Herbert amann, wolfgang arendt, matthias hieber, frank neubrander, serge nicaise, joachim below functional analysis and evolution equations the gunter lumer volume birkhäuser (2008)

www.pdfgrip.com


Functional Analysis
and Evolution Equations
The Günter Lumer Volume
Herbert Amann
Wolfgang Arendt
Matthias Hieber
Frank Neubrander
Serge Nicaise
Joachim von Below
Editors

Birkhäuser
Basel · Boston · Berlin

www.pdfgrip.com


Editors:
Frank M. Neubrander
Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803, USA
e-mail:

Herbert Amann
Institut für Mathematik
Universität Zürich

Winterthurerstrasse 190
8057 Zürich, Switzerland
e-mail:

Serge Nicaise
Université de Valenciennes et du Hainaut
Cambrésis
Le Mont Houy
59313 Valenciennes Cedex 9, France
e-mail:

Wolfgang Arendt
Abteilung Angewandte Analysis
Universität Ulm
89069 Ulm, Germany
e-mail:
Matthias Hieber
Technische Universität Darmstadt
Fachbereich Mathematik
Schloßgartenstr. 7
64289 Darmstadt, Germany
e-mail:

Joachim von Below
Université du Littoral-Côte d’Opale
Laboratoire de Mathématiques Pures et
Appliquées – LMPA “Joseph Liouville”
50, rue F. Buisson
62228 Calais Cedex, France
e-mail:

2000 Mathematical Subject Classification 35, 39B, 45G, 45K, 46F, 47D, 53C, 65M, 65N, 70G,
76D, 80A, 91B, 92D, 93D, 93E
Library of Congress Control Number: 2007933910
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <>.

ISBN 3-7643-7793-9 Birkhäuser Verlag, Basel – Boston – Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, re-use of illustrations,
recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data
banks. For any kind of use permission of the copyright owner must be obtained.
© 2008 Birkhäuser Verlag AG
Basel · Boston · Berlin
P.O. Box 133, CH-4010 Basel, Switzerland
Part of Springer Science+Business Media
Printed on acid-free paper produced of chlorine-free pulp. TCF ∞
Printed in Germany
ISBN 978-3-7643-7793-9

e-ISBN 978-3-7643-7794-6

9 8 7 6 5 4 3 2 1

www.birkhauser.ch

www.pdfgrip.com

Contents
Portrait of Gă
unter Lumer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

Life and Work of Gă
unter Lumer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

H. Kă
onig
In Remembrance of Gă
unter Lumer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ixx

F. Ali Mehmeti, R. Haller-Dintelmann and V. R´egnier
Expansions in Generalized Eigenfunctions of the
Weighted Laplacian on Star-shaped Networks . . . . . . . . . . . . . . . . . . . . . . . .

1

F. Andreu, V. Caselles and J.M. Maz´
on
Diffusion Equations with Finite Speed of Propagation . . . . . . . . . . . . . . . .

17

B. Baeumer, M. Kov´
acs and M.M. Meerschaert
Subordinated Multiparameter Groups of Linear Operators:
Properties via the Transference Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

A.V. Balakrishnan
An Integral Equation in AeroElasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

J. von Below and G. Fran¸cois
Eigenvalue Asymptotics Under a Non-dissipative
Eigenvalue Dependent Boundary Condition for
Second-order Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

J.A. Van Casteren
Feynman-Kac Formulas, Backward Stochastic Differential
Equations and Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

R. Chill, V. Keyantuo and M. Warma
Generation of Cosine Families on Lp (0, 1) by Elliptic
Operators with Robin Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . .

113

Ph. Cl´ement and R. Zacher
Global Smooth Solutions to a Fourth-order Quasilinear
Fractional Evolution Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
L. D’Ambrosio and E. Mitidieri
Positivity Property of Solutions of Some Quasilinear
Elliptic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

www.pdfgrip.com


vi

Contents

W. Desch and S.-O. Londen
On a Stochastic Parabolic Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . 157
P. Deuring
Resolvent Estimates for a Perturbed Oseen Problem . . . . . . . . . . . . . . . . . 171
O. Diekmann and M. Gyllenberg
Abstract Delay Equations Inspired by Population Dynamics . . . . . . . . .

187

T. Eisner and B. Farkas
Weak Stability for Orbits of C0 -semigroups on Banach Spaces . . . . . . . . 201
A.F.M. ter Elst and D.W. Robinson
Contraction Semigroups on L∞ (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
J. Escher and Z. Feng
On the Curve Shortening Flow with Triple Junction . . . . . . . . . . . . . . . . .

223

M. Farhloul, R. Korikache and L. Paquet
The Dual Mixed Finite Element Method for the Heat
Diffusion Equation in a Polygonal Domain, I . . . . . . . . . . . . . . . . . . . . . . . . . 239
R. Farwig, H. Kozono and H. Sohr
Maximal Regularity of the Stokes Operator in General
Unbounded Domains of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
H.O. Fattorini
Linear Control Systems in Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 273
ˇ Neˇcasov´
E. Feireisl and S.
a
On the Motion of Several Rigid Bodies in a Viscous
Multipolar Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
M. Geissert and Y. Giga
On the Stokes Resolvent Equations in Locally Uniform
Lp Spaces in Exterior Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

307

M. Giuli, F. Gozzi, R. Monte and V. Vespri
Generation of Analytic Semigroups and Domain
Characterization for Degenerate Elliptic Operators with
Unbounded Coefficients Arising in Financial Mathematics.
Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315

P. Guidotti
Numerical Approximation of Generalized Functions:
Aliasing, the Gibbs Phenomenon and a Numerical
Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
K. Gustafson and E. Ash
No Radial Symmetries in the Arrhenius–Semenov Thermal
Explosion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
V. Keyantuo and C. Lizama
Mild Well-posedness of Abstract Differential Equations . . . . . . . . . . . . . .

www.pdfgrip.com

371


Contents

vii

H. Koch and I. Lasiecka
Backward Uniqueness in Linear Thermoelasticity with Time
and Space Variable Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
H. Kă
onig
Measure and Integral: New Foundations after One Hundred Years . . . . 405
P.Ch. Kunstmann
Post-Widder Inversion for Laplace Transforms of Hyperfunctions . . . . . 423
L. Lorenzi
On a Class of Elliptic Operators with Unbounded Time- and
Space-dependent Coefficients in RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

R.H. Martin, Jr., T. Matsumoto, S. Oharu and N. Tanaka
Time-dependent Nonlinear Perturbations
of Analytic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
D. Mugnolo
A Variational Approach to Strongly Damped Wave Equations . . . . . . . . 503
S. Nicaise and C. Pignotti
Exponential and Polynomial Stability Estimates for the
Wave Equation and Maxwell’s System with Memory
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
J. Pră
uss and G. Simonett
Maximal Regularity for Degenerate Evolution Equations
with an Exponential Weight Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
J. Pră
uss, S. Sperlich and M. Wilke
An Analysis of Asian options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

547

W.M. Ruess
Linearized Stability and Regularity for Nonlinear
Age-dependent Population Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
B. Scarpellini
Space Almost Periodic Solutions of Reaction Diffusion Equations . . . .

