Before we formulate our results we give the following. Definition: Let We C~~ s) and let Ro(z ) denotes the free resolvent Ro(z ) = =(-A-. (Ro(z)f)(x) = fff]]Go(x, Y, ) ...
REGULARIZED POTENTIALS IN NONRELATIVISTIC Q U A N T U M MECHANICS II. THE THREE DIMENSIONAL CASE
P. Seba Nuclear Centre, Faculty o f Mathematics and Physics, Charles University, V Hole~ovi(kdch 2, 180 O0 Praha 8, Czechoslovakia Using scaling technique we describe various self-adjoint extensions of the three-dimensiona Schr6dinger operator with singular potential as a limit of Schr6dinger operators with regularized potentials. 1. INTRODUCTION
In a previous paper [1] we have considered the one-dimensional Schr6dinger operator with regularized singular potentials and we have shown that using various types of regularization we obtain, by removing the regularization, various operators as a limit. In this paper we continue the study of regularized singular potentials in the three-dimensional case. Let us have a potential V~ R n LI(R3), (R denotes the Rollnik class), i.e. a potential with
Iff+~ ffff~V(x) V(Y) lx -- Yl-2d3xd3y < ~176[2]) which is singular at origin (i.e. we define the operator
VeL~oo(R3\ {0})\ L~oc(R3)). Using
this potential,
H~ = -A + V(x), D(H~ = D ( - A ) n O ( V ) . In view of the fact that the potential V(x) is singular at 0 this operator is not essentially self-adjoint [3, 4]. In order to obtain a self-adjoint operator (the Hamiltonian) associated with the differential expression - A + V(x), we have to choose one of its self-adjoint extensions. (We describe this extensions in section 2.) The aim of this paper is to show that we get all these extensions as a limit of Schr6dinger operators with regularized potentials ~(x) if we remove the regularization, i.e. if we take the limit e ~ 0 where the regularized potentials 17,(x) fulfil lim ~(x) =
V(x)
pointwise almost everywhere.
This limit is computed in section 3. 2. THE SELF-ADJOINT EXTENSIONS OF //o
We define the operator Ho~ by
= _• Czech. J. Phys. B 36 [1986]
+ v(x);
D(/r ~ = D ( - a )
D(V), 559
P. Seba: Regularizedpotentials... where D ( - A ) = {f e L2(R3); h f e L2(R 3) in the sense of distributions} D(V) = {f E L2(R3); Vf ~ L2(R3)}. It can be easily seen that f ~ D(H ~ implies f(0) = 0 for potentials V(x) which are singular at origin. Using ([4], lemma 2.1) we get that no~ is a symmetric operator on L2(R 3) with deficiency indices (1, 1). Consequently there is a one-parameter family of its self-adjoint extensions which we denote H~. On the other hand, it is known for potentials Ve R n LI(R ~) that they are formbounded with respect to - A [5]. Using the K L M N theorem ([2], theorem X.17) we see that there is a unique se!f-adjoint operator H o associated with the quadratic form
h: h(I) = (S, -AS) + (S, VS). (H 0 is known as the Friedrichs extension of H~ Knowing H o we can also describe all the other self-adjoint extensions H~ as = -a
+ V(x)
o(n~) = {fe L'(Ra);f(x) = p(x) + e(G(x, 0, i) - e~=(a(x, 0, -i)), ; [0, where
G(x, y, z) is the Green's function of He
((Ho-Z)-lf)(x)
=ffffs
.
