September 2017, 16(5): 1719-1730. doi: 10.3934/cpaa.2017083

Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators

1. 

Technische Universität Dortmund, Fakultät für Mathematik, 44227 Dortmund, Germany

2. 

Technische Universität Chemnitz, Fakultät für Mathematik, 09126 Chemnitz, Germany

* Corresponding author

Received  September 2016 Revised  February 2017 Published  May 2017

We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrödinger operators. Let
$Λ_L = (-L/2, L/2)^d$
and
$H_L = -Δ_L + V_L$
be a Schrödinger operator on
$L^2 (Λ_L)$
with a bounded potential
$V_L : Λ_L \to \mathbb{R}^d$
and Dirichlet, Neumann, or periodic boundary conditions. Our main result is of the type
$ \int_{Λ_L} \lvert φ \rvert^2 \leq C_{\rm {sfuc}} \int_{W_δ (L)} \lvert φ \rvert^2,$
where
$φ$
is an infinite complex linear combination of eigenfunctions of
$H_L$
with exponentially decaying coefficients,
$W_δ (L)$
is some union of equidistributed
$δ$
-balls in
$Λ_L$
and
$C_{{\rm {sfuc}}} > 0$
an
$L$
-independent constant. The exponential decay condition on
$φ$
can alternatively be formulated as an exponential decay condition of the map
$λ \mapsto \lVert χ_{[λ, ∞)} (H_L) φ \rVert^2$
. The novelty is that at the same time we allow the function
$φ$
to be from an infinite dimensional spectral subspace and keep an explicit control over the constant
$C_{{\rm {sfuc}}}$
in terms of the parameters. Moreover, we show that a similar result cannot hold under a polynomial decay condition.
Citation: Matthias Täufer, Martin Tautenhahn. Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1719-1730. doi: 10.3934/cpaa.2017083
References:
[1]

L. Bakri, Carleman estimates for the Schrödinger operator. Applications to quantitative uniqueness, Commun. Part. Diff. Eq., 38 (2013), 69-91. doi: 10.1080/03605302.2012.736912.

[2]

D. I. Borisov, M. Tautenhahn and I. Veselić, Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations, preprint, arXiv: 1512.06347.,

[3]

J. Bourgain and C. E. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math., 161 (2005), 389-426. doi: 10.1007/s00222-004-0435-7.

[4]

J. Bourgain and A. Klein, Bounds on the density of states for Schrödinger operators, Invent. Math., 194 (2013), 41-72. doi: 10.1007/s00222-012-0440-1.

[5]

T. Carleman, Sur un probléme d'unicité pour les systémes d'équations aux dérivées partielles á deux variables indépendantes, Ark. Mat. Astron. Fysik, 26B (1939), 1-9.

[6]

H. Donnelly and C. Fefferman, Nodal sets for eigenfunctions on Riemannian manifolds, Invent. math., 93 (1988), 161-183. doi: 10.1007/BF01393691.

[7] L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅰ: Distribution Theory and Fourier Analysis, 2 edition, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.
[8]

D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in Harmonic Analysis and Partial Differential Equations (eds. M. Christ, C. E. Kenig and C. Sadosky), Chicago Lecture notes in Mathematics, The University of Chicago Press, (1999), 223-239.,

[9]

A. Klein, Unique continuation principle for spectral projections of Schrödinger operators and optimal Wegner estimates for non-ergodic random Schrödinger operators, Commun. Math. Phys., 323 (2013), 1229-1246. doi: 10.1007/s00220-013-1795-x.

[10]

I. Kukavica, Quantitative uniqueness for second-order elliptic operators, Duke Math. J., 91 (1998), 225-240. doi: 10.1215/S0012-7094-98-09111-6.

[11]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Contr. Optim. Ca., 18 (2012), 712-747. doi: 10.1051/cocv/2011168.

[12]

V. Z. Meshkov, On the possible rate of decay at infinity of solutions of second order partial differential equations, Math. USSR Sb., 72 (1992), 343-361.

[13]

I. NakićM. TäuferM. Tautenhahn and I. Veselić, Scale-free uncertainty principles and Wegner estimates for random breather potentials, C. R. Math., 353 (2015), 919-923. doi: 10.1016/j.crma.2015.08.005.

