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How could I compute the integral

$$ N(E)~=~ \int dx \int dp~ H(E-xp) $$

the 'Area' inside the Phase space is taken for $ x \ge 0 $ and $ p\ge 0 $? The result should be

$$ N(E)~=~ \frac{E}{2\pi}\log\frac{E}{2\pi}-\frac{E}{2\pi} ,$$

but I am unable to get it even with cut-offs :( What am I doing wrong to compute this? Here $H(x)$ is the Heaviside function. At least I would need help to get rid of the term involving the Heaviside function $H(E-xp)$ of the quadrant where both $x$ and $p$ are positive.

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You are basically asking for the area enclosed between the rectangular hyperbola $xp=E$ and its asymptotes (which happen to be the $x$ and $p$ axes). What are the limits on $x$ and $p$? –  Vijay Murthy Apr 23 '12 at 18:08
    
Shouldn't H be Hermitian? xp is not! –  valdo Apr 24 '12 at 11:44
    
H is Hermitian, remember weyl ordering the quantum operator reads $ H= \frac{xp+px}{2}$ and $ x^{T}p^{T}=px$ –  Jose Javier Garcia Apr 24 '12 at 11:54
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1 Answer

This question (v1) is discussed near eq. (8) in Ref. 1.

The simplest regularization is to truncate the variables $x\geq\ell_x$ and $p\geq\ell_p$ at cut-offs $\ell_x$ and $\ell_p$, respectively, in such a way that the product $\ell_x \ell_p = h$ is Planck's constant.

In an $(x,p)$ diagram, the truncated area under the hyperbola $p=\frac{E}{x}$ reads in Planck units

$$ N(E) ~\approx~\int_{\ell_x}^{\infty}\frac{dx}{h}\int_{\ell_p}^{\infty}dp~\theta(E-xp) ~=~\int_{\ell_x}^{\frac{E}{\ell_p}}\frac{dx}{h} \left(\frac{E}{x}-\ell_p\right) $$ $$~=~\frac{1}{h} \left[E\ln x-\ell_p x\right]_{x=\ell_x}^{x=\frac{E}{\ell_p}}~=~\frac{E}{h}\left(\ln\frac{E}{h}-1\right)+1,$$

where $\theta$ is the Heaviside step function.

References:

  1. M.V. Berry and J.P. Keating, H = xp and the Riemann zeros. In J.P. Keating, D.E. Khmelnitski and I.V. Lerner, Supersymmetry and Trace Formulae: Chaos and Disorder (1999) 355–367. The pdf file is available here.
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