577

Y. Shibata
On the Oseen Semigroup with Rotating Effect . . . . . . . . . . . . . . . . . . . . . . . 595
R. Triggiani

odinger Equation
Exact Controllability in L2 (Ω) of the Schră
in a Riemannian Manifold with L2 (1 )-Neumann
Boundary Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

www.pdfgrip.com


Gă
unter Lumer (19292005)

www.pdfgrip.com


Life and Work of Gă
unter Lumer
Gă
unter Lumer was born in Frankfurt, Germany in 1929. With Nazism on the
rise, the Lumer family left Germany in 1933 and settled in France, where Gă
unter
received his early education. Then, in 1941, the Lumer family ed once again, this
time to Uruguay, where Gă
unter would become a citizen.
Possessing what would be a life-long passion for mathematics, Gă
unter graduated in 1957 with a degree in electrical engineering from the University of Montevideo. In fact, while at Montevideo, he was in the research group of Paul Halmos,
who would later dedicate a page to Gă
unter in his book I Want to be a Mathematician: an Automathography. Gă
unters rst paper “Square roots of operators,”
a joint work with P. Halmos and J.J. Schă

aer, appeared in 1953 in the Proceedings
of the American Mathematical Society.
In 1956, Gă
unter received a Guggenheim fellowship to study at the University
of Chicago. There he received his Ph.D. in Mathematics in 1959; his dissertation
was entitled Numerical Range and States and was written under the supervision
of Irving Kaplansky, thus earning himself a place among a long lineage of mathematicians connected to Kaplansky.
Following Chicago, Gă
unter Lumer held positions at UCLA (19591960), Stanford University (1960–1961), University of Washington (1961–1974), University of
Mons-Hainaut (1973–2005), and the International Solvay Institutes for Physics
and Chemistry in Brussels (19992005).
Gă
unter Lumer was a creative and prolic mathematician whose works have
great influence on the research community in mathematical analysis and evolution
equations. His scientific activities greatly contributed to the standing of the Belgian Universities in general and the University of Mons-Hainaut in particular. In
1976, supported by the Belgium National Science Foundation, Gă
unter founded a
contact group with the goal of organizing research and exchange meetings in the
fields of Partial Differential Equations and Functional Analysis. From the 1990s on,
building on the success of this group, Gă
unter became a driving force and leading
contributor to several large-scale projects sponsored by the European Community. The resulting conferences on Evolution Equations created a lasting network
supporting international research collaboration. These activities, combined with
Gă
unters relentless energy and love for mathematics, were at the origin of the
breath-taking development of the field of evolution equations and the theory of
operator semigroups after the pioneering book of Hille and Phillips from 1957.

www.pdfgrip.com

x

Life and Work of Gă
unter Lumer

In particular, between 1992 and 1997 he co-organized the North West European
Analysis Seminar that was held in 1992 at Saint Amand les Eaux (France), in
1993 at Schloss Dagstuhl (Germany), in 1994 at Noordwijkerhout (The Netherlands), in 1995 at Lyon (France), in 1996 at Glasgow (United Kingdom) and in
1997 at Blaubeuren (Germany). Those seminars covered a broad range of topics
in analysis and were a reflection of the true spirit of Gă
unter Lumer, who always
enjoyed bringing together and working with a wide range of mathematicians and
scientists.
Although Gă
unter Lumer’s professional focus was on functional analysis, partial differential equations, and evolution equations, he nourished a broad interest
for almost all areas of mathematics and for science in general. He published more
than one hundred papers and edited many books. Probably his best known result is the celebrated Lumer-Phillips theorem, which gives necessary and sufficient
conditions on an operator to generate a strongly continuous semigroup of contractions on a general Banach space. This result, published in the Pacific Journal of
Mathematics in 1961, is a key contribution to the theory of operator semigroups.
Gă
unter Lumer deeply loved mathematics. He considered his work as the most
precious thing he could leave to future generations. He was an independent and
original person, never influenced by fashion or convention. He used to say, “If a
crowd of a thousand unanimously condemns someone, then he must be innocent.
For it is unlikely for a thousand people to honestly agree on the same thing.”
With Gă
unter Lumer we miss an inspiring teacher, a mentor and friend of
a generation of researchers, and a leader of our professional community. Gă
unter

Lumer: a mathematician to be honored.
List of Ph.D. students of Gă
unter Lumer
Charles Widger, Multiplicative perturbations of generators of semigroups of operators, U. Washington, 1970
David Neu, Summability of the linear predictor, U. Washington, 1972
Luc Paquet, Sur les ´equations d’´evolution en norme uniforme, U. Mons, 1978
Roger-Marie Dubois, Equations d’´evolution vectorielles, probl`emes mixte et formule de Duhamel, U. Mons, 1981
Serge Nicaise, Diffusion sur les espaces ramifi´es, U. Mons, 1986
Maryse Bourlard, M´ethodes d’´el´ements finis de bord raffin´es pour des probl`emes
aux limites concernant le laplacien et le bilaplacien dans des domaines polygonaux du plan, U. Mons, 1988
List of publications of Gă
unter Lumer
Wilansky, A. and Lumer, G., Advanced Problems and Solutions: Solutions: 4397, Amer.
Math. Monthly 58 (1951), no. 10, 706–708.
Butchart, J.H. and Lumer, G., Advanced Problems and Solutions: Solutions: 4403, Amer.
Math. Monthly 59 (1952), no. 2, 115.

www.pdfgrip.com


Life and Work of Gă
unter Lumer

xi

Grossman, G., Newman, D.J., Blumenthal, L.M., Venkataraman, C.S. and Lumer, G.,
Advanced Problems and Solutions: Problems for Solution: 4488–4492, Amer. Math.
Monthly 59 (1952), no. 5, 332–333.
Lumer, G. and Beesley, E.M., Advanced Problems and Solutions: Solutions: 4492, Amer.
Math. Monthly 60 (1953), no. 8, 557.

Halmos, P.R., Lumer, G. and Schăaer, J.J., Square roots of operators, Proc. Am. Math.
Soc. 4, 142–149 (1953).
Lumer, G., Fine structure and continuity of spectra in Banach algebras, Anais Acad.
Brasil. Ci. 26, 229–233 (1954).
Halmos, P.R. and Lumer, G., Square roots of operators. II, Proc. Am. Math. Soc. 5,
589–595 (1954).
Lumer, G., Sets with connected spherical section (Portuguese), Soc. Paranaense Mat.,
Anu´
ario 2, 12–17 (1955).
Jones, A. and Lumer, G., A note on radical rings (Spanish), Fac. Ing. Agrimensura
Montevideo, Publ. Inst. Mat. Estad. 3, 11–15 (1956).
Lumer, G., Polygons inscriptible in convex curves (Spanish), Rev. Un. Mat. Argentina
17, 97–102 (1956).
Lumer, G., The range of the exponential function, Fac. Ing. Agrimensura Montevideo,
Publ. Inst. Mat. Estad. 3, 53–55 (1957).
Lumer, G., Commutators in Banach algebras (Spanish) Fac. Ing. Agrimensura Montevideo, Publ. Inst. Mat. Estad. 3, 57–63 (1957).
Lumer, G. and Rosenblum, M., Linear operator equations, Proc. Am. Math. Soc. 10,
32–41 (1959).
Lumer, G. and Phillips, R.S., Dissipative operators in a Banach space, Pac. J. Math. 11,
679–698 (1961).
Lumer, G., Semi-inner-product spaces, Trans. Am. Math. Soc. 100, 29–43 (1961).
Lumer, G., Isometries of Orlicz spaces, Bull. Am. Math. Soc. 68, 28–30 (1962).
Lumer, G., Points extrˆemaux associ´es: fronti`eres de Silov et Choquet; principe du minimum (French), C.R. Acad. Sci. Paris 256, 858–861 (1963).
Lumer, G., Points extrˆemaux associ´es; fronti`eres de Silov et Choquet: application aux
cˆ
ones de fonctions semi-continues (French), C.R. Acad. Sci. Paris 256, 1066–1068
(1963).
Lumer, G., On the isometries of reflexive Orlicz spaces, Ann. Inst. Fourier 13, No. 1,
99–109 (1963).
Lumer, G., Analytic functions and Dirichlet problem, Bull. Am. Math. Soc. 70, 98–104

