The asymptotic behavior of the eigenvalues and eigenfunctions of the Laplace operator on a compact Riemannian manifold

The asymptotic properties of the spectrum of a Laplace operator on a Riemannian manifold are studied. New asymptotic formulas are derived for spectrum series, which are associated with stable geodesics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 448–453, March, 1972.The authors thank S. I. Al'ber for the statement of the problem and his interest in the work.     

On the kernel of the Laplace operator on two-dimensional polyhedra

The structure of spaces of harmonic functions on polyhedra is studied.     

On the expansion of diffracted electromagnetic fields in eigenfunctions of the D'Alembert operator

It is shown that under rather general assumptions on the character of the interaction of the field with an obstacle the scattering operator is invariant with respect to the eigenfunctions of the D'Alembert operator belonging to the zero eigenvalue. This situation makes it possible to write the field at points sufficiently far from the obstacle in the form of an expansion whose coefficients can be interpreted as matrix elements of the scattering operator in the representation of the eigenfunctions of the operator D. Conditions are obtained which are imposed on the coefficients by the law of energy conservation. A transition from integrals to sums is realized.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 29–34, July, 1972.     

On the continuity of generalized eigenfunctions of the Schrödinger operator

We prove that the generalized eigenfunctions of the Schrödinger operator are continuous for potentials obeing the following assumptions: V=V⁺?V^?,V^±≥0,$\text{V}{}^{\mathrm{+}}\text{∈L}{}^{\mathrm{p}}{}_{\mathrm{<mtext>loc</mtext>}}\text{(}\text{R}{}^{\mathrm{l}}\text{)}$, $\text{V}{}^{\mathrm{?}}\text{∈L}{}^{\mathrm{p}}\text{(}\text{R}{}^{\mathrm{l}}\text{)}$,$\text{p >}\text{l}\text{2}$.     

On the connection between the continuum eigenfunctions and the dispersion function of the vlasov operator

It is shown that a function may be extracted from the continuum eigenmode of the Vlasov operator which is related to the boundary values of the dispersion function on the real axis of the complex frequency plane. The precise content and possible consequences of this relationship are investigated in detail, especially the possibility of reconstructing the dispersion function and establishing stability criteria.     

Spectrum and eigenfunctions of the Frobenius-Perron operator of the tent map

The spectrum and eigenfunctions of the Frobenius-Perron operator induced by the tent map are discussed in detail. Special attention is paid to the case where the critical point of the map lies on an aperiodic trajectory and the differences from maps with a periodic critical trajectory are stressed. It is shown that the relevant eigenvalues of the spectrum are not sensitive to the aperiodicity of the critical trajectory. All other parts of the spectrum and all eigenfunctions in particular are changed drastically if the critical trajectory becomes aperiodic. The intimate connection between the point spectrum and the kneading invariant is established and the critical slowing down as well as the band splitting are a consequence of its properties. The structure of the infinite sequence of null spaces and its implications on the spectrum of the operator are discussed. It is shown that any initial distributionP(0,x) of bounded variation can be projected uniquely onto the eigenfunctions of the relevant eigenvalues and that the time dependence ofP(n, x) is determined by this expansion up to an errorO( ⁿ). From this the stationary and the asymptotic behavior of the correlation function x(n) x can be derived exactly.     

Feynman formulas generated by self-adjoint extensions of the Laplace operator

The Feynman formulas give a representation of a solution of the Cauchy problem for a Schrödinger-type equation (in a special case, for a heat-type equation) using the limit of integrals of finite multiplicity over Cartesian powers of the phase space (in the special case of the configuration space). The limit thus obtained, defining an explicit representation of a one-parameter unitary group e ^it? or a similar object (in our case, this concerns the semigroup e ^t? , which is often referred to in the literature as the Schrödinger semigroup) by integral operators, is interpreted by using Feynman integrals, whereas the expression thus obtained is referred in turn as the Feynman formula. As a rule, the Chernoff theorem, which is a generalization of the well known Trotter formula, is used in the derivation of the Feynman formula.In the paper, Feynman formulas for Schrödinger semigroups e ^t? are obtained, where the role of ? is played by the operator ? _a +V which is a perturbation of the self-adjoint extension of the Laplace operator (parametrized by some a ∈ (?∞, ∞]).     

