Celebratio Mathematica

I. Daubechies and E. H. Lieb: “One-electron relativistic molecules with Coulomb interaction,” Comm. Math. Phys. 90 : 4 (1983), pp. 497–510. MR 719430 Zbl 0946.81522 article

As an approximation to a relativistic one-electron molecule, we study the operator \[ H = (-\Delta + m^2)^{1/2} - e^2\sum_{j=1}^K Z_j|x-R_j|^{-1} \] with \( Z_j\geq 0 \), \( e^{-2} = 137.04 \). \( H \) is bounded below if and only if \[ e^2Z_j \leq \frac{2}{\pi} ,\] all \( j \). Assuming this condition, the system is unstable when \[ e^2\sum Z_j > \frac{2}{\pi} \] in the sense that \[ E_0 = \inf\operatorname{spec}(H)\to -\infty \] as the \( R_j\to 0 \), all \( j \). We prove that the nuclear Coulomb repulsion more than restores stability; namely \[ E_0+0.069\,e^2\sum_{i < j}Z_iZ_j\,|R_i-R_j|^{-1} \geq 0 .\] We also show that \( E_0 \) is an increasing function of the internuclear distances \( |R_i-R_j| \).

I. Daubechies and E. H. Lieb: “Relativistic molecules with Coulomb interaction,” pp. 143–148 in Differential equations (Birmingham, AL, 21–26 March 1983). Edited by I. W. Knowles and R. T. Lewis. North-Holland Mathematics Studies 92. North-Holland (Amsterdam), 1984. MR 799344 Zbl 0565.35101 incollection

Ian Walker Knowles	Related
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Roger Timothy Lewis	Related

I. Daubechies and J. C. Lagarias: “On the thermodynamic formalism for multifractal functions,” pp. 1033–1070 in Special issue dedicated to Elliott H. Lieb, published as Rev. Math. Phys. 6 : 5A. Issue edited by M. Aizenman and H. Araki. World Scientific (Singapore), 1994. Also published in The state of matter (1994). MR 1301365 Zbl 0843.58091 incollection

The thermodynamic formalism for “multifractal” functions \( \phi(x) \) is a heuristic principle that states that the singularity spectrum \( f(\alpha) \) (defined as the Hausdorff dimension of the set \( S_{\alpha} \) of points where \( \phi \) has Hölder exponent \( \alpha \)) and the moment scaling exponent \( \tau(q) \) (giving the power law behavior of \[ \int|\phi(x+t)-\phi(x)|^q\,dx \] for small \( |t| \)) should be related by the Legendre transform, \[ \tau(q) = 1 + \inf_{\alpha\geq 0}[q\alpha-f(\alpha)] .\] The range of validity of this heuristic principle is unknown. Here this principle is rigorously verified for a family of “toy examples” that are solutions of refinement equations. These example functions exhibit oscillations on all scales, and correspond to multifractal signed measures rather than multifractal measures; moreover, their singularity spectra \( f(\alpha) \) are not concave.

Jeffrey Clark Lagarias	Related
Michael Aizenman	Related
Elliott Hershel Lieb	Related
Huzihiro Araki	Related

I. Daubechies and J. C. Lagarias: “On the thermodynamic formalism for multifractal functions,” pp. 213–264 in The state of matter: A volume dedicated to E. H. Lieb (Copenhagen, July 1992). Edited by M. Aizenman and H. Araki. Advanced Series in Mathematical Physics 20. World Scientific (Singapore), 1994. Also published in Rev. Math. Phys. 6:5A (1994). MR 1462145 incollection

Jeffrey Clark Lagarias	Related
Michael Aizenman	Related
Elliott Hershel Lieb	Related
Huzihiro Araki	Related

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