(1964).
Lumer, G., Spectral operators, hermitian operators, and bounded groups, Acta Sci. Math.
25, 75–85 (1964).
Lumer, G., Remarks on n-th roots of operator, Acta Sci. Math. 25, 72–74 (1964).
Lumer, G., Herglotz transformation and H p theory, Bull. Am. Math. Soc. 71, 725–730
(1965).
Lumer, G., H ∞ and the imbedding of the classical H p spaces in arbitrary ones, Function
Algebras, Proc. Int. Symp. Tulane Univ. 1965, 285–286 (1966).

www.pdfgrip.com


xii

Life and Work of Gă
unter Lumer

Lumer, G., The Herglotz transformation and H p theory, Function Algebras, Proc. Int.
Symp. Tulane Univ. 1965, 287–291 (1966).
of, pour le disque unit´e et les
Lumer, G., Classes H et theor`eme de Phragmen-Lindelă
surfaces de Riemann hyperboliques (French), C.R. Acad. Sci. Paris, S´er. A 262,
1164–1166 (1966).
Lumer, G., Int´egrabilite uniforme dans les alg`ebres de fonctions, classes H ∞ et classe de
Hardy universelle (French), C.R. Acad. Sci. Paris, S´er. A 262, 1046–1049 (1966).
Lumer, G., Un th´eor`eme de modification de convergence, valable pour les mesures repr´esentatives arbitraires (French), C.R. Acad. Sci. Paris, S´er. A 266, 416–418 (1968).
Lumer, G., Une th´eorie des espaces de Hardy abstraits valable pour des alg`ebres de
fonctions arbitraires (French), C.R. Acad. Sci. Paris, S´er. A 267, 88–91 (1968).
Lumer, G., Alg`ebres de fonctions et espaces de Hardy (French), Springer-Verlag, BerlinHeidelberg-New York, 80 p. (1968).
Gamelin, T. and Lumer, G., Theory of abstract Hardy spaces and the universal Hardy

class, Adv. Math. 2, 118–174 (1968).
Lumer, G., On Wermer’s maximality theorem, Invent. Math. 8, 236–237 (1969).
Lumer, G., On some results concerning uniform approximation, Summer Gathering Function Algebras 1969, various Publ. Ser. 9, 63–66 (1969).
Lumer, G., On some results concerning uniform approximation, Invent. Math. 9, 246–248
(1970).
Lumer, G., Alg`ebres de fonctions, espaces de Hardy, et fonctions de plusieurs variables
complexes, Alg`ebres de Fonctions, Journ´ees Soc. Math. France 1970, 45–46 (1970).
Lumer, G., Espaces de Hardy en plusieurs variables complexes (French), C.R. Acad. Sci.
Paris, S´er. A 273, 151–154 (1971).
Lumer, G., Bounded groups and a theorem of Gelfand, Rev. Un. Mat. Argentina 25,
239–245 (1971).
Gustafson, K. and Lumer, G., Multiplicative perturbation of semigroup generators, Pac.
J. Math. 41, 731–742 (1972).
Lumer, G., Normes invariantes et caract´erisations des transform´ees de Fourier des mesures
(French), C.R. Acad. Sci. Paris, S´er. A 274, 749–751 (1972).
´
Lumer, G., Etats,
alg`ebres quotients et sous-espaces invariants (French), C.R. Acad. Sci.
Paris, S´er. A 274, 1308–1311 (1972).
Lumer, G., Quelques aspects de la th´eorie des alg`ebres uniformes et des espaces de
Hardy (French), S´eminaire d’analyse moderne. No. 6. Sherbrooke, Qu´ebec, Canada,
D´epartement de Math´ematiques, Universit´e de Sherbrooke, 66 p. (1972).
Lumer, G., Complex methods, and the estimation of operator norms and spectra from
real numerical ranges, J. Funct. Anal. 10, 482–495 (1972).
Lumer, G., Perturbations de g´en´erateurs infinit´esimaux du type “changement de temps”
(French), Ann. Inst. Fourier 23, No. 4, 271–279 (1973).
Lumer, G., Potential-like operators and extensions of Hunt’s theorem for σ-compact
spaces, J. Funct. Anal. 13, 410–416 (1973).
Lumer, G., Bochner’s theorem, states, and the Fourier transforms of measures, Stud.
Math. 46, 135–140 (1973).

www.pdfgrip.com


Life and Work of Gă
unter Lumer

xiii

Lumer, G., Probl`eme de Cauchy pour op´erateurs locaux, et “changement de temps”
(French), Ann. Inst. Fourier 25, No. 3-4, 409–446 (1975).
Lumer, G., Probl`eme de Cauchy avec valeurs au bord continues (French), C.R. Acad.
Sci. Paris, S´er. A 281, 805–807 (1975).
Lumer, G., Probl`eme de Cauchy pour op´erateurs locaux (French), C.R. Acad. Sci. Paris,
S´er. A 281, 763–765 (1975).
Lumer, G., Images num´eriques, principe du maximum g´en´eralis´e, et r´esolvantes (French),
S´eminaire de Th´eorie du Potentiel Paris 1972–74, Lect. Notes Math. 518, 107–119
(1976).
Lumer, G., Probl`eme de Cauchy avec valeurs au bord continues, comportement asymptotique, et applications (French), S´eminaire de Th´eorie du Potentiel Paris, No. 2,
Lect. Notes Math. 563, 193–201 (1976).
Lumer, G., Probl`eme de Cauchy et fonctions surharmoniques (French), S´eminaire de
Th´eorie du Potentiel Paris, No. 2, Lect. Notes Math. 563, 202–218 (1976).
´
Lumer, G., Equations
d’´evolution pour op´erateurs locaux non localement ferm´es (French),
C.R. Acad. Sci. Paris, S´er. A 284, 1361–1363 (1977).
Lumer, G. and Paquet, L., Semi-groupes holomorphes et ´equations d’´evolution (French),
C.R. Acad. Sci. Paris, S´er. A 284, 237–240 (1977).
´
Lumer, G., Equations