Remark: For a with Im a ee 0 H, is an accretive operator [7] and -iH~, generates
a contractive semigroup exp(itH,) (or exp(-itH=,)) t >__ 0 if Im ~ > 0 (or Im c~< 0) in LZ(R3) [8]. 3. THE SCALING LIMIT We show now that it is possible to get all these self-adjoint extensions H,. as a norm resolvent limit of three-dimensional Schr6dinger operators with regularized potentials. (The norm resolvent topology is defined in [9].) To show this we introduce a one-parameter family of Schr6dinger operators H ~ = - A + V,(x), where 17,(x) is a regular (continuous) potential for all e > 0 and lira P,(x) =
g(x)
a--+0
pointwise almost everywhere. We decompose the regularized potential ~(x) in the following way
~,(~) where 560
= ~(~) +
2(~) W(,~l~) ,
We C~(R3); 2(e)is continuous together with its first derivative, 2 ( 0 ) = 1, Czech. J. Phys. B 36 [1986]
P. Seba." Regularized potentials. . . 2'(0) 4= 0 and V, is a regularization of V fulfilling V~~ R c~ LI(R a)
I %1 I (x)l
Iv( )l for
> o
rv(x)l
and pointwise almost everywhere as e ~ 0. Before we formulate our results we give the following Definition: Let We C~~ s) and let Ro(z ) denotes the free resolvent Ro(z ) =
=(-A-
(Ro(z)f)(x) = fff]]Go(x, Y, )Y(Y)d Y Go(x, y, z) = exp (i x/(z)ix -
yl)/(4 ]x
- YF).
We say that H = - A + W has a zero energy resonance if - 1 is an eigenvalue of the operator Wz Ro(0) wi, where wl(x) = [W(x)] 1/2 ;
w2(x) : [W(x)lm sgn W(x).
We say that the resonance is simple if - 1 is a simple eigenvalue of w2 R0(0) wl. If the eigenvalue - 1 has a multiplicity N we say that H has a zero energy resonance with multiplicity N. Let further ~j(x) be the corresponding eigenvector
w2Ro(O) w l ~ j =
-~j
for j =
1,2 . . . . , N .
We call the function Oj = R o ( 0 )wl~) j
j = 1,2,...,N
the (zero energy) resonance function (cf. [ 1 0 - 1 2 ] ) . We distinguish the following cases [13, 14]. Case 1: H has no zero energy resonance. Case 2: H has a simple zero energy resonance and the corresponding resonance function ~ is not in LZ(RS). Case 3: H has a zero energy resonance with multiplicity N > 1 and all corresponding resonance functions 0~(x), j = 1, 2 , . . . , N are in L2(RS). Case 4: H has a zero energy resonance with multiplicity N => 2 and at least one resonance function ~, is not in L2(RS). We can now formulate our results as Theorem: 1) N.R.lim H~=Ho in cases 1 and 3. (N.R.lim denotes the norm resolvent limit) ~-.o 2) N.R.lim H~ = H~, where ~ is given by e"*0
Czech. d. Phys. B 36 [1986]
561
P. Seba." Re#ularized potentials...
exp (i~) =
_ ~/___ii+ G(0, 0, i) - Go(0, 0, - i ) - Z(0) (sgn W~; ~) 4re [(w,; ~)l 2 x/i + G(0, 0, - i ) - Go(0, 0, - i ) - 2'(0) (sgn W#; #) art ](wl; ~)]2
in case 2; N
exp (ie) = 1 + 2 (wa, Oj)(~b,, w,)(sgn WO, BO)~]' (G(O, O, i) - G(O, O, -i)) j,l= l
in case 4, where B denotes the operator
B = Z(O)w2Ro(O)w,+( - x/i - - + G(0, 0, - i ) - Go(0, 0, - i ) ) wz(w,, ") 4re
(sgn WO, B~)~ 1 is the inverse matrix of (sgn W~j, B4~1), and
G(O, O, z,) - G(O, O, zz) = lim (G(ex, ey, z,) - G(ex, ey, z2)) 1~-+0
G(0, 0, z,) - Go(0, 0, z2) = lim (G(ex, ~y, z,) - Go(eX, sy, z2)). ~0
Remark: This rather difficult mathematical theorem tell us the following: If we
remove the regularization from H~ (i.e. e ~ 0), we do not always obtain the operator H o though Ve --+ V pointwise almost everywhere. We can rather say that removing the regularization from H e means putting corresponding boundary conditions at 0 (the choice of the corresponding self-adjoint extension). Proof: We use the scaling technique developed in [15, 16] and follow [1]. Let Re(z) denote the resolvent of He
Re(z) = (H~- z)-*. Defining the unitary scaling group U e in
LZ(R3) by
(UJ)(x) = ~-3/Z f(xl~);e
>
O, f ~ L2(R3) ,
we get H~
where H(e) = - A +
=
~- 2
U~ H(~) U-f ~ ,
+
We denote H ~ = - A + ~z
