Selfadjoint extension of the Casimir operator

  • Roelof W. Bruggeman
Part of the Modern Birkhäuser Classics book series (MBC)


In 1.2.6 and 1.5.7 we discussed the selfadjoint extension of the differential operator L r . This concerned the modular case. The extension was an operator in a Hilbert space H(r) for r ∈ ℝ. Its eigenfunctions were stated to be modular forms, and \(\frac{{|r|}}{2}\left( {1 - \frac{{|r|}}{2}} \right)\) its smallest eigenvalue. In this chapter we prove these statements, in the more general setting of Part I. We work in a Hilbert space H(x,l) depending on a unitary character x of \(\widetilde \Gamma \), and a (real) weight l suitable for x. In Section 6.1 we define this Hilbert space as a completion of the space of all smooth x-l-equivariant functions with compact support in Y.


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Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Roelof W. Bruggeman
    • 1
  1. 1.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherland

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