Covariant Schrödinger Semigroups on Riemannian Manifolds

246 Pages·2017·2.459 MB·English

by  Batu Güneysu (auth.)

Operator Theory
Advances and Applications
264
Batu Güneysu
Covariant
Schrödinger
Semigroups
on Riemannian
Manifolds

Operator Theory: Advances and Applications
Volume 264
Founded in 1979 by Israel Gohberg

Editors:
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Linear Operators and Linear Systems
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Advances in Partial Differential Equations
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Batu Güneysu
Covariant Schrödinger
Semigroups on Riemannian
Manifolds
Batu Güneysu
Mathematisches Institut
Humboldt-Universität zu Berlin
Berlin, Germany
ISSN 0255-0156 ISSN 2296-4878 (electronic)
Operator Theory: Advances and Applications
ISBN 978-3-319-68902-9 ISBN 978-3-319-68903-6 (eBook)
https://doi.org/10.1007/978-3-319-68903-6

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Mathematics Subject Classification (2010): 31C12, 58J35, 58J65, 35P99

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♥
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
I Sobolev Spaces on Vector Bundles . . . . . . . . . . . . . . . . . . . . 1
I.1 Differential operators with smooth coefficients and
weak derivatives on vector bundles . . . . . . . . . . . . . . . . 1
I.2 Some remarks on covariantderivatives . . . . . . . . . . . . . . 10
I.3 Generalized Sobolev spaces and Meyers–Serrintheorems . . . . 12
II Smooth Heat Kernels on Vector Bundles . . . . . . . . . . . . . . . . 21
III Basic Differential Operators in Riemannian Manifolds . . . . . . . . . 27
III.1 Preliminaries from Riemannian geometry . . . . . . . . . . . . 27
III.2 Riemannian Sobolev spaces and Meyers–Serrintheorems. . . . 36
III.3 The Friedrichs realization of ∇†∇/2 . . . . . . . . . . . . . . . 47
IV Some Specific Results for the Minimal Heat Kernel . . . . . . . . . . 49
V Wiener Measure and Brownian Motion on Riemannian Manifolds . . 71
V.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
V.2 Path spaces as measurable spaces. . . . . . . . . . . . . . . . . 72
V.3 The Wiener measure on Riemannian manifolds . . . . . . . . . 76
VI Contractive Dynkin and Kato Potentials. . . . . . . . . . . . . . . . . 85
VI.1 Generally valid results . . . . . . . . . . . . . . . . . . . . . . . 85
VI.2 Specific results under some control on the geometry . . . . . . 99
VII Foundations of Covariant Schro¨dinger Semigroups . . . . . . . . . . . 101
VII.1 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . 101
VII.2 Scalar Schr¨odinger semigroups and the Feynman–Kac
(FK) formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
VII.3 Kato–Simon inequality. . . . . . . . . . . . . . . . . . . . . . . 111
VIII Compactness of V(H∇+1)−1 . . . . . . . . . . . . . . . . . . . . . 119
IX Lq-properties of Covariant Schr¨odinger Semigroups . . . . . . . . . . 127
vii
viii Contents
X Continuity Properties of Covariant Schro¨dinger Semigroups . . . . . 135
XI Integral Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
XII Essential Self-adjointness of Covariant Schr¨odinger Operators . . . . 153
XIII Smooth Compactly Supported Sections as Form Core . . . . . . . . . 163
XIV Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
XIV.1 Hamilton operators corresponding to magnetic fields . . . . . . 167
XIV.2 Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
XIV.3 Nonrelativistic hydrogen-type stability problems
on Riemannian 3-folds . . . . . . . . . . . . . . . . . . . . . . . 175
XIV.4 Riemannian total variation . . . . . . . . . . . . . . . . . . . . 182
XIV.5 C-positivity preservation . . . . . . . . . . . . . . . . . . . . . 192
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A Smooth Manifolds and Vector Bundles . . . . . . . . . . . . . . . . . 197
B Facts About Self-adjoint Operators. . . . . . . . . . . . . . . . . . . . 207
B.1 Self-adjoint operators and the spectral calculus . . . . . . . . . 207
B.1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . 207
B.1.2 Spectrum and resolvent set . . . . . . . . . . . . . . . . 208
B.1.3 A result from perturbation theory . . . . . . . . . . . . 209
B.1.4 Spectral calculus and the spectral theorem. . . . . . . . 210
B.2 Sesquilinear forms in Hilbert spaces . . . . . . . . . . . . . . . 214
B.3 Strong convergence results for semigroups . . . . . . . . . . . . 218
B.3.1 Semigroup convergence from convergence on a core . . . 218
B.3.2 Monotone convergence of sesquilinear forms . . . . . . . 218
B.4 Abstract Kato–Simon inequalities . . . . . . . . . . . . . . . . 219
B.5 Weak-to-strong differentiability theorem . . . . . . . . . . . . . 220
B.6 Trotter’s product formula . . . . . . . . . . . . . . . . . . . . . 220
C Some Measure Theoretic Results . . . . . . . . . . . . . . . . . . . . . 221
C.1 Monotone class theorem . . . . . . . . . . . . . . . . . . . . . . 222
C.2 A generalized convergence result for integrals . . . . . . . . . . 223
C.3 Regular conditional expectations . . . . . . . . . . . . . . . . . 223
