Quantum Mechanics

Harmonic Analysis and Semi-classical Bounds on the Number of Eigenvalues of Schrödinger Operators

Mathematically, bound states of quantum mechanical one-body systems correspond to negative eigenvalues of linear operators of the type \(T(-\mathrm{i}\nabla) + V(x)\), where \(T\) describes the kinetic energy of the system and \(V\) is the external potential in which the particle moves. For the one-particle Schrödinger operator \(-\Delta+V(x)\) with a real-valued potential \(V\), Cwikel, Lieb, and Rozenblum independently proved the bound $$ N(-\Delta+V) \leq L_{0,d} \int_{\mathbb{R}^d} V_-(x)^{d/2}\, \mathrm{d} x, \quad V_-(x) := \max\{-V(x), 0\}, $$ for the number of negative eigenvalues. This bound is a semi-classical bound since the right-hand side is a multiple of the classical phase-space volume of the region of negative total energy. The value of the CLR constant \(L_{0,d}\) has been of continued interest, because in a sense it describes by how much quantum and classical systems differ in terms of bound states. Using ideas from harmonic analysis we were able to connect this problem to maximal operator bounds on Fourier integral operators, and to establish an intriguing variational problem for the CLR constant. In dimensions greater or equal 5, this improves the so far best-known values due to Lieb , and is the first such improvement in about 40 years. Our method is flexible enough to apply also to the case of operator-valued potentials \(V\) and quite general weak trace-ideal estimates on operators if the type \(f(x) g(-\mathrm{i}\nabla)\).

Related publications

1. Cwikel's bound reloaded. Preprint arxiv:1809.05069, 2018.
   (with D. Hundertmark, P. Kunstmann, S. Vugalter)
2. Counting bound states with maximal Fourier multipliers. In Partial Differential Equations, Spectral Theory, and Mathematical Physics: The Ari Laptev Anniversary Volume (eds. P. Exner, R.L. Frank, F. Gesztesy, H. Holden, T. Weidl), EMS Series of Congress Reports Vol. 18, EMS Press, 2021, pp. 197–207.
   (with D. Hundertmark, P. Kunstmann, S. Vugalter)