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1D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS
EVGENY KOROTYAEV
Abstract.
We consider a Schr¨odinger operator with complex-valued potentials on the line.The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraicmultiplicity) in the complex plane without the positive half-line. We determine series of traceformulas. Here we have the new term: a singular measure, which is absent for real potentials.Moreover, we estimate of sum of Im part of eigenvalues plus singular measure in terms of thenorm of potentials. The proof is based on classical results about the Hardy spaces. Introduction and main results
Introduction.
We consider a Schr¨odinger operator H = − d dx + q ( x ) on the space L ( R ),where the potential q is complex and satisfies: Z R (1 + | x | ) | q ( x ) | dx < ∞ . (1.1)It is known that the spectrum of the operator H has two components: the essential spectrumwhich covers the half-line [0 , ∞ ) plus N ∞ eigenvalues (counted with multiplicity) in the cutspectral domain C \ [0 , ∞ ). We denote them by E j ∈ C \ [0 , ∞ ) , j = 1 , ..., N, according to theirmultiplicity. Note, that the multiplicity of each eigenvalue equals 1, but we call the multiplicityof the eigenvalue its algebraic multiplicity. Define the half-planes C ± = {± Im z > } . Insteadof the energy E ∈ C we define the momentum k = √ E ∈ C + . We call k j = √ E j ∈ C + alsothe eigenvalues of the operator H . Of course, E is really the energy, but since k is the naturalparameter, we will abuse terminology. We define the set k q = { k , ..., k N ∈ C + } and label k , .., k N ∈ C + by Im k > Im k > Im k > ... > Im k n > ... (1.2)We shortly describe results about trace formulas: • In 1960 Buslaev and Faddeev [BF60] determined the classical results about trace formulasfor Schr¨odinger operators with real decaying potentials on half-line. The case of the real linewas discussed by Faddeev and Zakharov in the nice paper [FZ71]. • There are a lot of results about one dimensional case, see [KS09] and references therein. • The multidimensional case was studied in [B66]. Trace formulas for Stark operators andmagnetic Schr¨odinger operators were discussed in [KP03], [KP04]. • The trace formulas for Schr¨odinger operators with real periodic potentials were obtainedin [KK95, K97]. They were used to obtain two-sided estimates of potential in terms of gaplengths (or the action variables for KdV) in [K00] via the conformal mapping theory for thequasimomentum. • Trace formulas for Schr¨odinger operators with complex potentials on the lattice Z d and on R are considered recently in [K17], [KL18], [MN15] and [K17x] respectively. In [K17], [KL18] Date : September 19, 2019.
Key words and phrases.
Complex potentials, trace formula. for the discrete case the main tool is the classical results about the Hardy spaces in the disc.Trace formulas for Schr¨odinger operators with complex potentials on R + with the Dirichletboundary condition are discussed in [K18]. In the case R and R + the Hardy spaces in theupper half-plane C + are used.1.2. The Hardy spaces.
Introduce the Jost solutions f ± ( x, k ) of the equation − f ±′′ + qf ± = k f ± , x ∈ R , k ∈ C + \ { } , (1.3)with the conditions f ± ( x, k ) = e ± ixk + o (1) as x → ±∞ , k ∈ R \ { } . (1.4)Here and in the following ′ denotes the derivative w.r.t. the first variable. For each x ∈ R the Jost solutions f ± ( x, k ) are analytic in C + , continuous up to the real line. Introduce theWronskian w and functions ψ, Ψ in C + by w ( k ) = { f − ( x, k ) , f + ( x, k ) }| x =0 , ψ ( k ) = w ( k )2 ik , Ψ = w i ( k + i ) , (1.5)where { y, f } = yf ′ − y ′ f . The function ψ satisfies (uniformly in arg k ∈ [0 , π ]): ψ ( k ) = 1 − q + o (1)2 ik as | k | → ∞ , q := Z R q ( t ) dt. (1.6)The function ψ has N > C + given by k j ∈ k q , counted with multiplicity.Define the Hardy space H p . Let a function F ( k ) , k = u + iv ∈ C + be analytic on C + . For0 < p ∞ we say F ∈ H p = H p ( C + ) if F satisfies k F k H p < ∞ , where k F k H p is given by k F k H p = sup v> π (cid:18) R R | F ( u + iv )) | p du (cid:19) p if < p < ∞ sup k ∈ C + | F ( k ) | if p = ∞ . Note that the definition of the Hardy space H p involves all v = Im k > k q k α = Z R | x | α | q ( x ) | dx, α > , k q k = k q k . Describe the properties of the functions w, ψ, Ψ: • The function w, ψ,
Ψ have the same zeros in C + \ { } , and these zeros { k j } in the upper-halfplane C + labeled by (1.2) satisfy (see e.g., [Sa10]): N X j =1 Im k j < ∞ . (1.7) • if w (0) = 0 and q = 0, then the function ψ − / ∈ H p for any p >
0, since ψ has asymptotics(1.6). • The properties of w give that Ψ − ∈ H p for any p > q . • If w (0) = 0, then the function ψ − ∈ H p for any p > • Due to (1.6) all zeros of w are uniformly bounded and satisfy (see e.g., [AAD01]) k q ⊂ { k ∈ C + : w ( k ) = 0 } ⊂ { k ∈ C + : | k | r c } , where r c := k q k . (1.8) D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 3
In order to study ψ ( k ) in the upper-half plane we defined the Blaschke product B by B ( k ) = N Y j =1 (cid:18) k − k j k − k j (cid:19) , k ∈ C + . (1.9)This product converges absolutely for each k ∈ C + , since all zeros of w are uniformly bounded,see (1.8). Moreover, it has an analytic continuation from C + into the domain {| k | > r c } , where r c = k q k and has the following Taylor serieslog B ( k ) = − i B k − i B k − i B k − ..., as | k | > r c ,B = 2 N X j =1 Im k j , B n = 2 N X j =1 Im k n +1 j , n > , (1.10)where each sum B n , n > | B n | N X j =1 | Im k n +1 j | π ( n + 1) r nc B . (1.11)We use asymptotics of B at large | k | to determine the trace formulas. Note that the function B has a complicated properties in the disk {| k | < r c } and very good properties for {| k | > r c } .We describe the basic properties of eigenvalues and the Blaschke product B . Proposition 1.1.
Let a potential q be complex and satisfy R R (1 + | x | ) | q ( x ) | dx < ∞ . Theni) The Blaschke product B ( k ) , k ∈ C + given by (1.9) belongs to H ∞ with k B k H ∞ .ii) Let q = R R q ( x ) dx, k q k = R R | xq ( x ) | dx and let A = k q kk q k e k q k . Then • If A < Re q , then the operator H does not have eigenvalues. • If A <
Re( − q ) , then the operator H has exactly one simple eigenvalue. Example.
Consider the potential q ( x ) = ctx t − , x ∈ (0 ,
1) and q ( x ) = 0 for x >
1, where c ∈ C , t >
0. We have k q k = | c | t , k q k = | c | t t , q = ct , A = | c | t exp | c | t t . If t is small, then the complex potential q is rather big and due to (1.8) all eigenvalues belongto the half-disk with the radius r c = k q k = | c | t . Let c = | c | e iφ and let t > φ = π , then Re q = | c | t > A and by Proposition 1.1, the operator H has not eigenvalues.If φ = − π , then Re( − q ) = | c | t > A and by Proposition 1.1, the operator H has one simpleeigenvalue.1.3. Trace formulas and estimates.