d’´evolution en norme uniforme pour op´erateurs elliptiques. R´egularit´e des solutions (French), C.R. Acad. Sci. Paris, S´er. A 284, 1435–1437 (1977).
´
Lumer, G., Equations
d’´evolution en norme uniforme (conditions n´ecessaires et suffisantes de r´esolution et holomorphie) (French), S´emin. Goulaouic-Schwartz 1976–
´
1977, Equat.
d´eriv. part. Anal. fonct., Expos´e No. V, 8 p. (1977).
Lumer, G., Evolution equations in sup-norm context and in L2 variational context, Lin.
Ră
aume und Approx., Abh. Tag. Oberwolfach 1977, ISNM 40, 547–558 (1978).
Lumer, G., Principe du maximum et ´equations d’´evolution dans L2 (French), S´eminaire
de Th´eorie du Potentiel Paris, No. 3, Lect. Notes Math. 681, 143–156 (1978).
Lumer, G., Approximation des solutions d’´equations d’´evolution pour op´erateurs locaux
en g´en´eral et pour op´erateurs elliptiques (French), C.R. Acad. Sci. Paris, S´er. A
288, 189–192 (1979).
Lumer, G., Perturbations additives d’op´erateurs locaux (French), C.R. Acad. Sci. Paris,
S´er. A 288, 107–110 (1979).
Lumer, G. and Paquet, L., Semi-groupes holomorphes, produit tensoriel de semi-groupes
et ´equations d’´evolution (French), S´eminaire de Th´eorie du Potentiel Paris, No. 4,
Lect. Notes Math. 713, 156–177 (1979).
´
Lumer, G., Equations
de diffusion sur des r´eseaux infinis (French), S´emin. Goulaouic´
Schwartz 1979–1980, Equat.
d´eriv. part., Expos´e No. 18, 9 p. (1980).
Lumer, G., Connecting of local operators and evolution equations on networks, Potential
theory, Proc. Colloq., Copenhagen 1979, Lect. Notes Math. 787, 219–234 (1980).
Lumer, G., Approximation d’op´erateurs locaux et de solutions d’´equations d’´evolution
(French), S´eminaire de Th´eorie du Potentiel Paris, No. 5, Lect. Notes Math. 814,
166–185 (1980).

Lumer, G., Espaces ramifi´es, et diffusions sur les r´eseaux topologiques (French), C.R.
Acad. Sci. Paris, S´er. A 291, 627–630 (1980).

www.pdfgrip.com


xiv

Life and Work of Gă
unter Lumer

Lumer, G., Local operators, regular sets, and evolution equations of diffusion type, Functional analysis and approximation, Proc. Conf., Oberwolfach 1980, ISNM 60, 51–71
(1981).
Lumer, G., Local dissipativeness and closure of local operators, Toeplitz centennial,
Toeplitz Mem. Conf., Tel Aviv 1981, Operator Theory, Adv. Appl. 4, 415–426
(1982).
Lumer, G., Redheffer, R. and Walter, W., Comportement des solutions d’in´equations
diff´erentielles d´eg´en´er´ees du second ordre, et applications aux diffusions (French),
C.R. Acad. Sci. Paris, S´er. I 294, 617–620 (1982).
´
Lumer, G., Equations
de diffusion g´en´erales sur des r´eseaux infinis (French), S´eminaire
de Th´eorie du Potentiel Paris, No. 7, Lect. Notes Math. 1061, 230–243 (1984).
Lumer, G., An exponential formula of Hille-Yosida type for propagators, Approximation
theory and functional analysis, Anniv. Vol., Proc. Conf., Oberwolfach 1983, ISNM
65, 527–542 (1984).
Dubois, R.M. and Lumer, G., Formule de Duhamel abstraite (French), Arch. Math. 43,
49–56 (1984).
´
Lumer, G., Equations

d’´evolution, semigroupes en espace-temps et perturbations
(French), C.R. Acad. Sci. Paris, S´er. I 300, 169–172 (1985).
Lumer, G., Op´erateurs d’´evolution, comparaison de solutions, perturbations et approximations (French), C.R. Acad. Sci. Paris, S´er. I 301, 351–354 (1985).
Lumer, G., Local operators, space-time methods, and evolution equations of diffusion
type, Aspects of positivity in functional analysis, Proc. Conf. in the occasion of
H.H. Schaefers Birthday, Tă
ubingen 1985, North-Holland Math. Stud. 122, 157168
(1986).
Lumer, G., Principes du maximum paraboliques pour des domaines (x, t) non-cylindriques
(French), S´eminaire de Th´eorie du Potentiel Paris, No. 8, Lect. Notes Math. 1235,
105–113 (1987).
Lumer, G., Perturbations “homotopiques”. Perturbations singuli`eres et non singuli`eres
de semi-groupes d’op´erateurs et de familles r´esolventes (French), C.R. Acad. Sci.
Paris, S´er. I 306, No. 13, 551–556 (1988).
Lumer, G., Redheffer, R. and Walter, W., Estimates for solutions of degenerate secondorder differential equations and inequalities with applications to diffusion, Nonlinear
Anal., Theory Methods Appl. 12, No. 10, 1105–1121 (1988).
Lumer, G., Applications de l’analyse non standard `
a l’approximation des semi-groupes
d’op´erateurs et aux ´equations d’´evolution (French), C.R. Acad. Sci. Paris, S´er. I
309, No. 3, 167–172 (1989).
Lumer, G., Singular perturbation and operators of finite local type, Semigroup theory
and applications, Proc. Conf., Trieste/Italy 1987, Lect. Notes Pure Appl. Math.
116, 291–302 (1989).
´
Lumer, G., Equations
de diffusion dans des domains (x, t) non-cylindriques et semigroupes “espace-temps” (French), S´eminaire de Th´eorie du Potentiel Paris, Lect.
Notes Math. 1393, 161–180 (1989).
Lumer, G., Homotopy-like perturbation: General results and applications, Arch. Math.
52, No. 6, 551–561 (1989).

www.pdfgrip.com


Life and Work of Gă
unter Lumer

xv

Lumer, G., New singular multiplicative perturbation results via homotopy-like perturbation, Arch. Math. 53, No. 1, 52–60 (1989).
Lumer, G., Solutions g´en´eralis´ees et semi-groupes int´egr´es (French), C.R. Acad. Sci.
Paris, S´er. I 310, No. 7, 577–582 (1990).
Lumer, G., Applications des solutions g´en´eralis´ees et semi-groupes int´egr´es `
a des probl`emes d’´evolution (French), C.R. Acad. Sci. Paris, S´er. I 311, No. 13, 873–878
(1990).
Lumer, G., Probl`emes dissipatifs et “analytiques” mal pos´es: Solutions et th´eorie asymptotique (French. Abridged English version), C.R. Acad. Sci. Paris, S´er. I 312, No.
11, 831–836 (1991).
Lumer, G., Generalized evolution operators and (generalized) C-semigroups, Semigroup
theory and evolution equations, Proc. 2nd Int. Conf., Delft/Neth. 1989, Lect. Notes
Pure Appl. Math. 135, Marcel Dekker, 337–345 (1991).
Lumer, G., Examples and results concerning the behavior of generalized solutions, integrated semigroups, and dissipative evolution problems, Semigroup theory and evolution equations, Proc. 2nd Int. Conf., Delft/Neth. 1989, Lect. Notes Pure Appl.
Math. 135, Marcel Dekker, 347–356 (1991).
Lumer, G., Semi-groupes irr´eguliers et semi-groupes int´egr´es: Application `
a l’identification de semi-groupes irr´eguliers analytiques et r´esultats de g´en´eration (French.
Abridged English version), C.R. Acad. Sci. Paris, S´er. I 314, No. 13, 1033–1038
(1992).
Lumer, G., Probl`emes d’´evolution avec chocs (changements brusques de conditions au
bord) et valeurs au bord variables entre chocs cons´ecutifs (French. Abridged English
version), C.R. Acad. Sci. Paris, S´er. I 316, No. 1, 41–46 (1993).
Lumer, G., Evolution equations. Solutions for irregular evolution problems via generalized
solutions and generalized initial values. Applications to periodic shocks models,

