Evans, W. D.

Lewis, R.

Saitó, Y.

English

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7W. D. Evans; Roger T. Lewis; Y. SaitóFiniteness of bound states of general n-body operatorsIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 159--162.Persistent URL: http://dml.cz/dmlcz/702343Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czFINITENESS OF BOUND STATES OF GENERAL N-BODY OPERATORS EVANS W.D., CARDIFF, United Kingdom LEWIS R.\ BIRMINGHAM, AL, U.S.A. SAITO Y.SBIRMINGHAM, AL, U.S.A. Consider the (N + l)-body Schrodinger operator of atomic type, <l> ^ -E( -A,- 1 | f )+ £ ---i--. i=i ' ' i<*<i<iV ' ' This is the Hamiltonian of an atom with an infinitely heavy nucleus of charge Z and N electrons of charge 1 and mass 1/2. Here x* 6 R 3 is the coordinate of the tth electron and A,- denotes the Laplacian in R 3 with respect to the variable x%. For the operator PN the next theorem gives a classic result. (For the appropriate references we refer the reader to the extensive reference list in [4] .) Theo rem 1. (Zhislin (1960, 1969,1971), Uchiyama (1969) and others) The operator PN given by (1) has at most a finite number of bound states if and only if Z < N — 1. In this paper we work towards a theory for general N-body operators which would include the results in Theorem 1 for atoms as well as results for molecules. A portion of this task has been accomplished in [2] for general atomic-type operators of the form (2) P=-JT(2mi)-1Ai + JTv<,i(Xi)+ VJ vtip-zi), t = l » = 1 l<i<3<N where m* is the mass of the ith electron. As in [2] we proceed from the foundation laid by Agmon[l] . Define n H = - £ ^^J + «<*)» X G Rn' . , . 7 = 1 where H satisfies the following (3)(i)-(iv): (i) Each a ,J is a bounded, continuous, real-valued function on R n . (II) The matrix A(x) = [a%*(x)] is symmetric and its smallest eigenvalue p,(x) is a positive continuous function on R n . (3) (iii) q 6 Ii(Rn)/oc. (iv) «7_ = max(-g,0) € Af/oc(Rn), where M ( R n ) = Af0(Rn) is the Stummel class of functions. The sesquilinear form which gives rise to II is (4) PWM= I {<VA<P,VA4>>+q<pt}dx (<p,i/>eCS°(Rn)), IR» w h e r e p[<f>] := p[<f>,<f>] a n d n (5) < VA<p, VAi}) >= J^ a*J'vVV .̂ f,j=i Define (6) A(/7) = ini{p[V]: v e <%>{*"), M = U-If A(II) > -oo , the sesquilinear form p[<p,ii>] on Go°(Rn) X Co°(Rn) is proved to be a densely defined symmetric form which is bounded below and closable in I2(Rn) - see Agmon[l] and [2] . 1 Supported by the U.S. National Science Foundation, NSF DMS-8719027. 159 Henceforth, let H denote the self-adjoint operator in L2(Rn) associated with the form p[<p^\. Then, the least point of the essential spectrum of H is given by (7) £(ff) = sup [inf {fW • V e C H R " - K),|M| = l } ] . i f .compact Agmon [1] gave an interesting, alternative expression for E(P) which is related to the celebrated HVZ Theorem of quantum physics - see the references in Sigal[3] or [4] . Let B(0;R) := {x G R n : ||x|| < R} and 5 n ~ 1 := {x G R n : \\x\\ = 1}. For u G Sn~\6 G (0 , r ) , and R > 0 define the truncated cone r(u;,$,R) = {arGRn : < x,u> » \x\cos6,\x\ > R}. Let E ( « , M ) = inf {,[?] : <p G C0°°(r(u,,*,R)),|M| = 1} and K(u : H) = lim lim E(u;,tf,R). Theorem 2. (Agmon[l], Lemma 2.7) K(-: H) is lower semicontinuous on 5 n _ 1 and E(B ) = min{K(o;: H) : u € 5 7 1 - 1}. Let M := {w G 5 n _ 1 : K(-: B) assumes its minimum at u}. Note that M is a closed subset of 571"1 since K(o; : H) is lower semicontinuous on 5 n _ 1 . For M we define the truncated conical region T(M : R) := {x G R n : x = tu for w G M and t > R}. Let d\f (x) := rfwt(x : M). The first part of our main hypothesis is given by W(l) : Let M be a proper subset of 5 n _ 1 which is the finite union of closed sets {Af J*.