## Analysis & PDE

• Anal. PDE
• Volume 11, Number 4 (2018), 1049-1081.

### Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

#### Abstract

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector $χ ( − ∞ , E ] ( H L )$ of a Schrödinger operator $H L$ on a cube of side $L ∈ ℕ$, with bounded potential. Previously, such estimates were known only for individual eigenfunctions and for spectral projectors $χ ( E − γ , E ] ( H L )$ with small $γ$. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. Our main application of such an estimate is to find lower bounds for the lifting of eigenvalues under semidefinite positive perturbations, which in turn can be applied to derive a Wegner estimate for random Schrödinger operators with nonlinear parameter-dependence. Another application is an estimate of the control cost for the heat equation in a multiscale domain in terms of geometric model parameters. Let us emphasize that previous uncertainty principles for individual eigenfunctions or spectral projectors onto small intervals were not sufficient to study such applications.

#### Article information

Source
Anal. PDE, Volume 11, Number 4 (2018), 1049-1081.

Dates
Revised: 11 August 2017
Accepted: 16 October 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.apde/1517454163

Digital Object Identifier
doi:10.2140/apde.2018.11.1049

Mathematical Reviews number (MathSciNet)
MR3749376

Zentralblatt MATH identifier
1383.35068

#### Citation

Nakić, Ivica; Täufer, Matthias; Tautenhahn, Martin; Veselić, Ivan. Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators. Anal. PDE 11 (2018), no. 4, 1049--1081. doi:10.2140/apde.2018.11.1049. https://projecteuclid.org/euclid.apde/1517454163

