arXiv subjects

I. G. Pirozhenko

Publications and source records attributed to I. G. Pirozhenko.

29 records · Page 2Linked to original sources

arXiv2005

Integral Equations for Heat Kernel in Compound Media

By making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the Laplace operator $-a^2Δ$ in the case of compound media. In each of the media the parameter $a^2$ acquires a certain constant value. At the interface of the media the conditions are imposed which demand the continuity of the `temperature' and the `heat flows'. The integration in the equations is spread out only over the interface of the media. As a result the dimension of the initial problem is reduced by 1. The perturbation series for the integral equations derived are nothing else as the multiple scattering expansions for the relevant heat kernels. Thus a rigorous derivation of these expansions is given. In the one dimensional case the integral equations at hand are solved explicitly (Abel equations) and the exact expressions for the regarding heat kernels are obtained for diverse matching conditions. Derivation of the asymptotic expansion of the integrated heat kernel for a compound media is considered by making use of the perturbation series for the integral equations obtained. The method proposed is also applicable to the configurations when the same medium is divided, by a smooth compact surface, into internal and external regions, or when only the region inside (or outside) this surface is considered with appropriate boundary conditions.

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arXiv2002

Nonsmoothness of the boundary and the relevant heat kernel coefficients

The contributions to the heat kernel coefficients generated by the corners of the boundary are studied. For this purpose the internal and external sectors of a wedge and a cone are considered. These sectors are obtained by introducing, inside the wedge, a cylindrical boundary. Transition to a cone is accomplished by identification of the wedge sides. The basic result of the paper is the calculation of the individual contributions to the heat kernel coefficients generated by the boundary singularities. In the course of this analysis certain patterns, that are followed by these contributions, are revealed. The implications of the obtained results in calculations of the vacuum energy for regions with nonsmooth boundary are discussed. The rules for obtaining all the heat kernel coefficients for the minus Laplace operator defined on a polygon or in its cylindrical generalization are formulated.

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arXiv2001

High Temperature Asymptotics in Terms of Heat Kernel Coefficients: Boundary Conditions with Spherical and Cylindrical Symmetries

The high temperature asymptotics of the Helmholtz free energy of electromagnetic field subjected to boundary conditions with spherical and cylindrical symmetries are constructed by making use of a general expansion in terms of heat kernel coefficients and the related determinant. For this, some new heat kernel coefficients and determinants had to be calculated for the boundary conditions under consideration. The obtained results reproduce all the asymptotics derived by other methods in the problems at hand and involve a few new terms in the high temperature expansions. An obvious merit of this approach is its universality and applicability to any boundary value problem correctly formulated.

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arXiv2001

High temperature asymptotics of thermodynamic functions of electromagnetic field subjected to boundary conditions on a sphere and cylinder

The high temperature asymptotics of thermodynamic functions of electromagnetic field subjected to boundary conditions with spherical and cylindrical symmetries are constructed by making use of a general expansion in terms of heat kernel coefficients and the related determinant. For this, some new heat kernel coefficients and determinants had to be calculated for the boundary conditions under consideration. The obtained results reproduce all the asymptotics derived by other methods in the problems at hand and involve a few new terms in the high temperature expansions. An obvious merit of this approach is its universality and applicability to any boundary value problem correctly formulated.

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arXiv2001

The heat kernel coefficients for the dielectric cylinder

We calculate the \hkks for the \elm field in the background of a dielectric cylinder with non equal speeds of light inside and outside. The coefficient $a_{2}$ whose vanishing makes the vacuum energy of a massless field unique, turns out to be zero in dilute order, i.e., in order $(\ep-1)^{2}$, and nonzero beyond. As a consequence, the vanishing of the vacuum energy in the presence of a dielectric cylinder found by Casimir-Polder summation must take place irrespectively of the methods by which it might be calculated.

