Commuting self-adjoint extensions of symmetric operators defined from the partial derivatives
Jorgensen, Palle E. T. ; Pedersen, Steen
arXiv, 0005248 / Harvested from arXiv
Text on arXiv
We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(\partial/\partial x_j):j=1,...,d} with domain C_c^\infty(\Omega) where the self-adjointness is defined relative to L^2(\Omega), and \Omega is a given open subset of R^d. The measure on \Omega is Lebesgue measure on R^d restricted to \Omega. The problem originates with I.E. Segal and B. Fuglede, and is difficult in general. In this paper, we provide a representation-theoretic answer in the special case when \Omega=I\times\Omega_2 and I is an open interval. We then apply the results to the case when \Omega is a d-cube, I^d, and we describe possible subsets \Lambda of R^d such that {e^(i2\pi\lambda \dot x) restricted to I^d:\lambda\in\Lambda} is an orthonormal basis in L^2(I^d).
Publié le : 2000-05-24
Classification:  Mathematics - Spectral Theory,  Mathematical Physics,  42C05, 22D25, 46L55, 47C05 
@article{0005248,
     author = {Jorgensen, Palle E. T. and Pedersen, Steen},
     title = {Commuting self-adjoint extensions of symmetric operators defined from
  the partial derivatives},
     journal = {arXiv},
     volume = {2000},
     number = {0},
     year = {2000},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0005248}
}

Jorgensen, Palle E. T.; Pedersen, Steen. Commuting self-adjoint extensions of symmetric operators defined from
  the partial derivatives. arXiv, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/0005248/