--_. Suppose that each <-M- is C 1 in a neighborhood of Mt-. When B is a generalized N-body operator, Agmon shows that K(u : H) < 0. In that special case, M = 5 n _ 1 implies that K(LJ : H) = 0. For 3-body operators this may result in a rather pathological case called the Efimov effect - see [4] and the references contained therein. An important ingredient in our proof is the introduction of a certain partition of unity {Jo,Ji,I2} satisfying the properties given in Lemma 3 below. To this end we first define Ms := W E 5 n _ 1 : dist(u>: M) < 6}. We associate with Ms the truncated conical region T(Ms : R) defined as above. In the following, suppJi denotes the support of J,-. LEMMA 3. There is a partition of unity {Jo,Ji,J2} satisfying (i) 0 < Ji < 1 for each i and each x G R n ; 00 £UIt?(*)--lforsGRn; (Hi) each Ji(x) is> Lipschitz in Kn; (iv) supp Jo C B(0; 1); (v) suppJx CT(M6:\); (vi) supp J2 C Rn \ (T(Ms/2 : 0) U B(0; I ) ) ; (vii) J\ and J2 are homogeneous of degree zero in Rn \B (0 ; 1); and (viii) for anye > 0 there exists Ct > 0 such that |VJi(x)|2 + |VJ2(x)|2 < (cJ,(x)2 +C<J2(x)2)\x\-2 for allxe R n \B (0 ; 1). P R O O F : For the basic ideas of the proof we refer the reader to either [2] or Sigal[3] . As the next lemma illustrates, this partition of unity allows us to separate the essential "parts" of the form p. This lemma gives the IMS localization formula[4] for H. 160 LEMMA 4. For any open set ficR" and any <j> 6 C£°(Rn) / [ |V^ | 2 + q\<f>\2] dx = Y f [ |V^(I^) |2 + q\J{<i>\2 - |V,4I.|2|</>|2] dx. Jn ~ ^ JQ PROOF: An elementary calculation shows that | V ^ ( I ^ ) | 2 = J 2 | v ^ | 2 + | V ^ | 2 M 2 + \ < VAJ2,vA\4>\2 > on using the fact that the matrix A(x) is symmetric. Since 2t=o J?(x) — -> ̂ n e identity follows. Let D be a bounded open subset of R n , which contains the unit ball B(0; 1), and for which the embedding HX(D) «-+ L2(D) is compact. This is the well-known Rellich property and is satisfied if D has a continuous boundary. As a consequence of Lemma 4, Pi*] = f [|V^|2 + qW2] dx+J2f [|V„(J,*)|J + q\Ji<f>\2 - \VAJi\2\<l>\2] dx. JD i=l JR»\D The finiteness of bound states of H, or equivalently the finiteness of eigenvalues of H below £(11), depends upon the behavior of q in the truncated conical regions T(Ms : R) for R arbitrarily large. Roughly speaking, we need a certain degree of positivity of the "part" of the potential q which does not determine S(B ) . The next part of our basic hypothesis can be interpreted in such a manner. H(2) : There exist €U€2 € (0,1) and C€2 > 0 such that / [\VA(M)\2 + (q--^)\Ji<l>\2)dx JR»\D \X\ > S(I/) / \J^\2dx -e2 f \VAcf>\2dx - Ce„ / \<j>\2dx. JR*\D JD JD The "part" of the form described in H(2) is the "essential" part mentioned above that determines the finiteness or infiniteness of bound states. The term —ci/|a;|2 arises from the error approximation associated with the partition of unity given in part (viii) of Lemma 3. At times it is helpful to replace *H(2) by the following two hypotheses: 7i(2a) : There exists ei > 0 and a function a defined on 3D such that / [\VA(Ji<t>)\2+(q~^)\Jicf>\2]dx JR»\D FY >£(/ .") / \Jx<f>\2dx-{- / a\JKl>\2da JR»\D JdD for all <f> £ C$°(Rn). H(2b) : For some e2 G (0,1) and C€2 > 0 / <r\JKf>\2dx>-€2 f \VA<j>\2dx-C,7 f \4>\2dx JdD JD JD for all <f> C£°(Rn). 