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arXiv1999

Casimir energy of a compact cylinder under the condition $εμ= c^{-2}$

The Casimir energy of an infinite compact cylinder placed in a uniform unbounded medium is investigated under the continuity condition for the light velocity when crossing the interface. As a characteristic parameter in the problem the ratio $ξ^2=(ε_1-ε_2)^2/ (ε_1+ε_2)^-2 = (μ_1-μ_2)^2/(μ_1+ μ_2)^2 \le 1$ is used, where $ε_1$ and $μ_1$ are, respectively, the permittivity and permeability of the material making up the cylinder and $ε_2$ and $μ_2$ are those for the surrounding medium. It is shown that the expansion of the Casimir energy in powers of this parameter begins with the term proportional to $ξ^4$. The explicit formulas permitting us to find numerically the Casimir energy for any fixed value of $ξ^2$ are obtained. Unlike a compact ball with the same properties of the materials, the Casimir forces in the problem under consideration are attractive. The implication of the calculated Casimir energy in the flux tube model of confinement is briefly discussed.

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arXiv1999

Justification of the zeta function renormalization in rigid string model

A consistent procedure for regularization of divergences and for the subsequent renormalization of the string tension is proposed in the framework of the one-loop calculation of the interquark potential generated by the Polyakov-Kleinert string. In this way, a justification of the formal treatment of divergences by analytic continuation of the Riemann and Epstein-Hurwitz zeta functions is given. A spectral representation for the renormalized string energy at zero temperature is derived, which enables one to find the Casimir energy in this string model at nonzero temperature very easy.

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arXiv1998

Open rigid string with the Gauss-Bonnet term in action

The effect of the Gaussian curvature in the rigid string action on the interquark potential is investigated. The linearized equations of motion and boundary conditions, following from the modified string action, are obtained. The equation, defining the eigenfrequency spectrum of the string oscillations is derived. On this basis, the interquark potential, generated by the string is calculated in one-loop approximation. A substantial influence of the topological term in the string action on the interquark potential at the distances of hadronic size order or less is revealed.

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arXiv1998

Direct mode summation for the Casimir energy of a solid ball

The Casimir energy of a solid ball placed in an infinite medium is calculated by a direct frequency summation using the contour integration. It is assumed that the permittivity and permeability of the ball and medium satisfy the condition $ε_1 μ_1=ε_2μ_2$. Upon deriving the general expression for the Casimir energy, a dilute compact ball is considered $(ε_1 -ε_2)^2/(ε_1+ε_2)^2\ll 1$. In this case the calculations are carried out which are of the first order in $ξ^2$ and take account of the five terms in the Debye expansion of the Bessel functions involved. The implication of the obtained results to the attempts of explaining the sonoluminescence via the Casimir effect is shortly discussed.

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arXiv1998

Is the Casimir effect relevant to sonoluminescence?

The Casimir energy of a solid ball (or cavity in an infinite medium) is calculated by a direct frequency summation using the contour integration. The dispersion is taken into account, and the divergences are removed by making use of the zeta function technique. The Casimir energy of a dielectric ball (or cavity) turns out to be positive, it being increased when the radius of the ball decreases. The latter eliminates completely the possibility of explaining, via the Casimir effect, the sonoluminescence for bubbles in a liquid. Besides, the Casimir energy of the air bubbles in water proves to be immensely smaller than the amount of the energy emitted in a sonoluminescent flash. The dispersive effect is shown to be inessential for the final result.

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arXiv1997

Simple method for calculating the Casimir energy for sphere

A simple method for calculating the Casimir energy for a sphere is developed which is based on a direct mode summation and counter integration in a complex plane of eigenfrequencies. The method uses only classical equations determining the eigenfrequencies of the quantum field under consideration. Efficiency of this approach is demonstrated by calculation of the Casimir energy for a perfectly conducting spherical shell and for a massless scalar field obeying the Dirichlet and Neumann boundary conditions on sphere. The possibility of rationalizing the removal of divergences in this problem as a renormalization of both the energy and the radius of the sphere is discussed.

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