We prove quantitative unique continuation estimates for relatively dense sets and spectral subspaces associated to small energies of Schrödinger operators on Riemannian manifolds with Ricci curvature bounded below. The upper bound for the energy range and the constant appearing in the estimate are given in terms of the lower bound of the Ricci curvature and the parameters of the relatively dense set.

1.
V.
Logvinenko
and
J.
Sereda
, “
Equivalent norms in spaces of entire functions of exponential type
,”
Teor. Funkts., Funkts. Anal. Prilozh.
20
,
102
111
(
1974
).
2.
B.
Panejah
, “
Some theorems of Paley–Wiener type
,”
Sov. Math. Dokl.
2
,
533
536
(
1961
).
3.
V.
Kacnel’son
, “
Equivalent norms in spaces of entire functions
,”
Math. USSR-Sb.
21
,
33
55
(
1973
), translation of the 1973 Russian original.
4.
O.
Kovrijkine
, “
Some estimates of Fourier transforms
,” Ph.D. thesis, (
California Institute of Technology
,
2000
).
5.
O.
Kovrijkine
, “
Some results related to the Logvinenko–Sereda theorem
,”
Proc. Am. Math. Soc.
129
,
3037
3047
(
2001
).
6.
G.
Lebeau
and
L.
Robbiano
, “
Contrôle exact de l’équation de la chaleur
,”
Commun. Partial Differ. Equ.
20
,
335
356
(
1995
).
7.
G.
Lebeau
and
E.
Zuazua
, “
Null-controllability of a system of linear thermoelasticity
,”
Arch. Ration. Mech. Anal.
141
,
297
329
(
1998
).
8.
D.
Jerison
and
G.
Lebeau
, “
Nodal sets of sums of eigenfunctions
,” in
Harmonic Analysis and Partial Differential Equations
,
Chicago Lectures in Mathematics
, edited by
M.
Christ
,
C.
Kenig
, and
C.
Sadosky
(
The University of Chicago Press
,
Chicago
,
1999
), pp.
223
239
.
9.
C.
Rojas-Molina
and
I.
Veselić
, “
Scale-free unique continuation estimates and applications to random Schrödinger operators
,”
Commun. Math. Phys.
320
,
245
274
(
2013
).
10.
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
,
1229
1246
(
2013
).
11.
A.
Klein
and
C.
Tsang
, “
Quantitative unique continuation principle for Schrödinger operators with singular potentials
,”
Proc. Am. Math. Soc.
144
,
665
679
(
2015
).
12.
I.
Nakić
,
M.
Täufer
,
M.
Tautenhahn
, and
I.
Veselić
, “
Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators
,”
Anal. PDE
11
,
1049
1081
(
2018
).
13.
I.
Nakić
,
M.
Täufer
,
M.
Tautenhahn
, and
I.
Veselić
, “
Unique continuation and lifting of spectral band edges of Schrödinger operators on unbounded domains
,”
J. Spectr. Theory
10
,
843
885
(
2020
), with Appendix A by A. Seelmann.
14.
A.
Dicke
,
C.
Rose
,
A.
Seelmann
, and
M.
Tautenhahn
, “
Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
,”
J. Differ. Equ.
369
,
405
423
(
2023
).
15.
M.
Egidi
, “
On null-controllability of the heat equation on infinite strips and control cost estimate
,”
Math. Nachr.
294
,
843
861
(
2021
).
16.
M.
Egidi
and
A.
Seelmann
, “
An abstract Logvinenko–Sereda type theorem for spectral subspaces
,”
J. Math. Anal. Appl.
500
,
125149
(
2021
).
17.
P.
Stollmann
and
G.
Stolz
, “
Lower bounds for Dirichlet Laplacians and uncertainty principles
,”
J. Eur. Math. Soc.
23
,
2337
2360
(
2021
).
18.
J.
Bourgain
and
C.
Kenig
, “
On localization in the continuous Anderson–Bernoulli model in higher dimension
,”
Invent. Math.
161
,
389
426
(
2005
).
19.
J.
Combes
,
P.
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
,
469
498
(
2007
).
20.
F.
Germinet
and
A.
Klein
, “
Bootstrap multiscale analysis and localization in random media
,”
Commun. Math. Phys.
222
,
415
448
(
2001
).
21.
A.
Klein
and
F.
Germinet
, “
A comprehensive proof of localization for continuous Anderson models with singular random potentials
,”
J. Eur. Math. Soc.
15
,
53
143
(
2012
).
22.
J.
Bourgain
and
A.
Klein
, “
Bounds on the density of states for Schrödinger operators
,”
Invent. Math.
194
,
41
72
(
2013
).
23.
P.
Müller
and
C.
Rojas-Molina
, “
Localisation for Delone operators via Bernoulli randomisation
,”
J.Anal. Math.
147
,
297
331
(
2022
).
24.
A.
Seelmann
and
M.