We describe the function ψ in terms of a canonicalfactorization. In general, the function ψ / ∈ H p for all p >
0, but we show that the function ψ has a canonical factorization for each potential q . Theorem 1.2.
Let a potential q satisfy (1.1). Then the function Ψ ∈ H ∞ ( C + ) and Ψ iscontinuous up to the real line. Moreover, ψ has a canonical factorization in C + given by ψ = ψ in ψ out . (1.12) EVGENY KOROTYAEV • ψ in is the inner factor of ψ having the form ψ in ( k ) = B ( k )2 ik e − iK ( k ) , K ( k ) = 1 π Z R dν ( t ) k − t , k ∈ C + . (1.13) • B is the Blaschke product defined by (1.9) and dν ( t ) > is some singular compactly supportedmeasure on R , which satisfies ν ( R ) = Z R dν ( t ) < ∞ , supp ν ⊂ { z ∈ R : w ( z ) = 0 } ⊂ [ − r c , r c ] , r c = k q k . (1.14) • The function K ( · ) has an analytic continuation from C + into the domain C \ [ − r c , r c ] andhas the following Taylor series K ( k ) = ∞ X j =0 K j k j +1 , K j = 1 π Z R t j dν ( t ) . (1.15) • the function log | ψ ( t + i | belongs to L loc ( R ) and ψ out is the outer factor given by ψ out ( k ) = e iM ( k ) , M ( k ) = 1 π Z R log | ψ ( t ) | k − t dt, k ∈ C + . (1.16) Remark.
1) These results are crucial to determine trace formulas in Theorem 1.3.2) Due to (1.6) the integral M ( k ) in (1.16) converges absolutely for each k ∈ C + .We recall the well-known results. Introduce the Sobolev space W m defined by W m = (cid:26) q ∈ L ( R ) : xq ( x ) ∈ L ( R ) , q ( j ) ∈ L ( R ) , j = 1 , .., m (cid:27) , m > . (1.17)If q ∈ W m +1 , m >
0, then the function ψ ( · ) satisfies i log ψ ( k ) = − Q k − Q k − Q k + · · · − Q m + o (1) k m +1 , (1.18)as | k | → ∞ , uniformly in arg k ∈ [0 , π ], where due to [FZ71] we have Q = q Z R q ( x ) dx, Q = 12 Z R q ( x ) dx, ... (1.19)Define constants I j , J j by I j = Im Q j , J = 1 π Z ∞ ( h ( t ) + h ( − t )) dt, J j = 1 π Z ∞ ( h j − ( t ) + h j − ( − t )) dt,h ( t ) = log | ψ ( t ) | , h j = t j +1 ( h ( t ) + P j ( t )) , P j ( t ) = I t + I t + ... + I j t j +1 . (1.20)In particular, we have I j +1 = 0 and J = 1 π Z ∞ (cid:0) t ( h ( t ) − h ( − t )) + 2 I (cid:1) dt. (1.21)The integral J converges absolutely since (1.6) gives ψ ( t ) ψ ( − t ) = 1 + O (1) t as t → + ∞ . D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 5
Theorem 1.3. ( Trace formulas ) If a potential q satisfies (1.1), then B + ν ( R ) π + 12 Z R Re q ( x ) dx = 1 π Z ∞ log | ψ ( t ) ψ ( − t ) | dt. (1.22) Let a potential q ∈ W m +1 for some m > . Then the following identities hold true: B j j + 1 + K j + Re Q j = J j , j = 1 , ..., m, (1.23) in particular, B K = J , (1.24) B K + 18 Z R Re q ( x ) dx = J . (1.25) Remark.
Recall that B > K = ν ( R ) π >
0. Thus in order to estimate B + ν ( R ) π > q we need to estimate the integral J in terms of the potential q . Theorem 1.4. ( Estimates ) Let a potential q satisfy (1.1). Then the following hold true B + ν ( R ) π + 12 Z R Re q ( x ) dx π (cid:0) k q k (cid:1) + 2 π r c (cid:0) C + log r c (cid:1) , (1.26) where k q k = R R | xq ( x ) | dx, r c = k q k and C = log e + e + e . Remark.
In Section 6 we prove Theorems 1.3 and 1.4 for the case of Schr¨odinger operators onthe half-line with the Neumann boundary condition. Recall that trace formulas of Schr¨odingeroperators on the half-line with Dirichlet boundary condition were discussed in [K18].Consider estimates for complex compactly supported potentials. In this case the Wronskian w ( k ) is the entire function and has a finite number of zeros in C + . Theorem 1.5.
Let q ∈ L ( R ) and let supp q ⊂ [0 , γ ] for some γ > . Then the number ofzeros N ( ρ ) of w (counted with multiplicity) in disk D ρ ( it ) with the center it = i k q k and theradius ρ > √ k q k satisfies N ( ρ ) (cid:18) γρπ + k q k ρ (cid:19) . (1.27) In particular, the number of zeros N + of w ( k ) (counted with multiplicity) in C + satisfies N + C + C γ k q k , (1.28) where the constants C , C (see more about C , C in Lemma 5.1). Note that the estimate of N + was obtained in [FLS16], when q decays exponentially atinfinity.Our main goal is to determine trace formulas for Schr¨odinger operators with complex po-tentials on the line. Our trace formula is the identity (1.22), where the left hand side is theintegral from the real part of potential, the sum of Im k j and the integral of the singularmeasure and the right hand side is the integral from log | ψ ( k + i ψ ( − k + i | on the realline. Here we have the new term, the singular measure, which is absent for real potentials.Moreover, in (1.26) we estimates the singular measure and the sum of Im k j in terms of thepotential. In our consideration the results and technique from [K17x], [K18] are important. EVGENY KOROTYAEV
In contrast to trace formulas for complex potentials, there are many results on estimatesof eigenvalues in terms of potentials, see recent articles [F18, FS17] and references therein.There exist many recent results about bounds on sums of powers of eigenvalues 1-dimensionalSchr¨odinger operators with complex-valued potentials in terms of L p -norms of the potentialspublished in [FLLS06, DHK09, LS09, Sa10, H11, F18].We shortly describe the plan of the paper. In Section 2 we present the main properties ofthe Jost functions and the Wronskian w . In Section 3 we prove main theorems. Section 4 is acollection of needed facts about Hardy spaces. In Section 5 we discuss the case of compactlysupported potentials. In Section 6 we consider Schr¨odinger operators on the half-line with theNeumann boundary condition.2. Fundamental solutions
Fundamental solutions.
