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A New Approach to Inverse Spectral Theory, II. General Real Potentials and the Connection to the Spectral Measure Annals of Mathematics, 152 (2000), 593–643 A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure By Fritz Gesztesy and Barry Simon Abstract We continue the study of the A-amplitude associated to a half-line d2 2 ∞ Schrodinger operator, dx2 + q in L ((0,b)),R b . A is related to the Weyl- 2 a 2 (2aε) Titchmarsh m-function via m( )= 0 A()e d+O(e ) for all ε>0. We discuss ve issues here. First, we extend the theory to general q in L1((0,a)) for all a, including q’s which are limit circle at innity. Second, we prove the followingR relation between√ the A-amplitude and the spectral measure ∞ 1 : A()=2 ∞ 2 sin(2 ) d() (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace trans- form representation for m without error term in the case b<∞. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly. 1. Introduction In this paper we will consider Schrodinger operators d2 (1.1) + q dx2 in L2((0,b)) for 0 <b<∞ or b = ∞ and real-valued locally integrable q. There are essentially four distinct cases. This material is based upon work supported by the National Science Foundation under Grant No. DMS-9707661. The government has certain rights in this material. 1991 Mathematics Subject Classication. Primary: 34A55, 34B20; Secondary: 34L05, 47A10. Key words and phrases. Inverse spectral theory, Weyl-Titchmarsh m-function, spectral measure. 594 FRITZ GESZTESY AND BARRY SIMON Case 1. b<∞. We suppose q ∈ L1((0,b)). We then pick h ∈ R ∪ {∞} and add the boundary condition at b 0 (1.2) u (b)+hu(b)=0, where h = ∞ is shorthand for the Dirichlet boundary condition u(b)=0. For Cases 2–4, b = ∞ and Z a (1.3) |q(x)| dx < ∞ for all a<∞. 0 Case 2. q is “essentially” bounded from below in the sense that Z a+1 (1.4) sup max(q(x), 0) dx < ∞. a>0 a Examples include q(x)=c(x+1) for c>0 and all ∈ R or q(x)=c(x+1) for all c>0 and 0. Case 3. (1.4) fails but (1.1) is limit point at ∞ (see [6, Ch. 9]; [33, Sect. X.1] for a discussion of limit point/limit circle), that is, for each z ∈ C+ = {z ∈ C | Im(z) > 0}, 00 (1.5) u + qu = zu has a unique solution, up to a multiplicative constant, which is L2 at ∞.An example is q(x)=c(x +1) for c>0 and 0 < 2. Case 4. (1.1) is limit circle at innity; that is, every solution of (1.5) 2 is L ((0, ∞)) at innity if z ∈ C+. We then pick a boundary condition by picking a nonzero solution u0 of (1.5) for z = i. Other functions u satisfying the associated boundary condition at innity then are supposed to satisfy 0 0 (1.6) lim [u0(x)u (x) u (x)u(x)]=0. x→∞ 0 Examples include q(x)=c(x +1) for c>0 and >2. The Weyl-Titchmarsh m-function, m(z), is dened for z ∈ C+ as follows. Fix z ∈ C+. Let u(x, z) be a nonzero solution of (1.5) which satises the boundary condition at b. In Case 1, that meansR u satises (1.2); in Case 4, it ∞ | |2 ∞ satises (1.6); and in Cases 2–3, it satises R u(x, z) dx < for some (and hence for all) R 0. Then, u0(0 ,z) (1.7) m(z)= + u(0+,z) and, more generally, u0(x, z) (1.8) m(z,x)= . u(x, z) INVERSE SPECTRAL THEORY, II 595 0 ∂m m(z,x) satises the Riccati equation (with m = ∂x ), 0 (1.9) m (z,x)=q(x) z m(z,x)2. m is an analytic function of z for z ∈ C+, and moreover: Case 1. m is meromorphic in C with a discrete set 1 <2 < of poles on R (and none on (∞,1)). Case 2. For some ∈ R, m has an analytic continuation to C\[,∞) with m real on (∞,). Case 3. In general, m cannot be continued beyond C+ (there exist q’s where m has a dense set of polar singularities on R). Case 4. m is meromorphic in C with a discrete set of poles (and zeros) on R with limit points at both +∞ and ∞. Moreover, if z ∈ C+ then m(z,x) ∈ C+; so m satises a Herglotz representation theorem, Z 1 (1.10) m(z)=c + d(), R z 1+2 where is a positive measure called the spectral measure, which satises Z d() (1.11) < ∞, R 1+||2 1 (1.12) d() = w-lim ↓ Im(m( + iε)) d, ε 0 where w-lim is meant in the distributional sense. All these properties of m are well known (see, e.g. [23, Ch. 2]). In (1.10), c (which is equal to Re(m(i))) is determined by the result of Everitt [10] that for each ε>0, (1.13) m(2)= + o(1) as ||→∞with + ε<arg() < ε<0. 