Curvature and the Eigenvalues of the Laplacian

Abstract

A famous formula of H. Weyl [17] states that if D is a bounded region of R ^d with a piecewise smooth boundary B, and if 0 > γ ₁ ≥ γ ₂ ≥ γ ₃ ≥ etc. ↓−∞ is the spectrum of the problem

$$\displaystyle\begin{array}{rcl} \varDelta f =\big (\partial ^{2}/\partial x_{ 1}^{2} + \cdots + \partial ^{2}/\partial x_{ d}^{2}\big)f =\gamma f\quad \mbox{ in }D,& &{}\end{array}$$

(6.1.1a)

$$\displaystyle\begin{array}{rcl} f \in C^{2}(D) \cap C(\overline{D}),& &{}\end{array}$$

(6.1.1b)

$$\displaystyle\begin{array}{rcl} f = 0\quad \mbox{ on }B,& &{}\end{array}$$

(6.1.1c)

then

$$\displaystyle\begin{array}{rcl} -\gamma _{n} \sim C(d)(n/\mbox{ vol }D)^{2/d}\quad (n \uparrow \infty ),& &{}\end{array}$$

(6.1.2)

or, what is the same,

$$\displaystyle\begin{array}{rcl} Z \equiv \mathop{\mathrm{sp}}\nolimits e^{t\varDelta } =\sum _{ n\geq 1}\exp \big(\gamma _{n}t\big) \sim (4\pi t)^{-d/2} \times \mathop{\mathrm{vol}}\nolimits D\quad (t \downarrow 0),& &{}\end{array}$$

(6.1.3)

where $C(d) = 2\pi [(d/2)!]^{d/2}$.

Communicated April 6, 1967. The partial support of the National Science Foundation under NSF GP-4364 and NSF GP-6166 is gratefully acknowledged.