It is well known that that the Jost solution f + ( x, k ) of equa-tion (1.3) satisfies the integral equation f + ( x, k ) = e ixk + Z ∞ x sin k ( t − x ) k q ( t ) f + ( t, k ) dt, ( x, k ) ∈ [0 , ∞ ) × C + . (2.1)We describe the main properties of the Jost solution. Due to (2.1) the function y + ( x, k ) = e − ikx f + ( x, k ) satisfies the integral equation y + ( x, k ) = 1 + Z ∞ x G ( t − x, k ) q ( t ) y + ( t, k ) dt, G ( t, k ) = sin ktk e ikt , (2.2) ∀ ( x, k ) ∈ [0 , ∞ ) × C + . The standard iterations give y + ( x, k ): y + ( x, k ) = 1 + X n > y + ,n ( x, k ) ,y + ,n ( x, k ) = Z ∞ x G ( t − x, k ) q ( t ) y + ,n − ( t, k ) dt, y + , = 1 . (2.3)The identity (2.2) gives f + (0 , k ) = y + (0 , k ) = 1 + Z ∞ sin ktk q ( t ) f + ( t, k ) dt,f + (0 , k ) ′ = ik − Z ∞ q ( t ) f + ( t, k ) cos ktdt. (2.4)Let q ± = qχ ± , where χ ± is the characteristic function of the set R ± . We recall well-knownproperties of the Jost solutions (see e.g., [F63]). Define k q k = R R | xq ( x ) | dx . Lemma 2.1.
Let R ∞ (1 + x ) | q ( x ) | dx < ∞ and let ς + ∈ {k q + k , k q + k| k | } . Then the functions f + ( x, · ) , f + ( x, · ) ′ , x > are analytic in C + and continuous up to the real line and satisfy | y + ( x, k ) | e ς + , | y + ( x, k ) − | ς + e ς + , | y + ( x, k ) − − y +1 ( x, k ) | ς e ς + , (2.5) D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 7 and | f + (0 , k ) − | ς + e ς + , | f + (0 , k ) − − f + , (0 , k ) | ς e ς + ,f + , (0 , k ) = Z ∞ sin ktk e ikt q + ( t ) dt. (2.6) Moreover, f + (0 , · ) ′ satisfies | f + (0 , k ) ′ − ik | k q + k e ς + , | f + (0 , k ) ′ − ik + f + , (0 , k ) ′ | k q + k ς + e ς + ,f + , (0 , k ) ′ = Z ∞ e ikx q ( x ) cos kxdx. (2.7) Proof.
Let ς + = k q + k| k | and D n ( x ) = { t = ( t j ) n ∈ R n : x = t < t < t < ... < t n } for x > | G ( t, k ) | | k | for all t > , k ∈ C + \ { } into the identity y + ,n ( x, k ) = Z D n ( x ) (cid:18) Y j n G ( t j − t j − , k ) q ( t j ) dt j (cid:19) , (2.8)we obtain | y + ,n ( x, k ) | | k | n Z D n ( x ) (cid:18) Y j n | q ( t j ) | dt j (cid:19) = k q + k n n ! | k | n . (2.9)This shows that for each x > C + \ {| k | > ε } for any ε >
0. Each term of this series is an analytic function in C + . Hence thesum is an analytic function in C + . Summing the majorants we obtain estimates (2.5)-(2.6)for ς + = k q + k| k | . Thus the functions f + ( x, · ) , f + ′ ( x, · ) , x > C + and continuousup to the real line without the point 0.Let in addition k q + k = R ∞ x | q ( x ) | dx < ∞ and let ς + = k q + k . The function G ( t, k ) = sin ktk e ikt satisfy | G ( t, k ) | t for all k ∈ C + , t >
0. Then using above arguments we obtain | y + ,n ( x, k ) | Z D n ( x ) (cid:18) Y j n | ( t j − t j − ) q ( t j ) | dt j (cid:19) Z D n ( x ) (cid:18) Y j n | t j q ( t j ) | dt j (cid:19) = k q + k n n ! . (2.10)This shows that for each x > C + .Each term of this series is an analytic function in C + . Hence the sum is an analytic functionin C + . Summing the majorants we obtain estimates (2.5)-(2.6) for ς + = k q + k . Thus thefunctions f + ( x, · ) , f + ( x, · ) ′ x > C + and continuous up to the real line. EVGENY KOROTYAEV
Consider f + (0 , k ) ′ . From (2.5) and (2.4) we get the first estimate in (2.7). From (2.4) wehave f ′ + (0 , k ) − ik = − Z ∞ (cos kx ) e ikx q ( x ) y + ( x, k ) dx = − f + , (0 , k ) ′ − F ( k ) ,f ′ + , (0 , k ) = Z ∞ (cos kx ) e ikx q ( x ) dx, F ( k ) = Z ∞ (cos kx ) e ikx q ( x )( y + ( x, k ) − dx, (2.11)where | F ( k ) | Z ∞ | q ( x ) || y + ( x, k ) − | dx k q + k ς + e ς + , which yields (2.7).We consider the Jost function f − ( x, · ), defined by (1.4), which satisfies the integral equation f − ( x, k ) = e − ikx − Z x −∞ sin k ( t − x ) k q ( t ) f − ( t, k ) dt, x , k ∈ C + , (2.12)and the function y − ( x, k ) = e ikx f − ( x, k ) also satisfies the integral equation y − ( x, k ) = 1 − Z x −∞ sin k ( t − x ) k q ( t ) e ik ( x − t ) y − ( t, k ) dt. (2.13) Lemma 2.2.
Let R −∞ (1 + | x | ) | q ( x ) | dx < ∞ and let x and let ς − ∈ (cid:26) k q − k , k q − k| k | (cid:27) . Thenthe functions f − ( x, · ) , f − ( x, · ) ′ are analytic in C + and continuous up to the real line and satisfy | y − ( x, k ) | e ς − , | y − ( x, k ) − | ς − e ς − , | y − ( x, k ) − − y − ( x, k ) | ς − e ς − , (2.14) and | f − (0 , k ) − | ς − e ς − , | f − (0 , k ) − − f − (0 , k ) | ς − e ς − ,f − , (0 , k ) = − Z −∞ sin ktk e − ikt q − ( t ) dt, (2.15) and f −′ (0 , · ) satisfies | f − (0 , k ) ′ − ik | k q − k e ς − , | f − (0 , k ) ′ − ik + f ′− (0 , k ) | k q − k ς − e ς − ,f ′− , (0 , k ) = Z −∞ q ( x ) e ikx cos kx dx (2.16) Proof.
The proof repeats the case of Lemma 2.1.Using Lemmas 2.1 and 2.2 we have the following decompositions: f + (0 , k ) = 1 + F + s ( k ) , F + s ( k ) = Z ∞ sin ktk q ( t ) f + ( t, k ) dt,f + (0 , k ) ′ = ik − kF + c ( k ) , F + c ( k ) = Z ∞ cos ktk q ( t ) f + ( t, k ) dt, (2.17) D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 9 and f − (0 , k ) = 1 − F − s ( k ) , F − s ( k ) = Z −∞ sin ktk q ( t ) f − ( t, k ) dt,f −′ (0 , k ) = − ik + kF − c ( k ) , F − c ( k ) = Z ∞ cos ktk q ( t ) f − ( t, k ) dt. (2.18) Lemma 2.3.