Ann. Univ. Sarav., Ser. Math. 5, No. 1, 102 p. (1994).
Cioranescu, I. and Lumer, G., Probl`emes d’´evolution r´egularis´es par un noyau g´en´eral
K(t). Formule de Duhamel, prolongements, th´eor`emes de g´en´eration (French.
Abridged English version), C.R. Acad. Sci. Paris, S´er. I 319, No. 12, 1273–1278
(1994).
Lumer, G., Models for diffusion-type phenomena with abrupt changes in boundary conditions in Banach space and classical context. Asymptotics under periodic shocks, in
Cl´ement, Ph. et al. (eds.), Evolution equations, control theory, and biomathematics,
Lect. Notes Pure Appl. Math. 155, Marcel Dekker, Basel, 337–351 (1993).
Lumer, G., On uniqueness and regularity in models for diffusion-type phenomena with
shocks, in Cl´ement, Ph. et al. (eds.), Evolution equations, control theory, and biomathematics, Lect. Notes Pure Appl. Math. 155, Marcel Dekker, Basel, 353–359
(1993).
Lumer, G., Singular problems, generalized solutions, and stability properties, in Lumer,
G. et al. (eds.), Partial differential equations. Models in physics and biology, Math.
Res. 82, Akademie Verlag, Berlin, 204–216 (1994).
Cioranescu, I. and Lumer, G., On K(t)-convoluted semigroups, in McBride, A.C. et al.
(eds.), Recent developments in evolution equations (Proceedings of a meeting held

www.pdfgrip.com


xvi

Life and Work of Gă
unter Lumer
at the University of Strathclyde, UK, 25–29 July, 1994), Pitman Res. Notes Math.
Ser. 324, Longman Scientific & Technical, Harlow, 86–93 (1995).

Fong, C.K., Lumer, G., Nordgren, E., Radjavi, H. and Rosenthal, P., Local polynomials
are polynomials, Stud. Math. 115, No. 2, 105–107 (1995).
Lumer, G., Transitions singuli`eres gouvern´ees par des ´equations de type parabolique

(French. Abridged English version), C.R. Acad. Sci. Paris, S´er. I 322, No. 8, 735–
740 (1996).
Lumer, G., Singular transitions and interactions governed by equations of parabolic type,
in Demuth, M. et al. (eds.), Differential equations, asymptotic analysis, and mathematical physics (Papers associated with the international conference on partial
differential equations, Potsdam, Germany, June 29–July 2, 1996), Math. Res. 100,
Akademie Verlag, Berlin, 192–217 (1997).
Lumer, G., Singular evolution problems, regularization, and applications to physics, engineering, and biology, in Janas, J. et al. (eds.), Linear operators, Proceedings of the
semester organized at the Stefan Banach International Mathematical Center, Warsaw, Poland, February 7–May 15, 1994, Banach Cent. Publ. 38, Polish Academy of
Sciences, Inst. of Mathematics, Warsaw, 205–216 (1997).
Lumer, G. and Neubrander, F., Signaux non-d´etactables en dimension N dans des
syst`emes gouvern´es par des ´equations de type parabolique (French. Abridged English version), C.R. Acad. Sci. Paris, S´er. I, Math. 325, No. 7, 731–736 (1997).
Lumer, G., Singular interaction problems of parabolic type with distribution and hyperfunction data, in Demuth, M. et al. (eds.), Evolution equations, Feshbach resonances, singular Hodge theory., Math. Top. 16, Wiley-VCH, Berlin, 11–36 (1999).
Lumer, G. and Neubrander, F., Asymptotic Laplace transforms and evolution equations,
in Demuth, M. et al. (eds.), Evolution equations, Feshbach resonances, singular
Hodge theory., Math. Top. 16, Wiley-VCH, Berlin, 37–57 (1999).
Lumer, G. and Schnaubelt, R., Local operator methods and time dependent parabolic
equations on non-cylindrical domains, in Demuth, M. et al. (eds.), Evolution equations, Feshbach resonances, singular Hodge theory., Math. Top. 16, Wiley-VCH,
Berlin, 58–130 (1999).
Lumer, G., An introduction to hyperfunctions and δ-expansions, in Antoniou, I. et al.
(eds.), Generalized functions, operator theory, and dynamical systems, Res. Notes
Math. 399, Chapman & Hall/CRC, Boca Raton, 1–25 (1999).
Bă
aumer, B., Lumer, G. and Neubrander, F., Convolution kernels and generalized functions, in Antoniou, I. et al. (eds.), Generalized functions, operator theory, and
dynamical systems, Res. Notes Math. 399, Chapman & Hall/CRC, Boca Raton,
68–78 (1999).
Lumer, G., Interaction problems with distributions and hyperfunctions data, in Antoniou,
I. et al. (eds.), Generalized functions, operator theory, and dynamical systems, Res.
Notes Math. 399, Chapman & Hall/CRC, Boca Raton, 299–307 (1999).
Lumer, G. and Neubrander, F., The asymptotic Laplace transform: New results and
relation to Komatsu’s Laplace transform of hyperfunctions, in Ali Mehmeti, F.

et al. (eds.), Partial differential equations on multistructures, Proceedings of the
conference, Luminy, France, Lect. Notes Pure Appl. Math. 219, Marcel Dekker,
New York, 147–162 (2001).

www.pdfgrip.com


Life and Work of Gă
unter Lumer

xvii

Lumer, G., Blow up and hovering in parabolic systems with singular interactions: Can
we “see” a hyperfunction?, in Lumer, G. et al. (eds.), Evolution equations and their
applications in physical and life sciences, Lect. Notes Pure Appl. Math. 215, Marcel
Dekker, New York, 387–393 (2001).
Lumer, G. and Schnaubelt, R., Time-dependent parabolic problems on non-cylindrical
domains with inhomogeneous boundary conditions, J. Evol. Equ. 1, No. 3, 291–309
(2001).
Lumer, G., A general “isotropic” Paley-Wiener theorem and some of its applications, in
Iannelli, M. et al. (eds.), Evolution equations: applications to physics, industry, life
sciences and economics, Prog. Nonlinear Dier. Equ. Appl. 55, Birkhăauser, Basel,
323332 (2003).

