Abstract
LetH 0≧0 be a self-adjoint operator acting in a spaceL 2(M, μ). It is assumed thatH 0 e=0, wheree is strictly positive, and that exp(−tH 0) is positivity preserving fort≧0. LetV be a real function onM such that its positive part is inL 2(M,e 2μ) and its negative part is relatively small with respect toH 0. ThenH=H 0+V is essentially self-adjoint on the intersection of the domains ofH 0 andV. This result is applied to Schrödinger operators and to quantum field Hamiltonians.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Davies, E. B.: Properties of the Green's functions of some Schrödinger operators, to appear.
Faris, W.: Product formula for semigroups defined by Friedrichs extensions. Pacific J. Math.22, 47–70 (1967).
Faris, W.: Quadratic forms and essential self-adjointness. Helv. Phys. Acta (to appear).
Glimm, J., Jaffe, A.: Singular perturbations of self-adjoint operators. Commun. Pure Appl. Math.22, 401–414 (1969).
Hoegh-Krohn, R.: A general class of quantum fields without cutoff in two space-time dimensions. Commun. math. Phys.21, 244–255 (1971).
Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966.
Kato, T.: Schrödinger operators with singular potentials, to appear.
Nelson, E.: A quartic interaction in two dimensions. In: Goodman, R., Segal, I.E. (Eds.): Mathematical theory of elementary particles. Cambridge: M.I.T. Press 1966.
Nelson, E.: Topics in dynamics I: Flows. Princeton, N.J.: Princeton University Press 1969.
Rosen, L.: A λ φ2n field theory without cutoffs. Commun. math. Phys.16, 157–183 (1970).
Schaefer, H. H.: Topological vector spaces. New York: Macmillan 1966.
Segal, I.: Construction of nonlinear local quantum processes: I. Ann. Math.92, 462–481 (1970).
Segal, I.: Construction of nonlinear local quantum processes: II. Inventiones math.14, 211–241 (1971).
Simon, B.: Essential self-adjointness of Schrödinger operators with positive potentials. Math. Ann., to appear.
Simon, B., Hoegh-Krohn, R.: Hypercontractive semigroups and two dimensional self-coupled Bose fields. J. Funct. Anal.9, 121–180 (1972).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Faris, W.G. Essential self-adjointness of operators in ordered hilbert space. Commun.Math. Phys. 30, 23–34 (1973). https://doi.org/10.1007/BF01646685
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01646685