Let R R (1 + | x | ) | q ( x ) | dx < ∞ and let ς ∈ (cid:26) k q k , k q k| k | (cid:27) . Then the Wronskian w isanalytic in C + and continuous up to the real line and satisfies | w ( k ) | (2 | k | + k q k ) e ς , (2.19) and w = 2 ik − w ( k ) − w ( k ) ,w ( k ) = Z ∞ q ( t ) y + ( t, k ) dt + Z −∞ q ( t ) y − ( t, k ) dt,w ( k ) = k ( F + s F − c − F + c F − s ) = Z ∞ dx Z −∞ (1 − e i k ( x − y ) )2 ik q ( x ) q ( y ) y + ( x, k ) y − ( y, k ) dy, (2.20) and | w ( k ) | k q k e ς , | w ( k ) − q | k q k e ς , | w ( k ) | ( k xq + kk q − k + k q + kk xq − k ) e ς , | w ( k ) | k q − kk q + k| k | e ς , (2.21) where q = R R qdx , and we have at ς = k q k| k | : | ψ ( k ) − | ( ς + ς )2 e ς , | ψ ( k ) − q ik | ς e ς . (2.22) Proof.
We show (2.19). From Lemmas 2.1, 2.2 we have | w ( k ) | = | f − (0 , k ) f + (0 , k ) ′ − f − (0 , k ) ′ f + (0 , k ) | e ς − ( | k | + k q + k e ς + ) + ( | k | + k q − k e ς − ) e ς + (2 | k | + k q k ) e ς . We show (2.20). Let q ∈ L ( R ). Using (2.17), (2.18) we obtain w = { f − , f + } = ( f − f + ′ − f −′ f + )(0 , k )= (1 − F − s )( ik − kF + c ) − ( − ik + kF − c )(1 + F + s )= 2 ik − k ( F + c − iF + s ) − k ( F − c + iF − s ) − k ( F + s F − c − F + c F − s ) , (2.23)where k ( F + c − iF + s ) = Z ∞ e − ikt q ( t ) f + ( t, k ) dt = Z ∞ q ( t ) y + ( t, k ) dt,k ( F − c + iF − s ) = Z ∞ e ikt q ( t ) f − ( t, k ) dt = Z ∞ q ( t ) y − ( t, k ) dt. (2.24)Thus we have w = 2 ik − Z ∞ q ( t ) y + ( t, k ) dt − Z −∞ q ( t ) y − ( t, k ) dt − k ( F + s F − c − F + c F − s ) . (2.25) Let c x = cos kx and s x = sin kx . We get w = k ( F + s F − c − F + c F − s ) = Z ∞ dx Z −∞ ( s x c y − c x s y ) k q ( x ) q ( y ) f + ( x, k ) f − ( y, k ) dy = − Z ∞ dx Z −∞ (1 − e i k ( x − y ) )2 ik q ( x ) q ( y ) y + ( x, k ) y − ( y, k ) dy, (2.26)which yields (2.20). We show (2.21). Let q = R R q ( x ) dx . From (2.5), (2.14) we have w ( k ) = q + Z ∞ q ( x )( y + ( x, k ) − dx + Z −∞ q ( x )( y − ( x, k ) − dx, | w ( k ) − q | Z ∞ | q ( x )( y + ( x, k ) − | dt + Z −∞ | q ( x )( y − ( x, k ) − | dx k q + k ς + e ς + + k q − k ς − e ς − k q k ςe ς . Similar arguments yield | w ( k ) | k q k e ς .From (2.5), (2.14) if | k | > | w ( k ) | Z ∞ dx Z −∞ | q ( x ) q ( y ) y + ( x, k ) y − ( y, k ) | dy | k | k q + kk q − k| k | e ς + + ς − ς k q k e ς , and if | k | | w ( k ) | Z R + dx Z R − ( x − y ) | q ( x ) q ( y ) y + ( x, k ) y − ( y, k ) | dy ( k xq + kk q − k + k q + kk xq − k ) e ς , where the simple estimate has been used : | − e ikz | z | k | for all ( k, z ) ∈ C + × R + .We show (2.22). From (2.21) we have w ( k ) − ik + q = ( q − w ) − w and | w ( k ) − ik + q | = | ( q − w ) − w | k q k ςe ς , which yields the first estimate in (2.22). Similar arguments give the second one in (2.22).3. Proof of main theorems
In order to study zeros of the function ψ ( k ) = w ( k )2 ik in the upper-half plane we need to studythe Blaschke product, defined by (1.9). Recall that in order to describe the basic propertiesof the Blaschke product B as an analytic function in C + we modify the function ψ and definethe modified function by Ψ( k ) = w ( k )2 i ( k + i ) , k ∈ C + . We recall the well-known identity ψ ( k ) = det( I + Y ( k )) , k ∈ C + , (3.1)where that Y ( k ) is a trace class operator given by Y ( k ) = | q | R ( k ) | q | e i arg q , R ( k ) = ( H − k ) − , k ∈ C + . Proof Proposition 1.1.
The zeros of w, ψ and Ψ in C + are the same. Due to estimate(2.19) the function Ψ ∈ H ∞ . The estimate (2.22) gives that all zeros of Ψ are uniformlybounded. Note that (see page 53 in [G81]), in general, in the upper half plane the condition(1.7) is replace by X Im k j | k j | < ∞ , (3.2) D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 11 and the Blaschke product with zeros k j has the form B ( z ) = ( k − i ) m ( k + i ) m N Y k j =0 | k j | k j (cid:18) k − k j k − k j (cid:19) , k ∈ C + . (3.3)If all moduli | k n | are uniformly bounded, the estimate (3.2) becomes P Im k j < ∞ and theconvergence factors in (3.3) are not needed, since Q Nk j =0 (cid:0) k − k j k − k j (cid:1) already converges.The statement i) is a standard fact for the function Ψ ∈ H ∞ , see Sect. VI in [Ko98]. Lemma2.3 and Lemma 4.3 imply ii).We describe the determinant ψ ( k ) , k ∈ C + in terms of a canonical factorization. Proof of Theorem 1.2.
From Proposition 1.1 we have that the modified function Ψ ∈ H ∞ and due to (2.22)the function Ψ has asymptotics Ψ( k ) = 1 + O (1 /k ) as | k | → ∞ uniformly inarg k ∈ [0 , π ]. Then from Theorem 4.1 we deduce that the function Ψ ∈ H ∞ has a canonicalfactorization in C + given byΨ = Ψ in Ψ out , Ψ in ( k ) = B ( k ) e − iK ( k ) , K ( k ) = 1 π Z R dν ( t ) k − t . (3.4) • dν ( t ) > R , which satisfies ν ( R ) = Z R dν ( t ) < ∞ , supp ν ⊂ { k ∈ R : Ψ( k ) = 0 } ⊂ [ − r c , r c ] , (3.5)for r c = k q k , since w satisfies (1.8). • The function K ( · ) has an analytic continuation from C + into the domain C \ [ − r c , r c ] andhas the following Taylor series K ( k ) = ∞ X j =0 K j k j +1 , K j = 1 π Z R t j dν ( t ) . (3.6) • B is the Blaschke product for Im k > • Ψ out is the outer factor given by Ψ out ( k ) = e iM ( k ) , where M ( k ) = π R R log | Ψ( t ) | k − t dt, k ∈ C + .We consider the function Ψ = ψξ , where ξ = kk + i . It is clear that ξ = kk + i has the followingfactorization ξ ( k ) = kk + i = exp 1 π Z R log | ξ ( t ) | dtk − t , k ∈ C + , (3.7)where log ξ ( k ) in C + is defined by log ξ ( k ) = O (1 /k ) as | k | → ∞ . This yields M ( k ) = 1 π Z R log | ψ ( t ) | k − t dt + 1 π Z R log | ξ ( t ) | k − t dt = 1 π Z R log | ψ ( t ) | k − t dt + log ξ ( k ) , for all k ∈ C + . Here log | Ψ | , log | ξ | ∈ L loc ( R ), which yields log | ψ | ∈ L loc ( R ), since Ψ = ψξ .Thus from the properties of Ψ we obtain all properties of ψ formulated in Theorem 1.2.Remark that a canonical factorization is the first trace formula. It is a generating function,the differentiation of a canonical factorization produces an identity for Tr( R ( k ) − R ( k )) , k ∈ C + , where R ( k ) = ( H − k ) − and R ( k ) = ( − ∂ x − k ) − is the free resolvent. Corollary 3.1.
