Perturbations of the covariance form of the scalar field ground state

T. A. Bolokhov<sup>ab

a</sup> Saint Petersburg State University
^b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: We discuss mathematical aspects of perturbations of the covariance of a square root of Laplacian corresponding to Gaussians which satisfy eigenstate equation for the free scalar quantum field. We show that corresponding operators can be represented as multiplicative perturbations of this square root with negative self-adjoint Hilbert–Schmidt addition to the unity operator. We show that domains of the perturbed covariance form and of its counterpart for the free quantum field coincide. Covariance forms, Schrödinger representation, ground-state Gaussian of a quantum field, singular perturbations of closed quadratic forms.

Key words and phrases: covariance of the ground state of quantum field, Schrödinger representation, Gaussian functional, singular perturbations of closable quadratic forms.

UDC: 517, 539.1.01

Received: 31.10.2025