#### References

• L. Bakri, “Carleman estimates for the Schrödinger operator: applications to quantitative uniqueness”, Comm. Partial Differential Equations 38:1 (2013), 69–91.
• J. M. Barbaroux, J. M. Combes, and P. D. Hislop, “Localization near band edges for random Schrödinger operators”, Helv. Phys. Acta 70:1-2 (1997), 16–43.
• J. Bourgain and C. E. Kenig, “On localization in the continuous Anderson–Bernoulli model in higher dimension”, Invent. Math. 161:2 (2005), 389–426.
• J. M. Combes, P. D. Hislop, and E. Mourre, “Spectral averaging, perturbation of singular spectra, and localization”, Trans. Amer. Math. Soc. 348:12 (1996), 4883–4894.
• J. M. Combes, P. D. Hislop, and S. Nakamura, “The $L^p$-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators”, Comm. Math. Phys. 218:1 (2001), 113–130.
• J.-M. Combes, P. D. Hislop, and F. Klopp, “Hölder continuity of the integrated density of states for some random operators at all energies”, Int. Math. Res. Not. 2003:4 (2003), 179–209.
• J.-M. Combes, P. D. Hislop, and F. Klopp, “An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators”, Duke Math. J. 140:3 (2007), 469–498.
• B. Davey, “Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator”, Comm. Partial Differential Equations 39:5 (2014), 876–945.
• A. Dietlein, M. Gebert, and P. Müller, “Bounds on the effect of perturbations of continuum random Schrödinger operators and applications”, preprint, 2017. To appear in J. Spectr. Theory.
• M. Egidi and I. Veselić, “Scale-free unique continuation estimates and Logvinenko–Sereda theorems on the torus”, preprint, 2016.
• S. Ervedoza and E. Zuazua, “Sharp observability estimates for heat equations”, Arch. Ration. Mech. Anal. 202:3 (2011), 975–1017.
• L. Escauriaza and S. Vessella, “Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients”, pp. 79–87 in Inverse problems: theory and applications (Cortona/Pisa, 2002), edited by G. Alessandrini and G. Uhlmann, Contemp. Math. 333, Amer. Math. Soc., Providence, RI, 2003.
• E. Fernández-Cara and E. Zuazua, “The cost of approximate controllability for heat equations: the linear case”, Adv. Differential Equations 5:4-6 (2000), 465–514.
• A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations, Lecture Notes Series 34, Seoul National University, Seoul, 1996.
• F. Germinet, “Recent advances about localization in continuum random Schrödinger operators with an extension to underlying Delone sets”, pp. 79–96 in Mathematical results in quantum mechanics, edited by I. Beltita et al., World Scientific, Hackensack, NJ, 2008.
• F. Germinet and A. Klein, “A comprehensive proof of localization for continuous Anderson models with singular random potentials”, J. Eur. Math. Soc. $($JEMS$)$ 15:1 (2013), 53–143.
• F. Germinet, P. D. Hislop, and A. Klein, “Localization for Schrödinger operators with Poisson random potential”, J. Eur. Math. Soc. $($JEMS$)$ 9:3 (2007), 577–607.
• F. Germinet, P. Müller, and C. Rojas-Molina, “Ergodicity and dynamical localization for Delone–Anderson operators”, Rev. Math. Phys. 27:9 (2015), art. id. 1550020.
• E. N. Güichal, “A lower bound of the norm of the control operator for the heat equation”, J. Math. Anal. Appl. 110:2 (1985), 519–527.
• M. Helm and I. Veselić, “Linear Wegner estimate for alloy-type Schrödinger operators on metric graphs”, J. Math. Phys. 48:9 (2007), art. id. 092107.
• D. Hundertmark, R. Killip, S. Nakamura, P. Stollmann, and I. Veselić, “Bounds on the spectral shift function and the density of states”, Comm. Math. Phys. 262:2 (2006), 489–503.
• D. Jerison and G. Lebeau, “Nodal sets of sums of eigenfunctions”, pp. 223–239 in Harmonic analysis and partial differential equations (Chicago, IL, 1996), edited by M. Christ et al., Univ. Chicago Press, 1999.
• J. Kellendonk, D. Lenz, and J. Savinien (editors), Mathematics of aperiodic order, Progress in Mathematics 309, Springer, 2015.
• C. E. Kenig, M. Salo, and G. Uhlmann, “Inverse problems for the anisotropic Maxwell equations”, Duke Math. J. 157:2 (2011), 369–419.
• W. Kirsch, “Wegner estimates and Anderson localization for alloy-type potentials”, Math. Z. 221:3 (1996), 507–512.
• W. Kirsch and I. Veselić, “Existence of the density of states for one-dimensional alloy-type potentials with small support”, pp. 171–176 in Mathematical results in quantum mechanics (Taxco, 2001), edited by R. Weder and P. Exner, Contemp. Math. 307, Amer. Math. Soc., Providence, RI, 2002.
• W. Kirsch and I. Veselić, “Lifshitz tails for a class of Schrödinger operators with random breather-type potential”, Lett. Math. Phys. 94:1 (2010), 27–39.
• W. Kirsch, P. Stollmann, and G. Stolz, “Localization for random perturbations of periodic Schrödinger operators”, Random Oper. Stochastic Equations 6:3 (1998), 241–268.
• A. Klein, “Unique continuation principle for spectral projections of Schrödinger operators and optimal Wegner estimates for non-ergodic random Schrödinger operators”, Comm. Math. Phys. 323:3 (2013), 1229–1246.