Let a potential q satisfy (1.1). Then the trace formula − k Tr (cid:18) R ( k ) − R ( k ) (cid:19) = X i Im k j ( k − k j )( k − k j ) + iπ Z R dµ ( t )( t − k ) , (3.8) holds true for any k ∈ C + \ k q , where the measure dµ ( t ) = log | ψ ( t ) | dt − dν ( t ) and the seriesconverges uniformly in every bounded disc in C + \ k q . Proof . We repeat arguments from the proof of Corollary 1.3 from [K17]. Differentiating(1.12) and using Theorem 1.2 we obtain ψ ′ ( k ) ψ ( k ) = B ′ ( k ) B ( k ) − iπ Z R dµ ( t )( k − t ) , B ′ ( k ) B ( k ) = X i Im k j ( k − k j )( k − k j ) , ∀ k ∈ C + , (3.9)where dµ ( t ) = h ( t ) dt − dν ( t ). The derivative of the determinant ψ = det( I + Y ( k )) definedby (3.1) satisfies ψ ′ ( k ) ψ ( k ) = − k Tr (cid:0) R ( k ) − R ( k ) (cid:1) , ∀ k ∈ C + , (3.10)see [GK69]. Combining (3.9), (3.10) we obtain (3.8). Note that the series converges uniformlyin every bounded disc in C + \ k q , since P Im k j < ∞ .We prove the first main result about the trace formulas. Proof of Theorem 1.3.
Let a potential q satisfy (1.1). Then due to Lemma 2.1 the function ψ ( t ) = 1 − Im Q + O (1 /t ) t as t → ±∞ and due to Theorem 4.1 the function h ( t ) = log | ψ ( t ) | , t ∈ R ,belongs to L loc ( R ). Then the function h ( t ) satisfies all conditions in Lemma 4.2 for m = 0and the function M ( k ) = π R R h ( t ) k − t dt, k ∈ C + satisfies M ( k ) = J + iI + o (1) k as k = iv, v → ∞ .From Lemma 4.2 and from asymptotics (1.18) we obtainexp (cid:20) i Q + o (1) k (cid:21) = exp (cid:20) − i B k − i K k + i J + iI k + o (1) k (cid:21) , as k = iv, v → ∞ , which yields Re Q = B + K − J , since I j = Im Q j . Thus we have(1.22). The proof of (1.23) is similar. Proof of Theorem 1.4.
We estimate the integral J = π R ∞ ξ ( k ) dk in (1.21), where ξ ( k ) =log | ψ ( k ) ψ ( − k ) | . We rewrite J in the following form J = 1 π ( J + J ) , J = Z ε ξ ( k ) dk, J = Z ∞ ε ξ ( k ) dk, (3.11)where ε := k q k for shortness. Consider J for the case ε >
1. We have J = Z ξ ( k ) dk + Z ε ξ ( k ) dk. (3.12)The estimate (2.19) gives with ς = k q k for k ∈ (0 , Z ξ ( k ) dk Z log (cid:18) k + r c k e ω (cid:19) dk = 2 k q k + 2 Z log (cid:18) r c k (cid:19) dk = 2 k q k + 2 log(1 + r c ) + 2 Z r c k dk r c k k q k + 2 log(1 + r c ) + 2 , (3.13) D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 13 and with ς = k q k| k | for k ∈ (1 , ε ): Z ε ξ ( k ) dk k q k Z ε dkk + 2 Z ε log (cid:18) r c k (cid:19) dk = 2 ε log ε + 2 k log (cid:18) r c k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ε + 2 Z ε r c k dk r c k ε log ε + 2 ε log 32 − r c ) + 2( ε − . This estimate and (3.13) give J k q k + 2 ε (cid:0) ε (cid:1) . (3.14)Consider J . Due to (2.21) we define g ( k ) , e g ( k ) by ψ = 1 + g and g = − q ik + e g ( k ). Theestimate (2.21) gives for k > ε : | e g ( k ) | ς e ς , | ψ ( k ) − | ς (1 + ς ) e ς ςe ς , (3.15)where ς = k q k| k | <
1. This yields f ( t ) = ψ ( t ) ψ ( − t ) = (1 + g ( k ))(1 + g ( − k )) = 1 + g ( k ) + g ( − k ) + g ( k ) g ( − k ) , (3.16)where | g ( k ) g ( − k ) | ς e ς ,g ( k ) + g ( − k ) = e g ( k ) + e g ( − k ) , | e g ( k ) + e g ( − k ) | ς e ς . (3.17)Thus we obtain f ( k ) = 1 + f ( k ) , f ( k ) = e g ( k ) + e g ( − k ) + g ( k ) g ( − k ) , | f ( k ) | ς e ς . (3.18)This yields f f = (1 + f )(1 + f ) = 1 + 2 Re f + | f | , (3.19)and then log | f | F, F = 2 Re f + | f | ς e ς + 9 ς e ς . (3.20)We have the identities V m = Z ∞ ε ε m t m e mεt dt = ε Z ∞ t m e mt dt = ε Z s m − e ms ds,V = ε e − , V = ε e − . Then we obtain J = 12 π Z ∞ ε log | f ( k ) | dk π Z ∞ ε F dk, (3.21)and Z ∞ ε F dk Z ∞ ε (6 ς e ς + 9 ς e ς ) dk = 6 V + 9 V = εC , (3.22)where C = 3( e −
1) + ( e − J k q k π + 2 επ (cid:0) ε (cid:1) + επ C = 2 k q k π + 2 επ (cid:0) C + log 32 + log ε (cid:1) . (3.23) Consider the case ε <
1. We need to estimate only J , since we obtain the estimate J for any ε >
0. The estimate (2.19) gives with ς = k q k for k ∈ (0 , ε ): Z ε ξ ( k ) dk Z ε log (cid:18) k + r c k e ω (cid:19) dk = 2 k q k ε + 2 Z ε log (cid:18) ε k (cid:19) dk = 2 ε k q k + 2 ε Z log (cid:18) t (cid:19) dt = 2 ε k q k + 2 ε log 3 , (3.24)since R log (cid:0) t (cid:1) dt = log . Thus collecting (3.24), (3.22) we obtain J επ k q k + 2 επ log 3 επ C . Appendix: Analytic functions in the upper half-plane
Let f ∈ H ∞ ( C + ) and let { k j } be its zeros, uniformly bounded by r . Then its Blaschkeproduct B ∈ H ∞ ( C + ) and satisfieslim v → +0 B ( u + iv ) = B ( u + i , | B ( u + i | = 1 almost e . w . for u ∈ R , (4.1)lim v → Z R log | B ( u + iv ) | du = 0 . (4.2)The function log B ( k ) is analytic in {| k | > r } and has the corresponding Tailor series givenby log B ( k ) = − iB k − iB k − iB k − .... − iB n − nk n − .... (4.3)where each sum B n = 2 P j Im k n +1 j , n > | B n | X | Im k n +1 j | π ( n + 1) r n B < ∞ ∀ n > , (4.4)see e.g., [K17x]. We recall the standard facts about the canonical factorization, see e.g. [G81],[Ko98] and in the needed specific form for us from [K17x]. Theorem 4.1.