2 Atkinson [3] improved (1.13) to read, Z a 0 (1.14) m(2)= + q()e 2 d + o( 1) 0 | |→∞ again as with 2 + ε<arg() < ε<0 (actually, he allows arg() → 0as||→∞as long as Re() > 0 and Im() > exp(D||) for suitable D). In (1.14), a0 is any xed a0 > 0. One of our main results in the present paper is to go way beyond the two leading orders in (1.14). 596 FRITZ GESZTESY AND BARRY SIMON Theorem 1.1. There exists a function A() for ∈ [0,b) so that A ∈ L1((0,a)) for all a<band Z a (1.15) m(2)= A()e 2 d + O˜(e 2) 0 | |→∞ ˜ → as with 2 + ε<arg() < ε<0. Here we say f = O(g) if g 0 and for all ε>0, (f/g)|g|ε → 0 as ||→∞. Moreover, A q is continuous and Z Z 2 (1.16) |(A q)()| |q(x) dx exp |q(x)| dx . 0 0 This result was proven in Cases 1 and 2 in [35]. Thus, one of our purposes here is to prove this result if one only assumes (1.3) (i.e., in Cases 3 and 4). Actually, in [35], (1.15) was proven in Cases 1 and 2 for real with ||→∞. Our proof under only (1.3) includes Case 2 in the general -region ∈ arg() ( 2 + ε, ε) and, as we will remark, the proof also holds in this region for Case 1. Remark. At rst sight, it may appear that Theorem 1.1 as we stated it does not imply the real result of [35], but if the spectral measure of (1.10) ∈ ∞ ∈ R | | has supp() [a, ) for some a , (1.15) extends to all in arg() < 2 ε, ||a + 1. To see this, one notes by (1.10) that m0(z) is bounded away from [a, ∞) so one has the a priori bound |m(z)|C|z| in the region Re(z) <a1. This bound and a Phragmen-Lindelof argument let one extend (1.15) to the real axis. Here is a result from [35] which we will need: Theorem 1.2 ([35, Theorem 2.1]). Let q ∈ L1((0, ∞)). Then there exists a function A() on (0, ∞) so that A q is continuous and satises (1.16) such 1 k k that for Re() > 2 q 1, Z ∞ (1.17) m(2)= A()e 2 d. 0 1 k k Remark. In [35], this is only stated for real with > 2 q 1, but (1.16) | |k k2 k k implies that A() q() q 1 exp( q 1) so the right-hand side of (1.17) 1 k k converges to an analytic function in Re() > 2 q 1. Since m(z) is analytic C\ ∞ { ∈ C | 1 k k } in [, ) for suitable , we have equality in Re() > 2 q 1 by analyticity. Theorem 1.1 in all cases follows from Theorem 1.2 and the following result which we will prove in Section 3. INVERSE SPECTRAL THEORY, II 597 Theorem 1.3. Let q1,q2 be potentials dened on (0,bj) with bj >a for j =1, 2. Suppose that q1 = q2 on [0,a]. Then in the region arg() ∈ | | ( 2 + ε, ε), K0, we have that 2 2 (1.18) |m1( ) m2( )|Cε, exp(2aRe()), R x+ | | where Cε, depends only on ε, , and sup0xa( x qj(y) dy), where >0 is any number so that a + bj, j =1, 2. Remarks. 1. An important consequence of Theorem 1.3 is that if q1(x) = q2(x) for x ∈ [0,a], then A1()=A2() for ∈ [0,a]. Thus, A() is only a function of q on [0,]. At the end of the introduction, we will note that q(x) is only a function of A on [0,x]. 2. This implies Theorem 1.1 by taking q1 = q and q2 = q[0,a] and using Theorem 1.2 on q2. ∈ 3. Our proof implies (1.18) on a larger region than arg() ( 2 + ε, ε). Basically, we will need Im() C1 exp(C2||)ifRe() →∞. We will obtain Theorem 1.3 from the following pair of results. Theorem 1.4. Let q be dened on (0,a+) and q ∈ L1((0,a+)).
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