Täufer
, “
Band edge localization beyond regular Floquet eigenvalues
,”
Ann. Henri Poincaré
21
,
2151
2166
(
2020
).
25.
H.
Donnelly
and
C.
Fefferman
, “
Nodal sets of eigenfunctions on Reimannian manifolds
,”
Invent. Math.
93
,
161
183
(
1988
).
26.
H.
Donnelly
and
C.
Fefferman
, “
Nodal sets of eigenfunctions: Riemannian manifolds with boundary
,” in
Analysis, Et Cetera
, edited by
P.
Rabinowitz
and
E.
Zehnder
(
Academic Press
,
Boston
,
1990
), pp.
251
262
.
27.
H.
Donnelly
and
C.
Fefferman
, “
Growth and geometry of eigenfunctions of the Laplacian
,” in
Analysis and Partial Differential Equations
,
Lecture Notes in Pure and Applied Mathematics
, edited by
C.
Sadosky
(
CRC Press
,
New York
,
1990
), Vol.
122
, pp.
635
655
.
28.
M.
Egidi
and
I.
Veselić
, “
Scale-free unique continuation estimates and Logvinenko-Sereda theorems on the torus
,”
Ann. Henri Poincaré
21
,
3757
3790
(
2020
).
29.
G.
Lebeau
and
I.
Moyano
, “
Spectral inequalities for the Schrödinger operator
,” arXiv:1901.03513 [math.AP] (
2019
).
30.
N.
Burq
and
I.
Moyano
, “
Propagation of smallness and spectral estimates
,” arXiv:2109.06654 (
2021
).
31.
N.
Burq
and
I.
Moyano
, “
Propagation of smallness and control for heat equations
,”
J. Eur. Math. Soc.
25
,
1349
1377
(
2022
).
32.
L.
Miller
, “
Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds
,”
Math. Res. Lett.
12
,
37
47
(
2005
).
33.
A.
Dicke
and
I.
Veselić
, “
Spherical Logvinenko-Sereda-Kovrijkine type inequality and null-controllability of the heat equation on the sphere
,” arXiv:2207.01369 (
2022
).
34.
S.
Gallot
,
D.
Hulin
, and
J.
Lafontaine
,
Riemannian Geometry
(
Springer
,
Berlin
,
1987
).
35.
T.
Kato
, in
Perturbation Theory for Linear Operators
,
Classics in Mathematics
, reprint of the 1980 ed. (
Springer
,
Berlin
,
1995
).
36.
P.
Stollmann
and
J.
Voigt
, “
Perturbation of Dirichlet forms by measures
,”
Potential Anal.
5
,
109
138
(
1996
).
37.
H.
Donnelly
, “
On the essential spectrum of a complete Riemannian manifold
,”
Topology
20
,
1
14
(
1981
).
38.
D.
Lenz
,
P.
Stollmann
, and
G.
Stolz
, “
An uncertainty principle and lower bounds for the Dirichlet Laplacian on graphs
,”
J. Spectr. Theory
10
,
115
145
(
2019
).
39.
A. B.
de Monvel
,
D.
Lenz
, and
P.
Stollmann
, “
An uncertainty principle, Wegner estimates and localization near fluctuation boundaries
,”
Math. Z.
269
,
663
670
(
2011
).
40.
I.
McGillivray
,
P.
Stollmann
, and
G.
Stolz
, “
Absence of absolutely continuous spectra for multidimensional Schrödinger operators with high barriers
,”
Bull. London Math. Soc.
27
,
162
168
(
1995
).
41.
W.
Hebisch
and
L.
Saloff-Coste
, “
On the relation between elliptic and parabolic Harnack inequalities
,”
Ann. Inst. Fourier
51
,
1437
1481
(
2001
).
42.
M.
Tautenhahn
and
I.
Veselić
, “
Sampling and equidistribution theorems for elliptic second order operators, lifting of eigenvalues, and applications
,”
J. Differ. Equ.
268
,
7669
7714
(
2020
).
43.
J.
Cheeger
,
M.
Gromov
, and
M.
Taylor
, “
Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
,”
J. Differ. Geom.
17
,
15
53
(
1982
).
44.
E.
Hebey
,
Sobolev Spaces on Riemannian Manifolds
,
Lecture Notes in Mathematics
(
Springer
,
Berlin
,
1996
), Vol.
1635
.
45.
S.
Yau
, “
On the heat kernel of a complete Riemannian manifold
,”
J. Math. Pures Appl.
57
(
9
),
191
201
(
1978
).
46.
A.
Grigor’yan
,
Heat Kernel and Analysis on Manifolds
,
Studies in Advanced Mathematics
(
American Mathematical Society; International Press
,
Providence
,
2009
), Vol.
47
.
47.
K.
Sturm
, “
Heat kernel bounds on manifolds
,”
Math. Ann.
292
,
149
162
(
1992
).
You do not currently have access to this content.