Let a function f ∈ H p and f ( t + i
0) = 1 + O ( t − a ) as t → ±∞ for some p > , a > . Then f has a canonical factorization in C + given by f = f in f out , f in ( k ) = B ( k ) e − iK ( k ) , K ( k ) = 1 π Z R dν ( t ) k − t . (4.5) • dν ( t ) > is some singular compactly supported measure on R , which satisfies ν ( R ) = Z R dν ( t ) < ∞ , supp ν ⊂ { k ∈ R : f ( k ) = 0 } ⊂ [ − r c , r c ] , (4.6) for some r c > . D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 15 • The function K ( · ) has an analytic continuation from C + into the domain C \ [ − r c , r c ] andhas the following Taylor series K ( k ) = ∞ X j =0 K j k j +1 , K j = 1 π Z R t j dν ( t ) . (4.7) • B is the Blaschke product for Im k > given by (1.9). • f out is the outer factor given by f out ( k ) = e iM ( k ) , M ( k ) = 1 π Z R log | f ( t ) | k − t dt, k ∈ C + , (4.8) where the function log | f ( t + i | belongs to L loc ( R ) . Remark.
1) These results are crucial to determine trace formulas in Theorem 1.3.2) The integral M ( k ) in (4.8) converges absolutely, since f ( t ) = 1 + O (1) t a as t → ±∞ .In order to determine trace formulas in Theorem 1.3 we recall asymptotics from [K17x]. Lemma 4.2.
Let h be a real function from L loc ( R ) and let h satisfy h ( t ) = − P m ( t ) − O (1) t m +2 , P m ( t ) = I t + I t + ... + I m t m + a , (4.9) as t → ±∞ , for some constants a > , I , I , ...., I m ∈ R and integer m > . Then π Z R h ( t ) k − t dt = J + iI k + J + iI k + ... + J m + iI m + o (1) k m +1 , (4.10) at k = iv, v → ∞ , where J j = 1 π Z ∞ ( h j ( t ) + h j ( − t )) dt, h = h, h j = t j ( h ( t ) + P j − ( t )) , j = 0 , , , ..., m. Lemma 4.3.
Let a function f ∈ H ∞ and be continuous up to the real line. Assume that itsatisfies | f ( k ) − (2 ik − τ o ) | τ ςe ς , ∀ k ∈ C + , ς = min { σ, τ | k | } , (4.11) for some positive constants σ, τ and τ o ∈ C \ R and | τ o | τ . Theni) If τ σe σ < Re τ o , then the function f does not have zeros in C + .ii) If τ σe σ < − Re τ o , then function f has exactly one simple zeros in C + . Proof.
Define the function e f ( k ) = f ( k ) − (2 ik − τ o ), where | e f ( k ) | τ σe σ . We define thenew variable t = kτ ∈ C + and rewrite f in the form f ( k ) = iτ g ( t ), where g ( t ) = t + iε + i e g ( t ) , ε = τ o τ , e g ( t ) = e f ( k ) τ , | e g ( t ) | σe σ , | ε | . (4.12)i) Consider the first case, and let σe σ < Re ε . Then we have for all t ∈ C + :Im g ( t ) = Im t + Re ε + Re e g ( t ) > Im t + Re ε − σe σ > . (4.13)ii) Consider the second case, let ε = − r + iν, r > σe σ < − Re ε = r . Then iε = − ( ν + ir ) and define the new variable z = t − ν and g = z − ir + e g ( t ) , g = z − ir. Introduce the contour C ρ = C + ρ ∪ I ρ , where C + ρ = {| z | = ρ, z ∈ C + } and I ρ = [ − ρ, ρ ] for ρ > r .Then, we have for all z ∈ C ρ and any ρ > r : | g ( t ) − g ( z ) | = | e g ( t ) | σe σ = | g ( z ) | ue u | g ( z ) | < | g ( z ) | , t = z + ν, where we have used σe σ | g ( z ) | σe σ r < C ρ , since | g ( t ) | > r for all t ∈ R , and | g ( t ) | = | ρe iφ − ir | > ρ − r > r ∀ t ∈ C ρ . Thus, by Rouche’s theorem, g has one simple root, as g in the region {| z | ρ, z ∈ C + } .5. Schr¨odinger operators with compactly supported potentials
Entire functions.
An entire function f ( k ) is said to be of exponential type if there isa constant β such that | f ( k ) | const e β | k | everywhere. The infimum of the set of β for whichsuch inequality holds is called the type of f . Definition.