List of books of Gă
unter Lumer
Clement, Ph. and Lumer, G., Evolution equations, control theory, and biomathematics,
Proceedings of the 3rd international workshop-conference held at the Han-sur-Lesse
Conference Center of the Belgian Ministry of Education, Lecture Notes in Pure and
Applied Mathematics 155, Marcel Dekker, Basel, 580 p. (1993).

Lumer, G., Nicaise, S. and Schulze, B.-W., Partial differential equations. Models in
physics and biology, Contributions to the conference held in Han-sur-Lesse (Belgium) in December 1993, Mathematical Research 82, Akademie Verlag, Berlin, 421
p. (1994).
Antoniou, I. and Lumer, G., Generalized functions, operator theory, and dynamical systems, Research Notes in Mathematics 399, Boca Raton, FL, Chapman & Hall/CRC
378 p. (1999).
Lumer, G. and Weis, L., Evolution equations and their applications in physical and life
sciences, Proceeding of the Bad Herrenalb (Karlsruhe) conference, Germany, 1999,
Lecture Notes in Pure and Applied Mathematics 215, Marcel Dekker, New York,
511 (2001).
Iannelli, M. and Lumer, G., Evolution equations: applications to physics, industry, life
sciences and economics, Proceedings of the 7th international conference on evolution equations and their applications, EVEQ2000 conference, Levico Terme, Italy,
October 30–November 4, 2000, Progress in Nonlinear Dierential Equations and
their Applications 55, Birkhă
auser, Basel, 423 p. (2003).

www.pdfgrip.com


H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise, J. von Below (eds):
Functional Analysis and Evolution Equations. The Gă
unter Lumer Volume. xixxx
c 2007 Birkhă
auser Verlag Basel/Switzerland

In Remembrance of Gă
unter Lumer
Heinz Kăonig
Gă
unter Lumer was a close friend of mine for several decades. We had the same
age: our dates of birth were but 13 days apart. We met for the first time in the

fall of 1962 at a functional analysis conference in Oberwolfach. The year before
Gă
unter had published two of his most important papers: the common paper with
Ralph Phillips on dissipative operators and the paper on semi-inner products.
The subsequent years were the grand period in the development of the functional analytic theory of abstract analytic functions, known under the key words of
uniform algebras and Hardy spaces. We were both deeply involved, with quite often
dierent methods but close results. Gă
unter obtained fundamental breakthroughs
in two situations: The first time in Bulletin Amer. Math. Soc. 70(1964), where he
was able to develop the abstract counterpart of the classical unit disk situation
on an arbitrary uniform algebra and for an individual multiplicative linear functional, under the basic assumption that the functional in question has a unique
representing measure. Before that one needed global assumptions on the algebra
like to be Dirichlet or logmodular. After his work then 1965 Kenneth HoffmanHugo Rossi and myself independently obtained the final abstract version of the
classical unit disk situation in terms of a fixed so-called Szeg˝
o measure for an
individual multiplicative linear functional.
The second breakthrough was in his 1968 Lecture Notes, this time for an
arbitrary multiplicative linear functional on any uniform algebra. Gă
unter dened
its universal Hardy class and was able to transfer the classical concepts and results
to an amazing extent, in particular to establish an abstract conjugation operation
via extension of the classical Kolmogorov estimations. He then left the field in
the early seventies. I myself returned to it in a common frame with the extended
concept of Daniell-Stone integration due to Michael Leinert 1982, which produced
a definitive theory around 1990. But it is clear that to an essential extent the basic
contributions are due to Gă
unter Lumer in the sixties.
In all these years we had close contacts. During the academic year 1967/68
Gă
unter stayed at Strasbourg University, thus close to my home University Saarbră

ucken. In the summer term 1967 he gave a series of lectures in Saarbră
ucken,
and in the winter term 1967/68, which I spent at Caltech in Pasadena, a little

www.pdfgrip.com


xx

In Remembrance of Gă
unter Lumer

bus supplied by our University brought my students to his lectures in Strasbourg
every week. In the academic year 1969/70 Gă
unter Lumer together with Irving
Glicksberg organized a Research Seminar on function algebras at their home
University, the University of Washington in Seattle. I had the good fortune to
participate for three months on his invitation.
After his move to Belgium in 1973/74 Gă
unter was a regular visitor to Saarbră
ucken, both private and for a further series of lectures and several colloquium
talks. He wrote a comprehensive survey article on evolution equations for our Annales Universitatis Saraviensis and published several papers in the Archiv
der Mathematik of which I had been the editor for abstract analysis. Our relations became even closer because of the sequence of the North-West European Analysis Seminars 1992–1997, of which Gă
unter was the unique creator
and driving force. We were common chairmen of the second seminar 1993 at Schloss
Dagstuhl in the Saar State, which is the Informatics counterpart of the Oberwolfach Institute. Thus we two are in the tiny group of “outside” mathematicians who
have ever been chairpersons of conferences at Schloss Dagstuhl. Unfortunately, in
1997 a serious hip joint operation forced Gă
unter to discontinue the beautiful enterprise. There was no successor.
For me the first of the seminars 1992 in Saint-Amand-les-Eaux near Lille was

a moving event: Near its end I fell into heart trouble, and my doctor said on the
telephone that I should come to his hospital right away but must not drive a car.
What then happened was that Gă
unter asked Luc Paquet to place his own car
next to his apartment in Brussels, and took the steering-wheel of my car (which
was new at the time) to drive us for at least 400 kilometers to Saarbră
ucken. We
arrived late at night, and my wife said later that I looked radiant with health but
Gă
unter grey with exhaustion. This was the deepest evidence of friendship which I
ever experienced in my life.
Heinz Kă
onig
Universită
at des Saarlandes
Fakultă
at fă
ur Mathematik und Informatik
D-66041 Saarbră
ucken, Germany
e-mail:

www.pdfgrip.com


H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise, J. von Below (eds):
Functional Analysis and Evolution Equations. The Gă
unter Lumer Volume. 116
c 2007 Birkhă
auser Verlag Basel/Switzerland

Expansions in Generalized Eigenfunctions
of the Weighted Laplacian
on Star-shaped Networks
F´elix Ali Mehmeti, Robert Haller-Dintelmann and Virginie Regnier
In memory of Gă
unter Lumer

Abstract. We are interested in evolution phenomena on star-shaped networks
composed of n semi-infinite branches which are connected at their origins.
Using spectral theory we construct the equivalent of the Fourier transform,
which diagonalizes the weighted Laplacian on the n-star. It is designed for
the construction of explicit solution formulas to various evolution equations
such as the heat, wave or the Klein-Gordon equation with different leading
coefficients on the branches.
Mathematics Subject Classification (2000). Primary 34B45; Secondary 42A38,
47A10, 47A60, 47A70.
Keywords. Networks, spectral theory, resolvent, generalized eigenfunctions,
functional calculus, evolution equations.

