The eigenvalue sum for a three-dimensional radial potential. (English) Zbl 0845.34086
Let \(H= -\Delta+ V\) be a Schrödinger operator on \(\mathbb{R}^3\), and let \(E_k\), \(\varphi_k(x)\) be the eigenvalues and (normalized) eigenfunctions of \(H\). We will study the eigenvalue sum and density, defined by
\[
\text{sneg}(H)= \sum_{E_k\leq 0} E_k,\tag{1}
\]
\[ \rho(x)= \sum_{E_k\leq 0} |\varphi_k(x)|^2\qquad (x\in \mathbb{R}^3).\tag{2} \] The standard “semiclassical approximations” to these quantities are \[ \text{sneg}(H)\approx- {1\over 15\pi^2} \int_{\mathbb{R}^3} (- V(x))^{5/2}_+ dx,\tag{3} \] and \[ \rho(x)\approx {1\over 6\pi^{22}} (- V(x))^{3/2}_+.\tag{4} \] Here, \(t^s_+= t^s\) if \(t> 0\), \(t^s_+= 0\) if \(t\leq 0\).
In [Bull. Am. Math. Soc., New Ser. 23, No. 2, 525-530 (1990; Zbl 0722.35072)], we announced the proof of a precise asymptotic formula for the ground-state energy of a non-relativistic atom. To give the proof, one must understand and refine (3) and (4) for a particular radial potential \(V^Z_{TF}\), the Thomas-Fermi potential for an atomic number \(Z\). In [Adv. Math. 107, No. 1, 1-185 (1994; Zbl 0822.35013)], we reduced the asymptotic formula of [FS1] to the task of proving that \[ \text{sneg}(H)= - {1\over 15\pi^2} \int_{\mathbb{R}^3} (- V(x))^{5/2}_+ dx+ {Z^2\over 8}+ {1\over 48\pi^2} \int_{\mathbb{R}^3} (- V(x))^{1/2}_+ \Delta V dx+ O(Z^{5/3- \alpha}),\tag{5} \] and \[ \int_{\mathbb{R}^3\times \mathbb{R}^3} \int\Biggl[ \rho(x)- {1\over 6\pi^2} (- V(x))^{3/2}_+\Biggr]\times\Biggl[ \rho(y)- {1\over 6\pi^2} (- V(y))^{3/2}_+\Biggr] {dx dy\over |x- y|}= O(Z^{5/3- \alpha})\tag{6} \] for \(V^Z_{TF}\), with \(\alpha> 0\) independent of \(Z\). The purpose of this paper is to prove (5) and (6), with \(a= {1\over 150}\), for a class of radial potentials that includes \(V^Z_{TF}\). This completes the proof of the results announced in [FS1].
Our proof of (5) and (6) is based on separation of variables. In [Adv. Math. 95, No. 2, 145-305 (1992; Zbl 0797.34083); ibid. 107, No. 2, 187-364 (1994; Zbl 0811.34063); ibid. 108, No. 2, 263-335 (1994; Zbl 0826.34070)], we made a careful study of ordinary differential operators. In [FS2] [ibid. 111, No. 1, 88-161 (1995; Zbl 0826.34071)], we used our ODE results to prove (6) for radial potentials \(V\) that satisfy a “non-resonance condition”. The non-resonance condition is related to the scarcity of periodic orbits of a classical particle in the potential \(V\). Here, we again use our results on ODE to show that (5) holds also, provided \(V\) satisfies another non-resonance condition, similar to that of [FS2]. Then we will show that the non-resonance conditions hold for a class of radial potentials including \(V^Z_{TF}\). Our proof of the non-resonance condition uses elementary number theory, together with an inequality for the Thomas-Fermi potential proved in [Rev. Math. Iberoam. 9, No. 3, 409-551 (1993; Zbl 0788.34004)].
\[ \rho(x)= \sum_{E_k\leq 0} |\varphi_k(x)|^2\qquad (x\in \mathbb{R}^3).\tag{2} \] The standard “semiclassical approximations” to these quantities are \[ \text{sneg}(H)\approx- {1\over 15\pi^2} \int_{\mathbb{R}^3} (- V(x))^{5/2}_+ dx,\tag{3} \] and \[ \rho(x)\approx {1\over 6\pi^{22}} (- V(x))^{3/2}_+.\tag{4} \] Here, \(t^s_+= t^s\) if \(t> 0\), \(t^s_+= 0\) if \(t\leq 0\).
In [Bull. Am. Math. Soc., New Ser. 23, No. 2, 525-530 (1990; Zbl 0722.35072)], we announced the proof of a precise asymptotic formula for the ground-state energy of a non-relativistic atom. To give the proof, one must understand and refine (3) and (4) for a particular radial potential \(V^Z_{TF}\), the Thomas-Fermi potential for an atomic number \(Z\). In [Adv. Math. 107, No. 1, 1-185 (1994; Zbl 0822.35013)], we reduced the asymptotic formula of [FS1] to the task of proving that \[ \text{sneg}(H)= - {1\over 15\pi^2} \int_{\mathbb{R}^3} (- V(x))^{5/2}_+ dx+ {Z^2\over 8}+ {1\over 48\pi^2} \int_{\mathbb{R}^3} (- V(x))^{1/2}_+ \Delta V dx+ O(Z^{5/3- \alpha}),\tag{5} \] and \[ \int_{\mathbb{R}^3\times \mathbb{R}^3} \int\Biggl[ \rho(x)- {1\over 6\pi^2} (- V(x))^{3/2}_+\Biggr]\times\Biggl[ \rho(y)- {1\over 6\pi^2} (- V(y))^{3/2}_+\Biggr] {dx dy\over |x- y|}= O(Z^{5/3- \alpha})\tag{6} \] for \(V^Z_{TF}\), with \(\alpha> 0\) independent of \(Z\). The purpose of this paper is to prove (5) and (6), with \(a= {1\over 150}\), for a class of radial potentials that includes \(V^Z_{TF}\). This completes the proof of the results announced in [FS1].
Our proof of (5) and (6) is based on separation of variables. In [Adv. Math. 95, No. 2, 145-305 (1992; Zbl 0797.34083); ibid. 107, No. 2, 187-364 (1994; Zbl 0811.34063); ibid. 108, No. 2, 263-335 (1994; Zbl 0826.34070)], we made a careful study of ordinary differential operators. In [FS2] [ibid. 111, No. 1, 88-161 (1995; Zbl 0826.34071)], we used our ODE results to prove (6) for radial potentials \(V\) that satisfy a “non-resonance condition”. The non-resonance condition is related to the scarcity of periodic orbits of a classical particle in the potential \(V\). Here, we again use our results on ODE to show that (5) holds also, provided \(V\) satisfies another non-resonance condition, similar to that of [FS2]. Then we will show that the non-resonance conditions hold for a class of radial potentials including \(V^Z_{TF}\). Our proof of the non-resonance condition uses elementary number theory, together with an inequality for the Thomas-Fermi potential proved in [Rev. Math. Iberoam. 9, No. 3, 409-551 (1993; Zbl 0788.34004)].
MSC:
| 34L05 | General spectral theory of ordinary differential operators |
| 81V45 | Atomic physics |
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