Let E γ , γ > denote the class of exponential type functions f satisfying | f ( k ) − (2 ik − τ o ) | τ ςe γk − + ς , ∀ k ∈ C + , ς ( k ) = min { σ, τ | k | } , (5.1) where k − = ( | Im k | − Im k ) > , for some constants ( σ, τ, τ o ) ∈ R × C and | τ o | v andeach its zero z in C + satisfies | z | τ . Note that if q ∈ L ( R + ) and supp q ⊂ [0 , γ ], then the Wronskian w ∈ E γ , see below theproof of Theorem 1.5. Define the disk D r ( t ) = { z : | t − z | < r } for t ∈ C and r > Lemma 5.1. i) Let f ∈ E γ , γ > . If k ∈ C + and | k | > τ , then | f ( k ) || k | > − √ e > . (5.2) Moreover, the number of zeros N ( ρ ) of f (counted with multiplicity) in disk D ρ ( it ) with thecenter it = i τ and the radius ρ > √ τ satisfies N ( ρ ) (cid:18) γρπ + τρ (cid:19) , (5.3) In particular, the number of zeros N + of f (counted with multiplicity) in C + satisfies N + C + C γτ, (5.4) where the constants C = 1 + √
17 log 2 + ε and C = √ π log 2 + ε , for any small ε , ε > . Proof. i) We have ς ( k ) and ψ = f ik satisfies | ψ ( k ) − (1 − τ o ik ) | τς | k | e ς and then | ψ ( it ) − (1 + τ o t ) | √ e for any k ∈ C + , which yields | ψ ( k ) | > − | τ o | τ − √ e > −√ e > .ii) Recall the Jensen formula (see p. 2 in [Koo88]) for an entire function F any r > | F (0) | + Z r N s ( F ) s ds = 12 π Z π log | F ( re iφ ) | dφ, (5.5)where N s ( F ) is the number of zeros of F in the disk D s (0). We take the function F ( z ) = f ( it + z ) and the disk D r ( it ) , t = 2 τ with the radius r > √ τ . Thus (5.5) implieslog | f ( it ) | + Z r N ( s ) s ds = 12 π Z π log (cid:12)(cid:12) f ( k φ ) (cid:12)(cid:12) dφ, k φ = it − ire iφ , (5.6) D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 17 where N ( s ) = N s ( f ( it + · )). We rewrite (5.6) in terms of a function ψ ( k ) := f ( k )2 ik :log | ψ ( it ) | + log | t | + Z r N ( s ) s ds = S + log | r | , S = 12 π Z π log (cid:12)(cid:12) ψ ( k φ ) (cid:12)(cid:12) dφ, (5.7)where we have used the identity Z T log | it − re iφ | dφ = Z T log (cid:18) r (cid:12)(cid:12)(cid:12)(cid:12) itr − e iφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) dφ = 2 π log r, since R T log | α − e iφ | dφ = 0 for any | α | <
1. From (5.1) we obtain at k φ = it − ire iφ and a ∈ [ π , π ] defined by cos a = tr : S = 12 π Z π log (cid:12)(cid:12) ψ ( k φ ) (cid:12)(cid:12) dφ = S + S , S = 12 π Z a − a log (cid:12)(cid:12) ψ ( k φ ) (cid:12)(cid:12) dφ,S = 12 π Z π − aa log (cid:12)(cid:12) ψ ( k φ ) (cid:12)(cid:12) dφ π Z π − aa log (cid:0) ς e ς (cid:1) dφ = π − aπ log (cid:0) ς e ς (cid:1) , (5.8)where on the circle { z : | it − z | = r } using (5.1) we get for k φ = it − ire iφ :min a φ π | k φ | = | t − re ia | = r sin a = √ r − t ,ς := max a φ π ς ( k φ ) = τr sin a τr √ , (5.9)and min φ a | k φ | = ( r − t ) = r (1 − cos a ) ,ς o := max φ a ς ( k φ ) = τr (1 − cos a ) τr (1 − √ ) . (5.10)Consider the integral S = γπ R a k − dφ , where we have k − = r cos φ − t = r (cos φ − cos a ) for φ ∈ ( − a, a ). Thus we obtain S = 2 γπ Z a r (cos φ − cos a ) dφ = 2 γrπ (sin a − a cos a ) γrπ , (5.11)and (5.1), (5.9) give S = 12 π Z a − a log (cid:12)(cid:12) ψ ( k φ ) (cid:12)(cid:12) dφ π Z a log (cid:0) ς o e γk − + ς o (cid:1) dφ S + 1 π Z a log (cid:0) ς o e ς o (cid:1) φ = S + 2 aπ ς o . (5.12)Collecting (5.8)-(5.12) we obtain S π ( γr + ( π − a ) ς + aς o ) . (5.13)We show (5.3) and let ρ = r . We have Z r N ( s ) dss > N ( ρ ) Z rρ dss = N ( ρ ) log 2 . Summing (5.8)-(5.13) we obtain − log 2 + N ( ρ ) log 2 < π ( γr + ( π − a ) ς + aς o ) π γr + 32 ς + 12 ς o π γr + 8 τr , which yields (5.3). We show (5.4). We take any r = √ τ and ρ = r > √ τ . Then all zerosof f in C + belong to the disk D ρ ( it ) and (5.3) implies N + C + γτ C , C = 1 + 8 √
17 log 2 + ε , C = 2 √ π log 2 + ε , for any small ε , ε >
0, which yields (5.4). ✲✻ Im k Re k ✛ rr rr rr Q − Q iQt = 2 iQrr Figure 1.
The case 2
Q < r < √ Q Fundamental solutions.
Consider the Schr¨odinger operator H , when the potential q ∈ L ( R ) is complex and supp q ⊂ [0 , γ ] for some γ >
0. In this case the equation (2.2) for y + ( x, k ) = e − ikx f + ( x, k ) for all ( x, k ) ∈ [0 , γ ] × C has the form y + ( x, k ) = 1 + Z γx sin k ( t − x ) k e ik ( t − x ) q ( t ) y + ( t, k ) dt. (5.14)We recall well-known some properties of the functions f + , y + (see e.g., [K16]). Lemma 5.2.
Let q ∈ L ( R ) and supp q ⊂ [0 , γ ] . Then the function f + (0 , k ) is entire andsatisfies | y + ( x, k ) | e γ − x ) k − + ς , | y + ( x, k ) − | ςe γ − x ) k − + ς , (5.15) for all k ∈ C , where k − = ( | Im k | − Im k ) and ς = min {k q k , k q k| k | } . Lemma 5.3.
Let q ∈ L ( R ) and supp q ⊂ [0 , γ ] . Then the Wronskian w is entire and satisfies w ( k ) = 2 ik − q − Z ∞ q ( t )( y + ( t, k ) − dt, (5.16) | w ( k ) − (2 ik − q ) | ς k q k e γk − + ς ( k ) , (5.17) D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 19 where q = R R qdx and ς ( k ) = min {k q k , k q k| k | } . Proof.
We show (2.19). From Lemmas 2.1, 2.2 we have w = { f − , f + } = f + ′ (0 , k ) + ikf + (0 , k ) = 2 ik + Z γ ( i sin kx − cos kx ) q ( x ) f + ( x, k ) dx,w − ik = − Z γ e − ikx q ( x ) f + ( x, k ) dx = − Z γ q ( x ) y + ( x, k ) dx. (5.18)Thus using (5.18) and Lemma 5.2 we obtain (5.16), (5.17).5.3. Proof of Theorem 1.5.
Consider the Schr¨odinger operator H , when the potential q ∈ L ( R ) is complex and supp q ⊂ [0 , γ ] for some γ >
0. Due to Lemma 5.3 the Wronskian w is entire and satisfies | w ( k ) − (2 ik − q ) | ς ( k ) e γk − + ς ( k ) for all k ∈ C , where ς ( k ) =min {k q k , k q k| k | } Thus due to this fact and (1.8) we obtain that the function w ∈ E γ with τ = k q k , τ o = q and σ = k q k . Then Lemma 5.1 gives the estimate (1.27), (1.28).6. Operators on a half-line with the Neumann boundary condition
Definitions.
