scholarly journals High orders of Weyl series for the heat content

2011 ◽  
Vol 467 (2133) ◽  
pp. 2479-2499 ◽  
Author(s):  
Igor Travěnec ◽  
Ladislav Šamaj
Keyword(s):  
Asymptotic Form ◽  
Smooth Boundary ◽  
Heat Content ◽  
Dirichlet Laplacian ◽  
Spectral Functions ◽  
Circular Symmetry ◽  
Exactly Solvable ◽  
Elliptic Domain ◽  
Dimensional Domain ◽  
Odd Dimensions

This article concerns the Weyl series of spectral functions associated with the Dirichlet Laplacian in a d -dimensional domain with a smooth boundary. In the case of the heat kernel, Berry and Howls predicted the asymptotic form of the Weyl series characterized by a set of parameters. Here, we concentrate on another spectral function, the (normalized) heat content. We show on several exactly solvable examples that, for even d , the same asymptotic formula is valid with different values of the parameters. The considered domains are d -dimensional balls and two limiting cases of the elliptic domain with eccentricity ε : a slightly deformed disk ( ε →0) and an extremely prolonged ellipse ( ε →1). These cases include two-dimensional domains with circular symmetry and those with only one shortest periodic orbit for the classical billiard. We also analyse the heat content for the balls in odd dimensions d for which the asymptotic form of the Weyl series changes significantly.

A power moment reformulation of the Nikiforov–Uvarov method for exactly solvable systems

2016 ◽  
Vol 94 (4) ◽  
pp. 410-424
Author(s):  
Carlos R. Handy ◽  
Daniel Vrinceanu
Keyword(s):  
Closed Form ◽  
Discrete Spectrum ◽  
Asymptotic Form ◽  
Moment Equation ◽  
Wave Functions ◽  
Moment Equations ◽  
Power Moment ◽  
Exactly Solvable ◽  
Uvarov Method ◽  
Polynomial Expansions

Exactly solvable (ES) systems are those for which the full, discrete spectrum can be solved in closed form. In this work, we argue that a moment’s representation analysis can generate these closed-form expressions for the energy in a more direct and transparent manner than the popular Nikiforov–Uvarov (NU) procedure. NU analysis strips the asymptotic form of the physical states. We retain these to generate appropriate moment equations. We show how the form of these moment equations leads to closed-form energy expressions. The wave functions can then be generated as well. Our analysis is extendable to quasi-exactly solvable systems (QES; those for which a subset of the discrete spectrum can be generated in closed form). Two formulations are presented. One of these affirms that a previously developed, general, moment quantization procedure is exact for ES and QES states. This method is referred to as the orthogonal polynomial projection quantization method. It combines moment equation representations for physical states with weighted polynomial expansions (Handy and Vrinceanu. J. Phys. A: Math. Theor. 46, 135202 (2013). doi:10.1088/1751-8113/46/13/135202 ). We also show that in implementing any numerical search procedure to determine the quantum parameter regimes corresponding to ES or QES states, our procedure is more reliable (i.e., numerically stable) than using a Hill determinant formulation. We develop our formalism, demonstrate its effectiveness, and prove its equivalence to the NU approach for ES systems.

scholarly journals SURVIVAL PROBABILITY (HEAT CONTENT) AND THE LOWEST EIGENVALUE OF DIRICHLET LAPLACIAN

2011 ◽  
Vol 25 (15) ◽  
pp. 1993-2007
Author(s):  
PAVOL KALINAY ◽  
LADISLAV ŠAMAJ ◽  
IGOR TRAVĚNEC
Keyword(s):  
Survival Probability ◽  
Heat Content ◽  
Simple Algorithm ◽  
Time Behavior ◽  
Dirichlet Laplacian ◽  
Geometric Characteristics ◽  
Long Time ◽  
Time Expansion ◽  
Short Time ◽  
Lowest Eigenvalue

We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this expansion, we look for a Padé interpolation between the short-time and the long-time behavior of the survival probability, i.e., between geometric characteristics of the boundary and the lowest eigenvalue of the Dirichlet Laplacian.

