This book reviews the recent progress in the calculation of the asymptotic
distribution of the eigenvalues for the wave equation in the limit of large
wavenumbers (Weyl's problem). It presents both the analytical and com-
putational methods and, particularly, smoothing procedures. The authors
consider a variety of boundary conditions and the corresponding applica-
tions in the physics of finite systems-acoustics of finite resonators and
elastic vibrations of finite samples, perfect quantum gases in finite en-
closures, nuclei in terms of the independent particle model, electro-
magnetism and black-body radiation in finite cavities, together with the
corresponding correlation functions, as well as size-, surface-, and shape-
effects in various branches of the physics of condensed matter. The
validity of the concept of the density of states in statistical mechanics
and of the underlying asymptotic expansions for finite eigenvalue regimes
is discussed.
Author(s): Heinrich P. Baltes
Edition: 1
Publisher: Bibliographisches Institut Mannheim
Year: 1976
Language: English
Pages: 112
City: Zurich
1. Introduction 11
iI. The Scalar, the Electromagnetic, and the Elastic Boundary Value Problems. and Weyl's Results. I3
I. The Scalar Problem 13
2. The Electromagnetic Problem 15
3. The Elastic Problem. 16
4. Some Pre-Weylian Results. 18
IiI. Mathematical Methods and General D-Results 19
1. The Methods of WeyI and Courant 19
2. The CarIeman Procedures 20
3. Refinements of the Asymptotic Laws 23
(a) The scalar problem. 23
(b) The electromagnetic problem 24
(c) The elastic problem 25
(d) Summary. 25
IV. Mathematical Methods Continued: Averaging Procedures. 27
I. Fluctuations and Average Mode Density 27
2. Analytical Averaging Procedures. 30
(a) Logarithmic Gaussian 30
(b) Lorentzian 32
(c) Integral transforms. 34
(0:) Fedosov's transform. 34
ai) Lambert transform 34
(r) Wentzel's method 35
(0) Pre- Tauberian results 36
3. Computational Methods 36
(3) Computation of the eigenvalues 37
(0:) The rectangular parallelepiped 37
({.i) The circular cylinder 37
(r) The sphere. 39
(b) Computational averaging. 39
(c) Computational versus analytical methods 40
v. 0- and a-Results for the Square and the Cube 42
1. Lattice-Type Spectra 42
2. The Square and the Rectangular Domain 43
(a) Eigenvalues and eigenfunctions for Dirichlet, Neumann, mixed, and general boundary conditions 43
(b) O-results and the circle problem 44
(c) O-results . 47
3. The Cube-Shaped Domain. 47
(a) The solutions of the scalar and the electromagnetic problems and their connection with the Iattice- point problem 47
(b) Q-results and the sphere problem 49
(c) O-results 51
(a) Brownell's average 51
(jJ) The Gaussian ensemble average 52
('y) The Lambert transform and the Poisson sum formula 52
(d) Computer results for the cube and the flat box 54
(a) Dirichlet and Neumann boundary conditions 54
(P) The electromagnetic problem. 55
(e) The number 130 and the mean level-spacing 56
VI. O-Results for General Domains 59
I. Two-Dimensional Domains (the Membrane Problem) 59
(a) Brownell's results 59
(b) Corners and cusps 60
(c) The circle and general smooth boundary curves 62
(d) On hearing the shape of a drum 64
2. Three-Dimensional Domains (the Resonant Cavity) 65
(a) Surface contributions. 65
(a) The scalar problem (Dirichlet, Neumann, and general boundary conditions) 65
(f3) The electromagnetic problem and the vanishing of the surface term. 66
(r) The elastic problem (surface specific heat and mode scrambling) 67
(b) Edge and curvature terms 68
(a) The scalar problem (smooth boundary surface, polyhedra, and general cylinders) 68
(jJ) The electromagnetic pro blem (general cylinders and the sphere) 71
(c) The constant term (comers and connectivity) 73
(d) Scalar versus electromagnetic 74
(e) The oscillations 75
VII. Physical Applications 77
I. Resonator Acoustics. 77
2. Perfect Gases 78
3. Nuclei 81
4. Black-Body Radiation 84
5. Correlation Functions 88
6. Condensed Matter 91
(a) The Bose-Einstein condensation in thin films 91
(b) The specific heat of grains at low temperatures 93
VIII. Some Open Problems and Extensions. 98
I. 0- Results for the Elastic Pro blem 98
2. The Schrodinger Wave Equation with Non-Rigid Walls 98
3. The Black-Body Cavity with Absorbing Walls 99
4. Inhomogeneous Media 100
5. The Inverse Problems 100
Post-Deadline Note 102
References 105
Index 113
