Bounds on the decay of electron densities with screening

Abstract

Let H = -\L + V be a general Schrödinger operator on R" (v~> 1), where A is the Laplace differential operator and V is a potential function on which we assume minimal hypotheses of growth and regularity, and in particular allow V which are unbounded below. We give a general survey of the properties of e~t H , t > 0, and related mappings given in terms of solutions of initial value problems for the differential equation du/dt + Hu = 0. Among the subjects treated are L ^-properties of these maps, existence of continuous integral kernels for them, and regularity properties of eigenfunctions, including Harnack's inequality. CONTENTS A. IntroductionAl. Overview A2. The class K v A3. Literature on larger classes B. L ^-properties BI. L^-smoothing of semigroups B2. Sobolev estimates B3. Continuity and derivative estimates B4. Localization B5. Growth of L ^-semigroup norms as t -> oo B6. Weighted L 2 -spaces B7. Integral kernels: General potentials B8. Integral kernels: Some special operators for some special potentials B9. Trace ideal properties BIO. Continuity in V Bl 1. Hypercontractive semigroups and all that B12. Some remarks on the case when H is unbounded below B13. The magnetic case C. Eigenfunctions CI. Harnack's inequality and subsolution estimates C2. Local estimates on v<p C3. Decay of eigenfunctions C4. Eigenfunctions and spectrum C5. Eigenfunction expansions 447 an integral kernel. This is especially beloved by physicists, in part because the asymptotics of this kernel at infinity are sometimes related to physically significant quantities like scattering amplitudes. The theorem of Dunford and Pettis (see Treves [202]) asserts that THEOREM A. 1.1. Let (M, /x) be a separable measure space and E a separable Banach space. Let A be a bounded operator from E to L°°(M, du). Then there exists a unique {up to sets of ^measure zero) weakly measurable function, a, from M to E* so that for each f E E and a.e.x G M (Af)(x)= (a(x),f). Moreover II A II ^ = Mil.In particular, choosing E = L P (M, dp); 1 <p < oo, so that E* = L%M, d[i) with q~x +/T 1 = 1, and noting the trivial converse of Theorem A.l.l, we have COROLLARY A. If A is a bounded operator on L P (M, dp) and A is bounded also from L p to L°° then there is a measurable function a on MX M obeying (A10) sup[/|a(x, j)r]' /9 (= MH, :80 )<oo so that, for any f E L p , (All) (Af)(x) = fa(x, y)f(y) d(i(y). Conversely, if A: L p -* L p has an internal kernel, a, in the sense of (All) obeying (A 10), then A is a bounded map from L p to L°°. Of especial interest is the case p = 2. Operators on L 2 obeying (A 10) are called Carleman (integral) operators and are of classical interest. Corollary A. 1.2 is often attributed to Korotkov [120] especially in the Russian literature. Although it is a special case of the much older theorem of Dunford and Pettis [202], Korotkov was the first to emphasize its usefulness in the context of Carleman operators.One can ask about continuity of the integral kernels. Let X be a separable locally compact metric space and let C(X), ?NL f (X) den...

Schrödinger semigroups

“…u is clearly in L 2 _ B for some 8 so that E is an eigenvalue of H_ 8 and thus in spec(i/_ fi ). But spec(if_ fi ) = spec(if ) (Theorem B.6.4).…”

“…Note that by Corollary C.2.3, there is decay of the local L 2 -norm of Vw if V_ G K p9 V + G K} 00 . There has developed a considerable literature on detailed estimates of decay of the eigenfunction Hu = Eu with E in the discrete spectrum of H ; for example Agmon [2, 4], Ahlrichs [6], Ahlrichs et al [7,8] [197]. Much of this involves detailed results on two special cases: (1) the iV-body case and (2) the case where V(x) -» oo at infinity especially in a regular way (e.g.…”

“…Although Trudinger's hypotheses are slightly stronger than V G K] 00 , recently Agmon [3] has used an analytical approach to obtain Harnack's inequality for V G K] 00 . Moreover, large parts of the theory of eigenfunctions, especially if one settles for strengthened hypotheses on V follow from clever uses of subharmonic comparison theorems; see Simon [182], M. and/or T. Hoffman-Ostenhof [86,[90][91][92][93][94][95]8] and Davies [48].…”

“…Because relativistic invariance becomes Euclidean invariance if //' is replaced by /, the associated theory is called Euclidean quantum field theory. These ideas are especially powerful in a path integration context; see (8) so long as a > 0.…”

Boundary condition determined wave functions for the ground states of one- and two-electron homonuclear molecules

“…It is well known [18][19][20][21][22] that the asymptotic behavior of the wave function of Eq. ͑2.22͒ has the form…”