[14]

I. Nakić, M. Täufer, M. Tautenhahn and I. Veselić, Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators, preprint, arXiv: 1609.01953.,

[15]

C. Rojas-Molina and I. Veselić, Scale-free unique continuation estimates and application to random Schrödinger operators, Commun. Math. Phys., 320 (2013), 245-274. doi: 10.1007/s00220-013-1683-4.

[16]

M. Täufer, M. Tautenhahn and I. Veselić, Harmonic analysis and random Schrödinger operators, in Spectral Theory and Mathematical Physics (eds. M. Mantoiu, G. Raikov and R. Tiedra de Aldecoa), Operator Theory: Advances and Applications, vol. 254, Birkhäuser, Basel, (2016), 223-255., doi: 10.1007/978-3-319-29992-1_11.

show all references

References:
[1]

L. Bakri, Carleman estimates for the Schrödinger operator. Applications to quantitative uniqueness, Commun. Part. Diff. Eq., 38 (2013), 69-91. doi: 10.1080/03605302.2012.736912.

[2]

D. I. Borisov, M. Tautenhahn and I. Veselić, Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations, preprint, arXiv: 1512.06347.,

[3]

J. Bourgain and C. E. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math., 161 (2005), 389-426. doi: 10.1007/s00222-004-0435-7.

[4]

J. Bourgain and A. Klein, Bounds on the density of states for Schrödinger operators, Invent. Math., 194 (2013), 41-72. doi: 10.1007/s00222-012-0440-1.

[5]

T. Carleman, Sur un probléme d'unicité pour les systémes d'équations aux dérivées partielles á deux variables indépendantes, Ark. Mat. Astron. Fysik, 26B (1939), 1-9.

[6]

H. Donnelly and C. Fefferman, Nodal sets for eigenfunctions on Riemannian manifolds, Invent. math., 93 (1988), 161-183. doi: 10.1007/BF01393691.

[7] L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅰ: Distribution Theory and Fourier Analysis, 2 edition, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.
[8]

D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in Harmonic Analysis and Partial Differential Equations (eds. M. Christ, C. E. Kenig and C. Sadosky), Chicago Lecture notes in Mathematics, The University of Chicago Press, (1999), 223-239.,

[9]

A. Klein, Unique continuation principle for spectral projections of Schrödinger operators and optimal Wegner estimates for non-ergodic random Schrödinger operators, Commun. Math. Phys., 323 (2013), 1229-1246. doi: 10.1007/s00220-013-1795-x.

[10]

I. Kukavica, Quantitative uniqueness for second-order elliptic operators, Duke Math. J., 91 (1998), 225-240. doi: 10.1215/S0012-7094-98-09111-6.

[11]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Contr. Optim. Ca., 18 (2012), 712-747. doi: 10.1051/cocv/2011168.

[12]

V. Z. Meshkov, On the possible rate of decay at infinity of solutions of second order partial differential equations, Math. USSR Sb., 72 (1992), 343-361.

[13]

I. NakićM. TäuferM. Tautenhahn and I. Veselić, Scale-free uncertainty principles and Wegner estimates for random breather potentials, C. R. Math., 353 (2015), 919-923. doi: 10.1016/j.crma.2015.08.005.

[14]

I. Nakić, M. Täufer, M. Tautenhahn and I. Veselić, Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators, preprint, arXiv: 1609.01953.,

[15]

C. Rojas-Molina and I. Veselić, Scale-free unique continuation estimates and application to random Schrödinger operators, Commun. Math. Phys., 320 (2013), 245-274. doi: 10.1007/s00220-013-1683-4.

[16]

M. Täufer, M. Tautenhahn and I. Veselić, Harmonic analysis and random Schrödinger operators, in Spectral Theory and Mathematical Physics (eds. M. Mantoiu, G. Raikov and R. Tiedra de Aldecoa), Operator Theory: Advances and Applications, vol. 254, Birkhäuser, Basel, (2016), 223-255., doi: 10.1007/978-3-319-29992-1_11.

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