3, with the four deepest local minima and their wells marked and labeled, and then a comparison with the actual first four eigenfunctions.

Fig. 7 .

 7 Fig. 7. For the same potential as Figure 6 the first 16 local minima locations accurately predict the locations of the corresponding eigenfunctions.

Fig. 8 .

 8 Fig. 8. First 20 local minima on the x-axis versus the maximum of the corresponding eigenfunction on the y-axis.

Fig. 9 .

 9 Fig. 9. The first plot shows the prediction for the location of the first four eigenfunctions for the Bernoulli potential of Figure 2. The second plot shows the actual positions of these eigenfunctions, superimposed. The final plot compares the actual positions to the predictions.

  2.4246 2.4763 2.4796 2.5370 eigenvalue: 3.6112 3.6618 3.7075 3.7717 3.9190Table 2

  012752 0.022966 0.024642 0.025647 0.025673 0.026626 0.028347 eig: 0.016534 0.030069 0.031620 0.032953 0.033822 0.034479 0.034735

Figure 12

 12 Figure 12 allows one to compare the two bounds for a random potential with 64 values chosen uniformly iid in the range [0, 8].The fundamental eigenvalue for this realization is 1.58, indicated on the plot in red. The infimum of V is, however, very near zero: 0.00009, and so the bound (7) is nearly worthless. (In this realization inf V happens to be particularly small, but the expected value of 1/65 = 0.015 is again of little use.) By contrast, inf W = 1.22, which is a useful lower bound. In fact, the fundamental eigenvalue is equal to about 1.3 inf W . We shall see below that this factor of roughly 1.25 or 1.3 applies for a wide range of random potentials in one dimension.

  the 1st, 10th, and 25th eigenvalues versus the corresponding minima values of W for numerous different realizations of a random potential. The first figure displays 64 realizations of a 1D potential on [0, 256] with 256 values selected uniformly iid from [0, 16], while the second figure displays 64 realizations of a 2D potential on [0, 40] × [0, 40] with 1, 600 random values again chosen uniformly iid from [0, 16]. We see that in the first case the points line up well

Fig. 13 .

 13 Fig. 13. A comparison of 100 eigenvalues with the corresponding minimum values of W for six different potentials with the ratio of the two shown beneath each one. Top two rows: (left) random 80 × 80 piecewise potential with values chosen uniformly in [0, 20]; (center) Bernoulli potential of Figure 2; (right) correlated potential of Figure 14. Bottom two rows: random 40 × 40 piecewise potential with values chosen uniformly in (left) [0, 4], (center) [0, 16], (right) [0, 64]. In all these cases, one can notice that the ratio of the minimum values of W to the corresponding eigenvalues remains remarkably close the value 1.5.

Fig. 14 .

 14 Fig. 14. A correlated potential with 80 × 80 pieces and the corresponding effective potential. The potential takes values ranging from 2.364 × 10 -9 to 60.56, while the effective potential's values range from 0.374 to 57.17.

Fig. 15 .

 15 Fig.15. A plot of all the minima of W versus the eigenvalues, and, on right, their ratio, for the same potential realization as the fifth case shown in Figure13.

Fig. 16 .

 16 Fig. 16. The 1st, 10th, and 25th eigenvalues versus the corresponding minima values of the effective potential, for 64 independent realizations of a random potential. Left in 1D where the black line shown is λ = 1.25W min , right in 2D with λ = 1.5W min .

Fig. 17 .

 17 Fig. 17. Density of states on [0, 1] for the case of a uniformly random potential with 2 19 pieces, displayed with 100 bins. On top left, histogram of the actual eigenvalues. On top right, histogram of 1.25 times the local minima of W . The bottom highlights the differences between the two.

Figure 18

 18 Figure 18 compares the true eigenvalue counting function N (shown in black), Weyl's law (green), and the effective Weyl's law (red), for four different types of potential.

Fig. 18 .

 18 Fig. 18. The eigenvalue counting function N , the Weyl's law approximation N V , and the effective Weyl's law approximation N W for some potentials in one dimension. Top row: uniform and Boolean random potentials. Bottom row: correlated and periodic Boolean potential.

Fig. 19 .

 19 Fig. 19. The eigenvalue counting function N , the Weyl's law approximation N V , and the effective Weyl's law approximation N W for the 2D potential of Figure 1, showing the first 100 eigenvalues on left and restricting to the first 10 on right.

Table 1

 1 is a plot of the sum of four eigenfunctions, each normalized in

	n W min eigfn max	n W min eigfn max
	1	90.57	90.55	11	47.43	47.43
	2	73.42	73.43	12	44.52	44.50
	3	204.50	204.50	13	180.52	180.52
	4	110.50	110.50	14	158.48	158.48
	5	100.48	100.48	15	41.50	41.50
	6	232.50	232.50	16	253.32	253.33
	7	225.48	225.48	17	177.50	17.43
	8	59.48	59.48	18	2.45	90.95
	9	18.22	18.18	19	146.53	177.50
	10	222.48	222.48	20	163.58	2.47

Table 4

 4 The mean and standard deviation for the ratio of first 10, 50, and 100 eigenvalues to the corresponding minima values of W . For a wide range of potentials in two dimensions the ratio is roughly 1.5 across many eigenvalues.

			10 eigs	50 eigs	100 eigs
	potential	n c	mean SD	mean SD	mean SD
	uniform [0, 20] 80	1.519 0.024 1.520 0.018 1.524 0.017
	Bernoulli	80	1.585 0.034 1.607 0.041 1.646 0.049
	correlated	80	1.519 0.018 1.532 0.028 1.530 0.022
	uniform [0, 4]	40	1.479 0.018 1.510 0.034 1.582 0.087
	uniform [0, 16] 40	1.518 0.041 1.515 0.026 1.516 0.021
	uniform [0, 64] 40	1.503 0.025 1.511 0.024 1.477 0.043
			10 eigs	25 eigs	50 eigs
	potential	n c	mean SD	mean SD	mean SD
	uniform [0, 4]	256	1.303 0.026 1.321 0.029 1.300 0.067
		1024	1.302 0.019 1.301 0.020 1.304 0.022
	uniform [0, 16]	256	1.322 0.023 1.308 0.031 1.240 0.099
		1024	1.274 0.018 1.296 0.031 1.294 0.027
	Bernoulli	256	1.301 0.033 1.316 0.050 1.296 0.130
		1024	1.262 0.014 1.272 0.026 1.266 0.073
	correlated	256	1.335 0.055 1.404 0.090 1.310 0.171
		1024	1.280 0.018 1.286 0.017 1.303 0.042

Table 5

 5 The mean and standard deviation for the ratio of first 10, 25, and 50 eigenvalues to the corresponding minima values of W , tabulated here for 8 different types of random potentials in one dimension.

Foundation (601937, DNA; 601941, GD; 601944, MF; 601948, Arnold was supported by NSF grant DMS-1719694, Mayboroda by NSF INSPIRE grant DMS-1344235 and a Simons Foundation Fellowship. ANR grant, programme blanc GEOMETRYA, ANR-12-BS01-0014. Supported by NSF grant DMS-1500771 and a Simons Foundation Fellowship.

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