• A. Klein and C. S. S. Tsang, “Quantitative unique continuation principle for Schrödinger operators with singular potentials”, Proc. Amer. Math. Soc. 144:2 (2016), 665–679.
• I. Kukavica, “Quantitative uniqueness for second-order elliptic operators”, Duke Math. J. 91:2 (1998), 225–240.
• J. Le Rousseau and G. Lebeau, “On Carleman estimates for elliptic and parabolic operators: applications to unique continuation and control of parabolic equations”, ESAIM Control Optim. Calc. Var. 18:3 (2012), 712–747.
• G. Lebeau and L. Robbiano, “Contrôle exact de l'équation de la chaleur”, Comm. Partial Differential Equations 20:1-2 (1995), 335–356.
• P. Lissy, “A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation”, C. R. Math. Acad. Sci. Paris 350:11-12 (2012), 591–595.
• L. Miller, “Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time”, J. Differential Equations 204:1 (2004), 202–226.
• L. Miller, “The control transmutation method and the cost of fast controls”, SIAM J. Control Optim. 45:2 (2006), 762–772.
• L. Miller, “A direct Lebeau–Robbiano strategy for the observability of heat-like semigroups”, Discrete Contin. Dyn. Syst. Ser. B 14:4 (2010), 1465–1485.
• A. Boutet de Monvel, S. Naboko, P. Stollmann, and G. Stolz, “Localization near fluctuation boundaries via fractional moments and applications”, J. Anal. Math. 100 (2006), 83–116.
• A. Boutet de Monvel, D. Lenz, and P. Stollmann, “An uncertainty principle, Wegner estimates and localization near fluctuation boundaries”, Math. Z. 269:3-4 (2011), 663–670.
• I. Nakić, C. Rose, and M. Tautenhahn, “A quantitative Carleman estimate for second order elliptic operators”, preprint, 2015. to appear in Proc. Roy. Soc. Edinburgh Sect. A.
• I. Nakić, M. Täufer, M. Tautenhahn, and I. Veselić, “Scale-free uncertainty principles and Wegner estimates for random breather potentials”, C. R. Math. Acad. Sci. Paris 353:10 (2015), 919–923.
• K.-D. Phung, “Note on the cost of the approximate controllability for the heat equation with potential”, J. Math. Anal. Appl. 295:2 (2004), 527–538.
• Y. Privat, E. Trélat, and E. Zuazua, “Complexity and regularity of maximal energy domains for the wave equation with fixed initial data”, Discrete Contin. Dyn. Syst. 35:12 (2015), 6133–6153.
• Y. Privat, E. Trélat, and E. Zuazua, “Optimal shape and location of sensors for parabolic equations with random initial data”, Arch. Ration. Mech. Anal. 216:3 (2015), 921–981.
• C. Rojas-Molina and I. Veselić, “Scale-free unique continuation estimates and applications to random Schrödinger operators”, Comm. Math. Phys. 320:1 (2013), 245–274.
• C. Schumacher and I. Veselić, “Lifshitz tails for Schrödinger operators with random breather potential”, preprint, 2017. To appear in C. R. Math. Acad. Sci. Paris.
• C. Shirley, “Decorrelation estimates for some continuous and discrete random Schr ödinger operators in dimension one, without covering condition”, preprint, 2015.
• P. Stollmann, Caught by disorder: bound states in random media, Progress in Mathematical Physics 20, Birkhäuser, Boston, 2001.
• M. Täufer and M. Tautenhahn, “Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators”, Commun. Pure Appl. Anal. 16:5 (2017), 1719–1730.
• M. Täufer and I. Veselić, “Conditional Wegner estimate for the standard random breather potential”, J. Stat. Phys. 161:4 (2015), 902–914.
• M. Täufer and I. Veselić, “Wegner estimate for Landau-breather Hamiltonians”, J. Math. Phys. 57:7 (2016), art. id. 072102.
• M. Täufer, M. Tautenhahn, and I. Veselić, “Harmonic analysis and random Schrödinger operators”, pp. 223–255 in Spectral theory and mathematical physics, edited by M. Mantoiu et al., Oper. Theory Adv. Appl. 254, Springer, 2016.
• G. Tenenbaum and M. Tucsnak, “New blow-up rates for fast controls of Schrödinger and heat equations”, J. Differential Equations 243:1 (2007), 70–100.
• M. Tucsnak and G. Weiss, Observation and control for operator semigroups, Birkhäuser, Basel, 2009.
• I. Veselić, Lokalisierung bei zufällig gestörten periodischen Schrödingeroperatoren in Dimension Eins, diploma thesis, Ruhr-Universität Bochum, 1996, http://www.ruhr-uni-bochum.de/mathphys/ivan/diplomski-www-abstract.htm.
• I. Veselić, “Lifshitz asymptotics for Hamiltonians monotone in the randomness”, Oberwolfach Rep. 4:1 (2007), 378–381.
• I. Veselić, Existence and regularity properties of the integrated density of states of random Schrödinger operators, Lecture Notes in Mathematics 1917, Springer, 2008.
• W. P. Ziemer, Weakly differentiable functions: Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120, Springer, 1989.
• E. Zuazua, “Controllability and observability of partial differential equations: some results and open problems”, pp. 527–621 in Handbook of differential equations: evolutionary equations, III, edited by C. M. Dafermos and E. Feireisl, North-Holland, Amsterdam, 2007.