An exactly solvable two component classical Coulomb system

1984 ◽  
Vol 26 (2) ◽  
pp. 119-128 ◽  
Author(s):  
Peter J. Forrester
Keyword(s):  
Coulomb Potential ◽  
Correlation Functions ◽  
Asymptotic Form ◽  
Coulomb System ◽  
Two Dimensional ◽  
Coupling Constant ◽  
Sum Rule ◽  
Exactly Solvable ◽  
Two Component ◽  
Special Value

AbstractA two component classical Coulomb system is considered, in which particles of charge +q and + 2q are constrained to lie on a circle and interact via the two-dimensional Coulomb potential. At a special value of the coupling constant the correlation functions are calculated exactly and the asymptotic form of the truncated charge-charge correlation is found to obey Jancovici's sum rule.

LAYERED SOLUTIONS WITH CONCENTRATION ON LINES IN THREE-DIMENSIONAL DOMAINS

2014 ◽  
Vol 12 (02) ◽  
pp. 161-194 ◽  
Author(s):  
WEIWEI AO ◽  
JUN YANG
Keyword(s):  
Small Parameter ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Smooth Boundary ◽  
Three Dimensional ◽  
Singularly Perturbed ◽  
Existence Of A Solution ◽  
Straight Line ◽  
Dimensional Domain ◽  
Exponentially Small

We consider the following singularly perturbed elliptic problem [Formula: see text] where Ω is a bounded domain in ℝ3with smooth boundary, ε is a small parameter, 1 < p < ∞, ν is the outward normal of ∂Ω. We employ techniques already developed in [39] to extend their result to three-dimensional domain. More precisely, let Γ be a straight line intersecting orthogonally with ∂Ω at exactly two points and satisfying a non-degenerate condition. We establish the existence of a solution uεconcentrating along a curve [Formula: see text] near Γ, exponentially small in ε at any positive distance from the curve, provided ε is small and away from certain critical numbers. The concentrating curve [Formula: see text] will collapse to Γ as ε → 0.

The heat content asymptotics of a time-dependent process

2000 ◽  
Vol 130 (2) ◽  
pp. 307-312 ◽  
Author(s):  
M. van den Berg ◽  
P. Gilkey
Keyword(s):  
Heat Source ◽  
Boundary Conditions ◽  
Compact Manifold ◽  
Smooth Boundary ◽  
Heat Content ◽  
Time Dependent ◽  
Time Dependent Process ◽  
Dependent Process ◽  
Heat Content Asymptotics

Let M be a compact manifold with smooth boundary. We study the heat content asymptotics on M defined by a time-dependent heat source and time-dependent boundary conditions.

HEAT CONTENT ASYMPTOTICS OF AN INHOMOGENEOUS TIME-DEPENDENT PROCESSES

2000 ◽  
Vol 15 (18) ◽  
pp. 1165-1179 ◽  
Author(s):  
PETER GILKEY ◽  
JEONG HYEONG PARK
Keyword(s):  
Heat Source ◽  
Boundary Conditions ◽  
Specific Heat ◽  
Compact Manifold ◽  
Smooth Boundary ◽  
Heat Content ◽  
Time Dependent ◽  
Heat Content Asymptotics ◽  
Dependent Processes

Let M be a compact manifold with smooth boundary. We study the heat content asymptotics on M which are defined by a time-dependent metric gt, by a time-dependent heat source, by time-dependent boundary conditions, and by a time-dependent specific heat.

The asymptotic form of the spectral functions associated with a class of Sturm–Liouville equations

1985 ◽  
Vol 100 (3-4) ◽  
pp. 343-360 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Inversion Formula ◽  
Asymptotic Form ◽  
Asymptotic Expression ◽  
Spectral Functions ◽  
Sturm Liouville

SynopsisWe use a modified form of the Stieltjes inversion formula due to Atkinson together with some recent results on the asymptotic form of Titchmarch–Weyl m-functions to obtain an asymptotic expression for the spectral functions τ(t) associated with the differential equation

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