In this section we consider Schr¨odinger operators H n on L ( R + ) and H d given by Case H n y = − y ′′ + q + y, with Neumann b . c . y ′ (0) = 0 ,Case H d y = − y ′′ + q + y, with Dirichlet b . c . y (0) = 0 , where the potential q + is complex and satisfies: Z ∞ (1 + x ) | q + ( x ) | dx < ∞ . (6.1)We define functions w n , ψ n , Ψ n and w d = ψ d in C + by w n = f ′ + (0 , k ) , ψ n ( k ) = w n ik , Ψ n = w n i ( k + i ) , ψ d = w d = f + (0 , k )It is known that the operator H a , a = n , d has two components: the essential spectrum whichcovers the half-line [0 , ∞ ) plus N a ∞ eigenvalues (counted with multiplicity) in the cutspectral domain C \ [0 , ∞ ). We denote them by E a,j ∈ C \ [0 , ∞ ) , j = 1 , ..., N a , accordingto their multiplicity. Note, that the multiplicity of each eigenvalue equals 1, but we call themultiplicity of the eigenvalue its algebraic multiplicity. We call k a,j = √ E a,j ∈ C + also theeigenvalues of the operator H a and label them by Im k a, > Im k a, > Im k a, > ... . We definethe set k a = { k a, , k a, , ... ∈ C + } and the Blaschke product B a = Y j > k − k a,j k − k a,j , k ∈ C + . This product converges absolutely for each k ∈ C + , since all zeros of w a are uniformly bounded,see (2.7). Moreover, it has an analytic continuation from C + into the domain {| k | > r c } , where r c = k q k and has the following Taylor serieslog B a ( k ) = − i B a, k − i B a, k − i B a, k − ..., as | k | > r c , (6.2)where B a, = 2 P Nj =1 Im k a,j , .... We show the relations between the operators H and H a . Here we use a standard trick. For q + ∈ L ( R + ) we define the Schr¨odinger operator e H on L ( R ) with an even potential e q by e Hy = − y ′′ + e qy, e q ( x ) = q + ( | x | ) , x ∈ R . (6.3)Recall that the Schr¨odinger equation − f ′′ + e qf = k f, has unique Jost solutions e f ± ( x, k ). Forthe case of the even potential e q they satisfy e f + ( x, k ) = e f − ( − x, k ) ∀ x > . This implies that the Wronskian e w ( k ) for the operator e H has the specific form: e w ( k ) = { e f + ( x, k ) , e f − ( x, k ) }| x =0 = 2 f + (0 , k ) f ′ + (0 , k ) = 2 w d w n . (6.4)This identity is very useful. For example, if we have estimate (1.27), (1.28) for the operator H , then (6.4) gives the estimate (6.5), (6.6) for the operators H n and H d . Consider estimatesfor complex compactly supported potentials q + . In this case the Jost function ψ ( k ) = f + (0 , k )is entire and due to (1.8) it has a finite number of zeros in C + . Corollary 6.1.
Let q + ∈ L ( R + ) and let supp q + ⊂ [0 , γ ] for some γ > . Then the numberof zeros N ρ ( w a ) of w a , a = n , d (counted with multiplicity) in disk D ρ ( it ) with the center it = i k q + k and the radius ρ > √ k q + k satisfies N ρ ( w n ) + N ρ ( w d ) (cid:18) γρπ + 2 k q + k ρ (cid:19) . (6.5) In particular, the number of zeros N + ( w a ) of w a (counted with multiplicity) in C + satisfies N + ( w n ) + N + ( w d ) C + 2 C γ k q + k , (6.6) where the constants C , C (see more about C , C in Lemma 5.1). Proof.
Note that we have supp e q ⊂ [ − γ, γ ] and the identity k e q k = 2 k q + k . Then usingthese relations and (6.4) and (1.27), (1.28) for the operator H , we obtain (6.5), (6.6) for theoperators H n and H d .We describe the basic properties of H d , H n and the Blaschke product B n . Proposition 6.2.
Let a potential q + be complex and let c := R R + | q + ( x ) | dx < ∞ . Theni) Each eigenvalue E of H n satisfies | E | c .ii) Let q + , = R ∞ q + ( x ) dx and c := R R + | xq + ( x ) | dx < ∞ , and let A = c c e c . Then • If A < Re q + , , then the operator H a does not have eigenvalues. • If A <
Re( − q + , ) , then H d does not have eigenvalues and the operator H n has one simpleeigenvalue.iii) The Blaschke product B n ( k ) , k ∈ C + given by (1.9) belongs to H ∞ with k B n k H ∞ . Proof.
The proof is similar to the case of Proposition 1.1.6.2.
Trace formulas and estimates.
We describe the Jost function ψ n in terms of a canon-ical factorization, which in general ψ n / ∈ H ∞ . Theorem 6.3.
Let q + satisfy (6.1). Then ψ n has a standard canonical factorization: ψ n = ψ n ,in ψ n ,out , D SCHR ¨ODINGER OPERATORS WITH COMPLEX POTENTIALS 21 where the inner factor ψ n ,in and the outer factor ψ n ,out are given by ψ n ,in ( k ) = B n ( k ) ik e − iK n ( k ) , K n ( k ) = 1 π Z R dν n ( t ) k − t ,ψ n ,out ( k ) = e iM a ( k ) , M n ( k ) = 1 π Z R log | ψ n ( t + i | k − t dt, (6.7) for all k ∈ C + and the function log | ψ n ( t + i | belongs to L loc ( R ) . • dν n ( t ) > is some singular compactly supported measure on R , which satisfies ν n ( R ) = Z R dν n ( t ) < ∞ , supp ν n ⊂ { z ∈ R : w n ( z ) = 0 } ⊂ [ − r c , r c ] , (6.8) • The function K n ( · ) has an analytic continuation from C + into the cut domain C \ [ − r c , r c ] and has the following Taylor series in the domain {| k | > r c } : K n ( k ) = ∞ X j =0 K n ,j k j +1 , K n ,j = 1 π Z R t j dν n ( t ) . (6.9) Proof.
The proof is similar to the case of Theorem 1.2.The Jost function ψ d = f + (0 , k ) for the operator H d on R + with the Dirichlet boundarycondition at x = 0 also the has a standard canonical factorization similar to ψ n in Theorem6.3, see [K18]. Moreover, the corresponding inner factor ψ d ,in is expressed in terms of somesingular compactly supported measure dν d ( t ) > R , which satisfies ν d ( R ) = Z R dν d ( t ) < ∞ , supp ν d ⊂ { z ∈ R : ψ d ( z ) = 0 } ⊂ [ − r c , r c ] , (6.10)Then from (6.8) and (6.10) we obtain the simple fact:supp ν n ∩ supp ν d = ∅ . (6.11) Theorem 6.4.
Let a potential q + satisfy (6.1). Then B n , + ν n ( R ) π + 12 Z ∞ Re q + ( x ) dx = v . p . π Z R log | ψ n ( t ) | dt, (6.12) where the integral in the r.h.s converges. Moreover, the following hold true B n , + ν n ( R ) π + 12 Z ∞ Re q + ( x ) dx π (cid:0) k q + k (cid:1) + r + (cid:0) C o + log r + (cid:1) , (6.13) where k q + k = R ∞ | xq + ( x ) | dx and r + = k q + k and C o is some absolute constant. Proof.
The proof is similar to the case of Theorem 1.3 and 1.4.
Acknowledgments.
EK is grateful to Ari Laptev (London) for discussions about the Schr¨odinger operatorswith complex potentials and to Alexei Alexandrov (St.Petersburg) for discussions and useful comments aboutHardy spaces. Our study was supported by the RSF grant No 18-11-00032.
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