An unusual self-adjoint linear partial differential operator

Authors:
W. N. Everitt, L. Markus and M. Plum

Journal:
Trans. Amer. Math. Soc. **357** (2005), 1303-1324

MSC (2000):
Primary 35J40, 35J67, 35P05; Secondary 32A36, 32A40, 47B25.

DOI:
https://doi.org/10.1090/S0002-9947-04-03719-5

Published electronically:
November 4, 2004

MathSciNet review:
2115367

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Abstract: In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region.

This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the ``Harmonic'' operator.

The boundary value problems considered in the Memoir (see above) and in this paper are called *regular* in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions.

In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty.

This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty.

Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent.

In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.

**1.**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****2.**Alberto Alonso and Barry Simon,*The Birman-Kreĭn-Vishik theory of selfadjoint extensions of semibounded operators*, J. Operator Theory**4**(1980), no. 2, 251–270. MR**595414****3.**J. M. Anderson and Loren D. Pitt,*On recurrence properties of certain lacunary series. I. General results*, J. Reine Angew. Math.**377**(1987), 65–82. MR**887400**

J. M. Anderson and L. D. Pitt,*On recurrence properties of certain lacunary series. II. The series ∑ⁿ_{𝑘=1}𝑒𝑥𝑝(𝑖𝑎^{𝑘}𝜃)*, J. Reine Angew. Math.**377**(1987), 83–96. MR**887401****4.**J.M. Anderson. Personal communication: April 2002.**5.**Stefan Bergman,*The kernel function and conformal mapping*, Second, revised edition, American Mathematical Society, Providence, R.I., 1970. Mathematical Surveys, No. V. MR**0507701****6.**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523**

Nelson Dunford and Jacob T. Schwartz,*Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space*, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR**0188745****7.**W. Norrie Everitt and Lawrence Markus,*Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators*, Mathematical Surveys and Monographs, vol. 61, American Mathematical Society, Providence, RI, 1999. MR**1647856****8.**W. N. Everitt and L. Markus,*Elliptic partial differential operators and symplectic algebra*, Mem. Amer. Math. Soc.**162**(2003), no. 770, x+111. MR**1955204**, https://doi.org/10.1090/memo/0770**9.**Gaetano Fichera,*Linear elliptic differential systems and eigenvalue problems*, Lecture Notes in Mathematics, vol. 8, Springer-Verlag, Berlin-New York, 1965. MR**0209639****10.**Avner Friedman,*Partial differential equations*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR**0445088****11.**G. H. Hardy,*Divergent Series*, Oxford, at the Clarendon Press, 1949. MR**0030620****12.**Yitzhak Katznelson,*An introduction to harmonic analysis*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0248482****13.**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157****14.**S. Timoshenko.*Theory of plates and shells*(McGraw-Hill, Inc., New York:1959).**15.**J. Wloka,*Partial differential equations*, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas. MR**895589****16.**Kôsaku Yosida,*Functional analysis*, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR**0180824****17.**A. Zygmund,*Trigonometric series: Vols. I, II*, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR**0236587**

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Additional Information

**W. N. Everitt**

Affiliation:
School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT, England, United Kingdom

Email:
w.n.everitt@bham.ac.uk

**L. Markus**

Affiliation:
School of Mathematics, Universty of Minnesota, Minneapolis, Minnesota 55455-0487

Email:
markus@math.umn.edu

**M. Plum**

Affiliation:
Mathematisches Institut I, Universität Karlsruhe, D-76128 Karlsruhe, Germany

Email:
michael.plum@math.uni-karlsruhe.de

DOI:
https://doi.org/10.1090/S0002-9947-04-03719-5

Keywords:
Linear partial differential equations,
self-adjoint partial differential operators,
spectral theory.

Received by editor(s):
April 15, 2003

Published electronically:
November 4, 2004

Dedicated:
Dedicated to Professor Johann Schröder

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.