1. Introduction
We study the foundations for the understanding of evolution phenomena on starshaped networks composed of n semi-infinite branches which are connected at their
origins. To this end, we construct the equivalent of the Fourier transform which
diagonalizes the weighted Laplacian on the n-star, using spectral theory. This
allows us to formulate a functional calculus for the weighted Laplacian, designed
to construct explicit solution formulas to various evolution equations such as the
heat, wave or the Klein-Gordon equation with different leading coefficients on
the branches. The model of the n-star should lead to a comprehension of the
phenomena happening locally in time and space near the ramification nodes of
Parts of this work were done, while the second author visited the University of Valenciennes. He

wishes to express his gratitude to F. Ali Mehmeti and the LAMAV for their hospitality.

www.pdfgrip.com


2

F. Ali Mehmeti, R. Haller-Dintelmann and V. R´egnier

more complicated networks. The investigation of evolution equations on networks
starts with G. Lumer [17] and subsequent papers. See [1, 4, 9] and the references
mentioned therein.
Let N1 , . . . , Nn be n disjoint copies of (0; +∞) (n ∈ N, n ≥ 2) and ck > 0,
for k ∈ {1, . . . , n}. A vector (u1 , . . . , un ) of functions uk : Nk → C is said to satisfy
the transmission conditions
(T0 ), if ui (0) = uk (0) for all (i, k) ∈ {1, . . . , n}2 ,
n

c2k ∂x uk (0+ ) = 0.

(T1 ), if
k=1

A vector (uk )k=1,...,n satisfying (T0 ) can also be viewed as a function on N :=
n
k=1 Nk , where the n boundary points corresponding to 0 ∈ Nk are identified. This
domain is called a star-shaped network or n-star with the branches N1 , . . . , Nn .
In this paper, we study the weighted Laplacian submitted to (T0 ) and (T1 ):
⎧
n

⎪
⎨ D(A) := (uk ) ∈
H 2 (Nk ) | (uk ) satisfies (T0 ) and (T1 ) ,
⎪
⎩

A(uk ) :=

(−c2k

·

k=1
2
∂x uk )k=1,...,n .

This operator can be inserted for example in the abstract wave equation
u
ă(t) + Au(t) = 0,
u(0) = u0 , u(0)

= v0 ,
which means in concrete terms:
⎧ 2
⎪ [∂t − c2k ∂x2 ]uk (t, x) = 0,
⎪
⎪
⎪
⎪
⎪

ui (t, 0) = uk (t, 0),
⎪
⎪
⎪
⎪
n
⎨
c2k ∂x uk (t, 0+ ) = 0,
⎪
⎪
k=1
⎪
⎪
⎪
⎪
⎪
uk (0, x) = u0k (x),
⎪
⎪
⎪
⎩
∂t uk (0, x) = vk0 (x),

∀ k ∈ {1, . . . , n},
∀ (i, k) ∈ {1, . . . , n}2 ,

∀ k ∈ {1, . . . , n},
∀ k ∈ {1, . . . , n}

for x, t ≥ 0, where u0 = (u0k )k=1,...,n , v0 = (vk0 )k=1,...,n and u(t) = (uk (t, ·))k=1,...,n .

The operator A is self-adjoint, its spectrum is [0; +∞) and has multiplicity n
(in the sense of ordered spectral representations, see Definition XII.3.15, p. 1216
of [14]). The analytical core of this paper is a representation of the kernel of the
resolvent of A in terms of a special choice of a family of n generalized eigenfunctions
parametrized by λ ∈ [0; +∞).
After having proved a limiting absorption principle for the resolvent, we insert
A in Stone’s formula to obtain a representation of the resolution of the identity of A
in terms of the generalized eigenfunctions. This classical procedure (see for example
n
[3]) should lead to an expansion formula for functions in H = k=1 L2 (Nk ) in
terms of the family of generalized eigenfunctions.

www.pdfgrip.com


Generalized Eigenfunctions on Star-shaped Networks

3

We observe that the transition from the formula for the resolution of the
identity to an expansion formula involving a generalized Fourier transform, which
diagonalizes A, is not straightforward in the case of the n-star. This comes from the
fact that the resolvent kernel, which is defined on N ×N , changes its structure when
crossing the n diagonals of Nk × Nk , k = 1, . . . , n. These diagonals cut N × N into
n connected pieces in accordance with the structure of the resolvent. Our special
choice of the generalized eigenfunctions allows us to recombine the inner integral of
the formula for the resolution of the identity across the diagonals of Nk × Nk to an
integral over all of N , furnishing the desired generalized Fourier transformation V
as well as its left inverse Z. It is not obvious, whether this recombination is possible
for all choices of generalized eigenfunctions, although theoretical results imply that

an expansion in generalized eigenfunctions always exists [11, 19]. Now, V can be extended to an isometry on H, which diagonalizes A, and an explicit functional calculus for A can be given. We plan to give explicit expressions for the solutions of evolution equations like the weighted wave, heat and Klein-Gordon equations on the
n-star and to derive results on their qualitative behaviour in a subsequent paper.
Such expressions can be obtained (at least formally) also from representations
of the resolution of the identity which are not recombined to Fourier-type transformations. But these expressions would be sums of terms with very poor regularity
although their sum, representing the solution, is regular (like a decomposition of
a C ∞ -function by multiplying it with characteristic functions on sub-domains).
These artificial singularities are totally undesirable for any kind of investigations.
They occur for example in [13], a pioneering paper of theoretical physics explaining the phenomenon of advanced transmission of dispersive wave packets crossing
a potential barrier. The authors obtain a solution formula using Laplace transform in time, but which splits up into irregular terms. They do not attempt to
prove that their formula represents a solution of the original problem, which should
be possible only in some very weak sense. But this (artificial) lack of regularity
permits only to study the advanced transmission phenomenon for gaussian wave
packets using a highly special method.
In [7], the authors study the similar phenomenon of delayed reflection occurring at semi-infinite barriers. They construct an expansion in generalized eigenfunctions and thus avoid those artificial singularities. This expansion is used to
define wave packets in frequency bands adapted to the transmission conditions.
Thus it is possible to study the dependence of propagation patterns, in particular
the delayed reflection, on the main frequency of the wave packets. In [8] it is pointed
out using similar methods, that classical causality is valid for nonlinear dispersive
waves hitting a semi-infinite barrier. In [6] a solution formula for the Klein-Gordon
equation on the n-star but with one finite branch with an end with prescribed excitation is presented using Laplace transform in time. This result is not comparable
with the present paper, because it does not concern an initial value problem.
There remains an unsatisfactory point in the present paper: our Fouriertype transformation V is not a spectral representation of A in the classical sense
although it diagonalizes this operator: the natural norm on the range of V making

