On positive eigenvalues of one‐body schrödinger operators

The last twenty years have produced a rather extensive literature on the exact mathematical treatment of general features of the Schrodinger equation for one or many particles. One of the more intriguing questions concerns the presence of discrete eigenvalues of positive energy (that is square-integrable eigenfunctions with positive eigenvalues) . There is a highly non-rigorous but physically appealing argument which assures us that such positive energy “bound states” cannot exist (c.f. [l], pages 30 and 51). On the other hand, there is an ancient, explicit example due to von Neumann and Wigner [2] which presents a fairly reasonable potential V, with V(r) -+ 0 as r .--f CO, and which possesses an eigenfunction with E = 1 (in units with fi2/2m = 1). According to the excellent review article of Kato [3], Section 8, there are two general results which yield cases where H = A + V(r) has no positive eigenvalues : (a) In [4], Kato has proven that if V = o(l/r), then H has no positive eigenvalue. (b) In [ 5 ] , Odeh has proven a similar result in case V(r) .--f 0 as r + 00 and aV/ar < 0 for sufficiently large r . Both of the above statements require certain regularity conditions along with the indicated asymptotic behavior. For the case of spherically symmetric V, stronger results do exist (see e.g. [9]). In this paper, we prove