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We show how Pauli-Villars regularization works in the construction of renormalized Hamiltonian for two exemplars of quantum systems with singular perturbations. The systems are the scalar 3-dimensional particle interacting with [delta]-potential and the infrared extensions of the quadratic forms of the Gaussian functional of the ground state in the quantum field theory. Bibliography: 11 titles., Introduction Quantum mechanical models with singular perturbation represent an interesting example. Indeed the scattering operator of the corresponding system can be calculated in terms of the resolvent of the energy [...]

We extend the quadratic form of the Gaussian functional of the free quantum scalar field theory to the set of functions decreasing in the infinity as [|x|.sup.-1]. We use the momentum-space representation (after the Fourier transform) and the scalar product is generated by the quadratic form of the Laplace operator (potential term of the quantum Hamiltonian). Bibliography: 6 titles., INTRODUCTION Formulation of the quantum field theory in the terms of the Schroedinger picture is a convenient basement in the description of boundary effects in the manifold of field configurations. [...]

It is demonstrated that within the framework of the second quantization, the quantum Hamiltonian operator for a free transverse field reveals an alternative set of states satisfying the eigenstate functional equations. The construction is based on extensions of the quadratic form of the transverse Laplace operator, which are used as a source of spherical basis functions with singularity at the origin. This basis naturally replaces the basis of plane or spherical waves, which is used to separate variables with the help of the Fourier transform or transition to spherical coordinates. Bibliography: 15 titles., INTRODUCTION The second quantization approach [1,2] has been the basic mechanism for constructing the quantum field theory since its inception in the first half of the 20th century. Later, in [...]

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing at the origin together with first derivatives is a symmetric operator with deficiency indices (3, 3). Krein's theory allows one to derive an expression for the resolvent kernel of a self-adjoint extension of the operator in question as a sum of the Green's function of the vector Laplace operator and some additional kernel of finite rank. Bibliography: 11 titles., UDC 517.98 INTRODUCTION In [2], it was shown that the three-dimensional solenoidal vector functions of three variables in spherical coordinates can be represented (parametrized) by two sets of radial functions [...]

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing at the origin together with first derivatives is a symmetric operator with deficiency indices (3, 3). Krein's theory allows one to derive an expression for the resolvent kernel of a self-adjoint extension of the operator in question as a sum of the Green's function of the vector Laplace operator and some additional kernel of finite rank. Bibliography: 11 titles., UDC 517.98 INTRODUCTION In [2], it was shown that the three-dimensional solenoidal vector functions of three variables in spherical coordinates can be represented (parametrized) by two sets of radial functions [...]

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing together with their first derivatives at selected points [x.sub.n], n = 1, ..., N, is a symmetric operator with deficiency indices (3N, 3N). The calculation of the scalar products of its regular analytic vectors is the key step in the construction of the resolvents of its self- adjoint extensions via Krein's formula. Bibliography: 11 titles., UDC 517.9 Dedicated to M. A. Semenov-Tian-Shansky on the occasion of his 70th birthday INTRODUCTION In [2], we calculated the scalar products of regular analytic vectors of the symmetric Laplace [...]

Determinant and higher-loop terms, usually treated by the Pauli-Villars and higher covariant derivatives methods, in the background field method can hardly be regularized simultaneously. At the same time, we observe that introducing a scalar multiplier in front of the quadratic form, which is equivalent to changing the measure in the functional integral, influences only the determinant part of the effective action. This allows one to choose the integration measure and the function in the regularized propagator in such a way as to make all terms in the expansion finite. Bibliography: 15 titles., UDC 517.9 Introduction Originally introduced in [1,2], the background field method significantly simplifies the calculation of the effective action and the [beta]-function in quantum field models. In the general case, [...]

We consider the set of closable homogeneous extensions of the quadratic form of the Laplace operator generated by interaction with two point-like sources. We show that this set consists of the trivial (maximal) extension, one point, and a subset equivalent to the Riemann sphere [S.sup.2]. Bibliography: 7 titles., UDC 517.9 Introduction Quadratic forms defined by the expression [mathematical expression not reproducible] (1) where {[x.sup.n]} are N points in general position in the space [R.sup.3], can be regarded as [...]

We consider the action of the quadratic form of the Laplace operator and its extensions in subspaces of linear combinations of the 'transverse ' and 'longitudinal' functions with the fixed orbital momentum with respect to the coordinate origin. In the statement of the problem, it is required that the extensions obtained, after the transfer back to the space of vector functions, can be represented as simple limit expressions with two coefficients. We study the behavior of these coefficients with respect to the initial choice of the linear subspace. Bibliography: 5 titles., UDC 517.9 INTRODUCTION We consider the behavior of quadratic forms of the type [mathematical expression not reproducible] (1) on vector functions [mathematical expression not reproducible] in the 3-dimensional space with [...]

We develop self-adjoint extensions of the radial part of the Laplace operator for l = 1 in a special scalar product. The product arises under the passage of the standard product from [R.sup.3] to the set of functions parametrizing one of two components of the transverse vector field. Similar extensions are treated for the square of the inverse operator of the radial part in question. Bibliography: 8 titles., UDC 517.9 Introduction The radial part of the Laplace operator [T.sub.l] = -[d.sup.2]/[dr.sup.2]+l(l+1)/[r.sup.2], r [greater than or equal to] 0, l = 0,1, ..., appears as a result of separation [...]

We study the quadratic form of the Laplace operator in 3 dimensions written in spherical coordinates and acting on transverse components of vector-functions. Operators which act on parametrizing functions of one of the transverse components with angular momentum 1 and 2 appear to be fourth-order symmetric operators with deficiency indices (1,1). We consider self-adjoint extensions of these operators and propose the corresponding extensions for the initial quadratic form. The relevant scalar product for angular momentum 2 differs from the original product in the space of vector-functions, but, nevertheless, it is still local in radial variable. Eigenfunctions of the operator extensions in question can be treated as stable soliton-like solutions of the corresponding dynamical system whose quadratic form is a functional of the potential energy. Bibliography: 7 titles., UDC 517.9 To Petr Petrovich Kulish on the occasion of his 70th anniversary INTRODUCTION The quadratic form of the Laplace operator acting on transverse components of a vector field [MATHEMATICAL [...]

We study algebraic properties of matrix operators which act on the torsion in the Einstein-Cart an action. Multiplication rules are written and the corresponding equations of motion are explored in different variables. Bibliography: 6 titles., UCD 517.9 INTRODUCTION The Hilbert-Einstein action for the gravitational field [1] is written in terms of the Riemannian connection [Γ.sub.λ,νμ]: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1) This connection (also [...]