C.4 Riesz–Thorin interpolation . . . . . . . . . . . . . . . . . . . . 225
C.5 Pitt’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Introduction
Many problems in the analysis on Riemannian manifolds naturally lead to the
studyofoperatorsemigroupshavingtheform(e−tHV∇)t≥0,wherethegeneratorHV∇
isacovariantSchr¨odingeroperator.Inotherwords,H∇isaself-adjointrealization
V
of ∇†∇+V in the underlying Hilbert space of square integrable sections, where
∇ is a metric covariant derivative on a metric vector bundle over a Riemannian
manifold, ∇† is its formal adjoint, and V is a smooth self-adjoint endomorphism.
For example, if we assume that V is bounded from below by a constant, then a
canonical choice for H∇ is given by the Friedrichs realization of ∇†∇+V.
V
Besides the semigroups that are induced by the scalar Laplace–Beltramiop-
erator−Δandtheperturbationof−Δbypotentials,probablythemostprominent
examples of such covariant Schro¨dinger semigroups are provided by “heat kernel
proofs”oftheAtiyah–Singerindextheoremoncompactmanifolds[16].Inthissit-
uation,suchasemigrouparisesnaturallythroughaBochner–Weitzenb¨ockformula
D2 =∇†∇+V, with D the underlying geometric Dirac operator. More precisely,
the McKean-Singer formula states that the Z -graded index of D is equal to the
2
Z2-graded trace of e−tHV∇ for all t>0, and so one can take t →0+ to obtain (in
fact a stronger local version of) the Atiyah–Singer index theorem.
In this work, however, we are particularly interested in the analysis on non-
compact Riemannian manifolds. In the context of geometric problems on such
manifolds,the abovecovariantSchro¨dingersemigroupsoftenappearinadisguise:
Namely, one is often interested in estimating the gradient detΔ of the underlying
“heat semigroup”. Noticing now that, at least formally, one has detΔ = etΔ(1)d
withΔ(1) the Hodge-Laplacianactingon1-forms,oneis backinthe abovecovari-
ant Schro¨dinger semigroup case, since by another Bocher–Weitzenbo¨ck formula
the operator Δ(1) is again a covariant Schr¨odinger operator. A typical situation
of this type appears in connection with heat kernel characterizations of the total
variation: Namely, being motivated by E. de Giorgi’s Euclidean result [38], one
would like to establish the equality of the (total) variationVa(cid:2)r(f) o(cid:2)f a function f
(cid:2) (cid:2)
on a noncompact Riemannian manifold to the limit limt→0+ detΔ L1.
In the above geometric situations, the potential terms of the underlying co-
variantSchro¨dingeroperatorsare smooth. One shouldnotice that in the noncom-
pact case many technical difficulties arise due to the behaviour of the potentials
ix
x Introduction
at ∞. On the other hand, it is well known that in the standard representation
of quantum mechanics one has to deal with Schro¨dinger operators in R3 whose
potential terms typically have local 1/|x|-type Coulomb singularities.
The aim of this work is to establish the foundations of a general theory of
covariant Schro¨dinger semigroups on noncompact Riemannian manifolds whichis
flexible enough to deal with all above situations simultaneously.
To this end, we introduce the concept of covariant Schro¨dinger bundles:
These are data of the form (E,∇,V)→M, where
• M is a smooth Riemannian m-manifold
• E →M is a smooth complex metric vector bundle with a finite rank
• ∇ is a smooth metric covariant derivative on E →M
• V : M → End(E) is Borel measurable with V(x) : E → E a linear self-
x x
adjoint map for every x. We will also refer to such an endomorphism V as a
potential on E →M.
As usual, E → M and the corresponding metrics determine the complex
Hilbert space ΓL2(M,E) of square integrable sections. Using sesquilinear-form-
techniques, one finds that under mild additional assumptions on V, such a Schr¨o-
dingerbundle canonicallyinduces acovariantSchro¨dingeroperatorinΓL2(M,E),
which is semibounded from below. To explain this operator construction, let us
denotebyp(t,x,y),t>0,x,y ∈M,theminimalnonnegativeheatkernelonM.In
other words, for every fixed y ∈M the function (t,x)(cid:7)→p(t,x,y) is the pointwise
minimal nonnegative solution of the heat equation
(∂/∂t)p(t,x,y)=(1/2)Δ p(t,x,y), lim p(t,•,y)=δ .
x y
t→0+
Then the contractive Dynkin class D(M) of the Riemannian manifold M is given
by all Borel functions w on M such that
(cid:3) (cid:3)
t
lim sup p(s,x,y)|w(y)|dμ(y)ds <1, (1)
t→0+x∈M 0 M
with μ the Riemannianvolumemeasure.Standardpropertiesofthe minimal non-
negative heat kernel imply L∞(M)⊂D(M). By definition, the Kato class K(M)
iscontainedinD(M)aswell,withK(M)beingdefinedasthespaceofallw’sthat
satisfy limt→0+···=0 in (1), instead of only <1. In view of L∞(M)⊂D(M), in
the sequel a contractive Dynkin condition should always be understood as a very
weak (and operator theoretic) form of boundedness.
With these definitions, we call a potential V as above contractively Dynkin
decomposable, if it can be decomposed V =V+−V− into potentials with V± ≥0
1, such that V+ is L1loc, and such that the fiberwise taken operator norm of V−
1Thatis,forallx∈M theeigenvalues ofV±(x):Ex→Ex arenonnegative.