11 Kurzweil, Equadiff 7 161 «'):=Ll: Hypothesis H(2b) holds if <r_ G L7(#D) for 7 = n - 1 when n > 2 and 7 6 (1,oo] when n = 2. The final part of our hypothesis assures us that the remainder of the form is under control. First, it is helpful to define the weight for x € D (x) for x $ D. Now, we introduce the notation / \VAw<f>\2dx:= f \VA<f>\2dx + f \VAh<t>?dx, IR- JD JR»\D for all <f> ~ CS°(Rn). Note that VA(w<f>) is not defined on dD. H(3): Given €3 there exists C63 > 0 such that / q\w<f>\2dx < €3 / \VA(w<j>)\2dx + C€3 / \w<f>\2dx IR» IR» IR» for all <f> e CS°(Rn). This last part of the hypothesis is satisfied by the N-body potentials briefly mentioned above. Now, we state our main theorem. Due to space limitations, we cannot give a full proof here. We will give a sketch of the main ideas of the proof. THEOREM 5. If (3)(i)-(iv) and H(l),H(2),kH(3) hold, then H has only a finite number of bound states. PROOF(SKETCH) : If we interpret H(2) in terms of operators, then we might expect that B>Ii(x)£(ff)Ii(a:) + Br for some "remainder" operator Hr. The sesquilinear form for such an operator is defined for CQ°(D U «ttppI2)-functions. This form is much like p with some additional error terms. Part H(3) of the hypothesis is a key ingredient in the proof of the fact that this form is closeable. Now, Hr arises as the operator associated with this form. A result like Theorem 2 applies to Hr. Part H(3) of the hypothesis assures that £(11r) > £(11). It follows that Hr can have only a finite number of eigenvalues below £(11). Finally, we show that the number of eigenvalues of H below £(11) can be no more than the number of eigenvalues of Hr below £(11), which completes the proof. A version of Theorem 5 with a more restrictive hypothesis was used in [2] to establish results for atomic-type operators (2). There it was assumed that K(- : P) assumes its minimum at no more than a finite number of points. That assumption excluded the consideration of molecules and an atom with a nucleus which was not assumed to be infinitely heavy. In these latter cases it is common to first remove the motion of the center of mass via a Jacobi coordinate change restricting the operator to "configuration space" X - see [4] . In A" we can think of the motion of the center of mass as being held fixed at the orgin. Now, letting H be the Hamiltonian in L2(X) = L2(K3^N~^), then A'(-: 11) assumes its minimum on curves in S3N~A or on their intersections. These curves correspond to x = (xl,x2,.. .,xN~*) € S3N~4 where xl = x* for distinct t and j . Theorem 5 now allows us to consider these other cases. References 1. S.Agmon, "Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations: Bounds on Eigenfunctions of N-body Schrodinger Operators," Mathematical Notes 29, Princeton University Press and the University of Tokyo Press, 1982. 2. D. Evans and R. Lewis, N-Body Schrodinger Operators with Finitely Many Bound States, to appear in the Trans. Amer. Math. Soc.(1989 or 1990). 3. I. Sigal, Geometric Methods in the Quantum Many-body Problem. Nonexistence of Very Negative Ions, Communications in Math. Phys. 85 (1982), 309-324. 4. H. Cycon, R. Froese, W. Kirsch and B. Simon, "Schrodinger Operators with Application to Quantum Mechanics and Global Geometry," Springer-Verlag, 1987. 162 

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