Zeta function determinant of the Laplace operator on theD-dimensional ball

We present a direct approach for the calculation of functional determinants of the Laplace operator on balls. Dirichlet and Robin boundary conditions are considered. Using this approach, formulas for any value of the dimension,D, of the ball, can be obtained quite easily. Explicit results are presented here for dimensionsD=2,3,4,5 and 6.     

On the nodal behaviour of eigenfunctions

The generic behaviour of the nodal pattern of the eigenfunctions of the Schrödinger equation depending on parameters is discussed. Numerical examples are presented for the case of a hard-walled parallelogram.     

Superscaling of percolation on rectangular domains

For percolation on (RL)xL two-dimensional rectangular domains with a width L and aspect ratio R, we propose that the existence probability of the percolating cluster E(p)(L,epsilon,R) as a function of L, R, and deviation from the critical point epsilon can be expressed as F(epsilonL(y(t))R(a)), where y(t) identical with1/nu is the thermal scaling power, a is a new exponent, and F is a scaling function. We use Monte Carlo simulation of bond percolation on square lattices to test our proposal and find that it is well satisfied with a=0.14(1) for R>2. We also propose superscaling for other critical quantities.     

Correlations due to localization in quantum eigenfunctions of disordered microwave cavities

Statistical properties of experimental eigenfunctions of quantum chaotic and disordered microwave cavities are shown to demonstrate nonuniversal correlations due to localization. Varying energy E and mean free path l enable us to experimentally tune from localized to delocalized states. Large level-to-level inverse participation ratio ( I2) fluctuations are observed for the disordered billiards, whose distribution is strongly asymmetric about . The spatial density autocorrelations of eigenfunctions are shown to spatially decay exponentially and the decay lengths are experimentally determined. All the results are quantitatively consistent with calculations based upon nonlinear sigma models.     

On the second eigenfunctions of the Laplacian in R2

A conjecture about the nodal line of a second eigenfunction states that the nodal line of a second eigenfunction divides the domain by intersecting with the boundary of transversely, where is a bounded convex domain ofR ². We prove this conjecture provided has a symmetry. Also, we prove the multiplicity of the second eigenvalue is two at most provided is a bounded convex domain ofR ².Supported in part by NSF DMS 84-09447Home Institution: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA     

On the q-analog of the Laplace transform

In this paper, we consider the q-analog of the Laplace transform and investigate some properties of the q-Laplace transform. In our investigation, we derive some interesting formulas related to the q-Laplace transform.     

Efficient discretization of Laplace boundary integral equations on polygonal domains

We describe a numerical procedure for the construction of quadrature formulae suitable for the efficient discretization of boundary integral equations over very general curve segments. While the procedure has applications to the solution of boundary value problems on a wide class of complicated domains, we concentrate in this paper on a particularly simple case: the rapid solution of boundary value problems for Laplace’s equation on two-dimensional polygonal domains. We view this work as the first step toward the efficient solution of boundary value problems on very general singular domains in both two and three dimensions. The performance of the method is illustrated with several numerical examples.     

On completeness of eigenfunctions of the one-speed transport equation

It is shown that the set of Case's eigenfunctions of the one speed transport equation is complete in the rigged Hilbert spaceW ₂ ¹ ([?1, 1])?L ₂(?1, 1)?W ₂ ^?1 ([?1, 1]).     

Zeta function for the Laplace operator acting on forms in a ball with gauge boundary conditions

The Laplace operator acting on antisymmetric tensor fields in a D-dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions of the operator are found explicitly for all values of D. Using in a row a number of basic techniques, as Mellin transforms, deformation and shifting of the complex integration contour and pole compensation, the zeta function of the operator is obtained. From its expression, in particular, ζ(0) and ζ'(0) are evaluated exactly. A table is given in the paper for D=3,4,...,8. The functional determinants and Casimir energies are obtained for D=3,4,...,6. Received: 2 May 1996 / Accepted: 20 July 1996     

On the off-diagonal operator equivalents

A closed polynomial formula for the qth component of any off-diagonal operator equivalent of order k (integer or half-integer) is derived in terms of two-dimensional harmonic oscillator creation and annihilation operators.