Exponential Decay of Quantum Wave Functions

I’ve no doubt that for ODEs, the questions and techniques for exponential decay of solutions go back a long way, maybe even to the nineteenth century. For one body systems with decaying potential, I think they are at least in Titchmarsh’s book. For ODEs with explicit polynomially growing potentials there are precise asymptotics going back to the at least the middle of the twentieth century. My work concerns N–body systems and qualitative polynomially growth where instead of exact asymptotics on the potential one supposes lower bounds on the growth. Slaggie-Wichmann [45] used integral equation ideas to prove some kind of exponential decay in certain three body systems with decaying potentials. I gave looking at N–body systems to Tony O’Connor, my first graduate student (who began working with me when I was a first year instructor). He had the idea of looking at analyticity of the Fourier transform and obtains results in the L2 sense (i.e. ea|x|ψ ∈ L2) that were optimal in that you couldn’t do better in terms of isotropic decay. Here |x| is a mass weighted measure of the spread of the N particles, explicitly if X = ∑N j=1mjxj/ ∑N j=1mj is the center of mass, then |x|2 = ∑N j=1mj(xj −X)2/ ∑N j=1mj . His paper [33] motivated Combes–Thomas [15] to an approach that has now become standard of using boost analyticity. Independently of O’Connor, Ahlrichs [4] found pointwise isotropic bounds but his result was not optimal and restricted to Coulomb systems since he used the explicit |r|−1 form. All these results, except Ahlrichs, obtained L2–decay. In three papers [37, 38, 39], I looked at getting pointwise bounds. In the first paper, I obtained optimal pointwise isotropic bounds for N–body systems. In the second paper, I considered the case where V goes to infinity at infinity and proved pointwise exponential decay by every exponential (Sch’nol [36] earlier had a related result). In the third paper, I assumed |x|2m lower bounds and got exp(−|x|m+1) pointwise upper bounds. When one has an upper bound on V of this form, one gets lower bounds of the same form on the ground state. Papers 1-2 were written during my fall 1972 visit to IHES, one of my most productive times when Lieb and I did most of the Thomas–Fermi work and I developed new aspects of correlation inequalities and Lee–Yang for EQFT. Two of my students used these bounds in their work. Jay Rosen [35] needed them in his thesis proof of supercontractive estimates. Harrell [19, 20] while a postdoc followed my suggestions to study 1D double wells using these bounds and he and I [21] then used his techniques to prove the Oppenheimer formula for the width of the Stark Hamiltonian and the Bender–Wu formula for the asymptotics of the anharmonic oscillator perturbation coefficients. Optimal decay estimates for N–body systems were obtained in the fourth paper in my series [17] jointly with Deift, Hunziker and Vock. 1982 saw two big breakthroughs. Agmon [1] introduced his metric as a way to