www.pdfgrip.com


4

F. Ali Mehmeti, R. Haller-Dintelmann and V. R´egnier

V an isometry, as in the theorem of Plancherel, is not just a weighted L2 -norm on
some measure space. This is due to the fact that the back transformation Z has a
different expression on each branch, and this is caused by the ramification of the
domain.
It is not clear to us how one could find a family of generalized eigenfunctions
leading to a spectral representation of A. The existing general literature on expansions in generalized eigenfunctions ([11, 19, 20] for example) does not seem to be
helpful for this kind of problem: their constructions start from an abstractly given
spectral representation. But in concrete cases you do not have an explicit formula
for it at the beginning.
In [10] the relation of the eigenvalues of the Laplacian in a L∞ -setting on infinite, locally finite networks to the adjacency operator of the network is studied.
The question of the completeness of the corresponding eigenfunctions, viewed as
generalized eigenfunctions in an L2 -setting, could be asked. The n-star we consider
is a particular case of the geometry studied by J. von Below and the completeness of
the eigenfunctions is established in a way. In a recent paper ([15]), the authors consider general networks with semi-infinite ends. They give a construction to compute
some generalized eigenfunctions from the coefficients of the transmission conditions
(scattering matrix). The eigenvalues of the associated Laplacian are the poles of
the scattering matrix and their asymptotic behaviour is studied. But no attempt
is made to show the completeness of a given family of generalized eigenfunctions.
Spectral theory for the Laplacian on finite networks has been studied since the
1980ies for example by J.P. Roth, J.v. Below, S. Nicaise, F. Ali Mehmeti (see [1]).
Natural perspectives for our expansion result are investigations on the qualitative behaviour of solutions of evolution equations on the n-star. For the weighted
heat equation on the n-star, our expansion permits to prove Gaussian estimates
(this feature shall be treated in a subsequent paper). For bounded networks and
variable coefficients this has already been proved by D. Mugnolo ([18]) using different methods. In [16] the transport operator is considered on finite networks.
The connection between the spectrum of the adjacency matrix of the network
and the (discrete) spectrum of the transport operator is established. By adding
semi-infinite branches to the finite network, continuous parts of the spectrum and
generalized eigenfunctions might appear.
Many results have been obtained in spectral theory for elliptic operators on

various types of unbounded domains in Rn . Using the existing results on stratified
bands [12] for example, one could reduce the spectral analysis of the Laplacian on
networks of bands locally near the nodes to the case of the n-star. Time asymptotics
for the associated evolution equations have also been studied extensively. For the
Klein-Gordon equation on the n-star we conjecture that the maximum of the
absolute value of the solutions decays as t−1/2 when t tends to infinity as on
the real line. For two branches with potential step this has been already proved
using generalized eigenfunctions in [2]. An example for a three-dimensional coupled
domain with singularities is treated in [5]. See also the other literature mentioned
therein and in [3].

www.pdfgrip.com


Generalized Eigenfunctions on Star-shaped Networks

5

2. Data and functional analytic framework
Let us introduce some notation which will be used throughout the rest of the
paper:
• Domain and functions: Let N1 , . . . , Nn be n disjoint sets identified with
n
(0; +∞) (n ∈ N, n ≥ 2) and put N := k=1 Nk . Furthermore, we write
[a, b]Nk for the interval [a, b] in the branch Nk . For the notation of functions
two viewpoints are used:
– functions f on the object N taking their values in R and fk is then the
restriction of f to Nk .
– n-tuples of functions on the branches Nk ; then sometimes we write f =
(f1 , . . . , fn ).

• Transmission conditions:
n

(T0 ): (uk )k=1,...,n ∈

C 0 (Nk ) satisfies ui (0) = uk (0), ∀ (i, k) ∈ {1, . . . , n}2 .
k=1
n

(T1 ): (uk )k=1,...,n ∈

n

c2k · ∂x uk (0+ ) = 0.

C 1 (Nk ) satisfies
k=1

k=1

• Definition of the operator: Define the real Hilbert space
n

n

L2 (Nk ) with scalar product ((uk ), (vk ))H =

H=
k=1

(uk , vk )L2 (Nk )
k=1

and the operator A : D(A) −→ H by
⎧
n
⎪
⎨ D(A) = (uk ) ∈
H 2 (Nk ) | (uk ) satisfies (T0 ) and (T1 ) ,
⎪
⎩

k=1

A(uk ) = (Ak uk )k=1,...,n = (−c2k · ∂x2 uk )k=1,...,n .

Note that, if ck = 1 for every k ∈ {1, . . . , n}, A is the Laplacian in the sense
of the existing literature.
• Notation for the resolvent: The resolvent of an operator T is denoted by R,
i.e., R(z, T ) = (zI − T )−1 for z ∈ ρ(T ).
Proposition 2.1 (spectrum of A). The operator A : D(A) → H defined above is
self-adjoint and satisfies σ(A) = [0; +∞).
Proof. Simple adaptation of the proof of Lemma 1.1.5 in [3].

3. Expansion in generalized eigenfunctions
The aim of this section is to find an explicit expression for the kernel of the resolvent
of the operator A on the star-shaped network defined in the previous section.

www.pdfgrip.com

6

F. Ali Mehmeti, R. Haller-Dintelmann and V. R´egnier

Definition 3.1 (generalized eigenfunction). Let λ ∈ C be fixed. An element f ∈
n
∞
k=1 C (Nk ) is called generalized eigenfunction of A if it satisfies (T0 ), (T1 ) and
the formal differential expression Af = λf .
Proposition 3.2 (an expression of the resolvent). Let λ ∈ C be fixed. Let Im(λ) = 0
λ
(x)
and eλ1 , eλ2 be generalized eigenfunctions of A such that the Wronskian w1,2
satisfies for every x in N
λ
w1,2
(x) = det W (eλ1 (x), eλ2 (x)) = eλ1 (x) · (eλ2 ) (x) − (eλ1 ) (x) · eλ2 (x) = 0.

If for some k ∈ {1, . . . , n} we have eλ1 |Nm ∈ H 2 (Nm ) for all m = k and eλ2 |Nk ∈
H 2 (Nk ), then we have for any f ∈ H, λ ∈ ρ(A) and x ∈ Nk
[R(λ, A)f ](x) =

1
λ )(x)
c2k (w1,2

·

eλ1 (x)eλ2 (x )f (x ) dx

(1)

[x;+∞)Nk

+
N \[x;+∞)Nk

eλ2 (x)eλ1 (x )f (x ) dx

.

Note that by integral over N , we mean the sum of the integrals over Nk , k =
1, . . . , n.
Proof. The arguments are the same as in the proof of Theorem 1.3.4 of [3] (see
also [2]) and the calculations are analogous. The integration by parts is replaced
here by the Green formula for the star-shaped network that is given in the next
lemma.
Lemma 3.3 (Green’s formula on the star-shaped network with n semi-infinite
branches). Denote by Va1 ,...,an the subset of the network N defined by
Va1 ,...,an = {x ∈ N | x ∈ [0; ak ), where k is the index such that x ∈ Nk }.
Then u, v ∈ D(A) implies
n

n

u(x)v (x) dx −

u (x)v(x) dx =
Va1 ,...,an

Va1 ,...,an

u(ak )v (ak ) +
k=1

u (ak )v(ak ).
k=1

Proof. Two successive integrations by parts are used and since both u and v belong
to D(A), they both satisfy the transmission conditions (T0 ) and (T1 ). So
n

n

uk (0)vk (0) = u1 (0)
k=1

Idem for

n
k=1

vk (0) = 0.
k=1

uk (0)vk (0).

Definition 3.4 (generalized eigenfunctions of A). For j ∈ {1, . . . , n} let
sj := −c−1

j ·

cl ,

d1,j := (1 + sj )/2

l=j

www.pdfgrip.com

and

d2,j := (1 − sj )/2.