On the harmonic measure of stable processes

Using three hypergeometric identities, we evaluate the harmonic measure of a finite interval and of its complementary for a strictly stable real L{\'e}vy process. This gives a simple and unified proof of several results in the literature, old and recent. We also provide a full description of the corresponding Green functions. As a by-product, we compute the hitting probabilities of points and describe the non-negative harmonic functions for the stable process killed outside a finite interval

One-dimensional quasi-relativistic particle in the box

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional quasi-relativistic Hamiltonian (-h^2 c^2 d^2/dx^2 + m^2 c^4)^(1/2) + V_well(x) (the Klein-Gordon square-root operator with electrostatic potential) with the infinite square well potential V_well(x) is given: the n-th eigenvalue is equal to (n pi/2 - pi/8) h c/a + O(1/n), where 2a is the width of the potential well. Simplicity of eigenvalues is proved. Some L^2 and L^infinity properties of eigenfunctions are also studied. Eigenvalues represent energies of a `massive particle in the box' quasi-relativistic model.Comment: 40 pages, 4 figures; minor correction

Quantization of the elastic modes in an isotropic plate

We quantize the elastic modes in a plate. For this, we find a complete, orthogonal set of eigenfunctions of the elastic equations and we normalize them. These are the phonon modes in the plate and their specific forms and dispersion relations are manifested in low temperature experiments in ultra-thin membranes.Comment: 14 pages, 2 figure

Solving the difference initial-boundary value problems by the operator exponential method

We suggest a modification of the operator exponential method for the numerical solving the difference linear initial boundary value problems. The scheme is based on the representation of the difference operator for given boundary conditions as the perturbation of the same operator for periodic ones. We analyze the error, stability and efficiency of the scheme for a model example of the one-dimensional operator of second difference

Critical strength of attractive central potentials

We obtain several sequences of necessary and sufficient conditions for the existence of bound states applicable to attractive (purely negative) central potentials. These conditions yields several sequences of upper and lower limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant (strength), $g$, of the potential, $V(r)=-g v(r)$, for which a first $\ell$-wave bound state appears, which converges to the exact critical value.Comment: 18 page

Third post-Newtonian dynamics of compact binaries: Noetherian conserved quantities and equivalence between the harmonic-coordinate and ADM-Hamiltonian formalisms

A Lagrangian from which derive the third post-Newtonian (3PN) equations of motion of compact binaries (neglecting the radiation reaction damping) is obtained. The 3PN equations of motion were computed previously by Blanchet and Faye in harmonic coordinates. The Lagrangian depends on the harmonic-coordinate positions, velocities and accelerations of the two bodies. At the 3PN order, the appearance of one undetermined physical parameter \lambda reflects an incompleteness of the point-mass regularization used when deriving the equations of motion. In addition the Lagrangian involves two unphysical (gauge-dependent) constants r'_1 and r'_2 parametrizing some logarithmic terms. The expressions of the ten Noetherian conserved quantities, associated with the invariance of the Lagrangian under the Poincar\'e group, are computed. By performing an infinitesimal ``contact'' transformation of the motion, we prove that the 3PN harmonic-coordinate Lagrangian is physically equivalent to the 3PN Arnowitt-Deser-Misner Hamiltonian obtained recently by Damour, Jaranowski and Sch\"afer.Comment: 30 pages, to appear in Classical and Quantum Gravit

The antiferromagnetic phi4 Model, II. The one-loop renormalization

It is shown that the four dimensional antiferromagnetic lattice phi4 model has the usual non-asymptotically free scaling law in the UV regime around the chiral symmetrical critical point. The theory describes a scalar and a pseudoscalar particle. A continuum effective theory is derived for low energies. A possibility of constructing a model with a single chiral boson is mentioned.Comment: To appear in Phys. Rev.

Neumark Operators and Sharp Reconstructions, the finite dimensional case

A commutative POV measure $F$ with real spectrum is characterized by the existence of a PV measure $E$ (the sharp reconstruction of $F$) with real spectrum such that $F$ can be interpreted as a randomization of $E$. This paper focuses on the relationships between this characterization of commutative POV measures and Neumark's extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of $F$ and the sharp reconstruction of $F$. The relevance of this result to the theory of non-ideal quantum measurement and to the definition of unsharpness is analyzed.Comment: 37 page

Dimensional regularization of the third post-Newtonian dynamics of point particles in harmonic coordinates

Dimensional regularization is used to derive the equations of motion of two point masses in harmonic coordinates. At the third post-Newtonian (3PN) approximation, it is found that the dimensionally regularized equations of motion contain a pole part [proportional to 1/(d-3)] which diverges as the space dimension d tends to 3. It is proven that the pole part can be renormalized away by introducing suitable shifts of the two world-lines representing the point masses, and that the same shifts renormalize away the pole part of the "bulk" metric tensor g_munu(x). The ensuing, finite renormalized equations of motion are then found to belong to the general parametric equations of motion derived by an extended Hadamard regularization method, and to uniquely determine the heretofore unknown 3PN parameter lambda to be: lambda = - 1987/3080. This value is fully consistent with the recent determination of the equivalent 3PN static ambiguity parameter, omega_s = 0, by a dimensional-regularization derivation of the Hamiltonian in Arnowitt-Deser-Misner coordinates. Our work provides a new, powerful check of the consistency of the dimensional regularization method within the context of the classical gravitational interaction of point particles.Comment: 82 pages, LaTeX 2e, REVTeX 4, 8 PostScript figures, minor changes to reflect Phys. Rev. D versio

Dimensional regularization of the gravitational interaction of point masses

We show how to use dimensional regularization to determine, within the Arnowitt-Deser-Misner canonical formalism, the reduced Hamiltonian describing the dynamics of two gravitationally interacting point masses. Implementing, at the third post-Newtonian (3PN) accuracy, our procedure we find that dimensional continuation yields a finite, unambiguous (no pole part) 3PN Hamiltonian which uniquely determines the heretofore ambiguous ``static'' parameter: namely, $\omega_s=0$. Our work also provides a remarkable check of the perturbative consistency (compatibility with gauge symmetry) of dimensional continuation through a direct calculation of the ``kinetic'' parameter $\omega_k$, giving the unique answer compatible with global Poincar\'e invariance ($\omega_k={41/24}$) by summing $\sim50$ different dimensionally continued contributions.Comment: REVTeX, 8 pages, 1 figure; submitted to Phys. Lett.