Boundary Data Maps for Schrodinger Operators on a Compact Interval
aa r X i v : . [ m a t h . SP ] F e b BOUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORSON A COMPACT INTERVAL
STEPHEN CLARK, FRITZ GESZTESY, AND MARIUS MITREA
Abstract.
We provide a systematic study of boundary data maps, that is,2 × , R ] with separated boundary conditions at 0 and R . Most of ourresults are formulated in the non-self-adjoint context.Our principal results include explicit representations of these boundarydata maps in terms of the resolvent of the underlying Schr¨odinger operatorand the associated boundary trace maps, Krein-type resolvent formulas re-lating Schr¨odinger operators corresponding to different (separated) boundaryconditions, and a derivation of the Herglotz property of boundary data maps(up to right multiplication by an appropriate diagonal matrix) in the specialself-adjoint case. Introduction
To briefly set the stage for this paper, let
R >
0, introduce the strip S π = { z ∈ C | ≤ Re( z ) < π } , and consider the boundary trace map γ θ ,θ R : C ([0 , R ]) → C ,u " cos( θ ) u (0) + sin( θ ) u ′ (0)cos( θ R ) u ( R ) − sin( θ R ) u ′ ( R ) , θ , θ R ∈ S π , (1.1)where “prime” denotes d/dx . In addition, assuming that V ∈ L ((0 , R ); dx ) (1.2)(we emphasize that V is not assumed to be real-valued for most of this paper),one can introduce the family of one-dimensional Schr¨odinger operators H θ ,θ R in L ((0 , R ); dx ) by H θ ,θ R f = − f ′′ + V f, θ , θ R ∈ S π ,f ∈ dom( H θ ,θ R ) = (cid:8) g ∈ L ((0 , R ); dx ) (cid:12)(cid:12) g, g ′ ∈ AC ([0 , R ]); γ θ ,θ R ( g ) = 0; (1.3)( − g ′′ + V g ) ∈ L ((0 , R ); dx ) (cid:9) , were AC ([0 , R ]) denotes the set of absolutely continuous functions on [0 , R ]. Date : November 21, 2018.2000
Mathematics Subject Classification.
Primary: 34B05, 34B27, 34B40, 34L40; Secondary:34B20, 34L05, 47A10, 47E05.
Key words and phrases. (non-self-adjoint) Schr¨odinger operators on a compact interval, sep-arated boundary conditions, boundary data maps, Robin-to-Robin maps, linear fractional trans-formations, Krein-type resolvent formulas.Based upon work partially supported by the US National Science Foundation under Grant No.DMS-0653180.
Math. Modelling Natural Phenomena (to appear).
Assuming that z ∈ C \ σ ( H θ ,θ R ) (with σ ( T ) denoting the spectrum of T ) and θ , θ R ∈ S π , we recall that the boundary value problem given by − u ′′ + V u = zu, u, u ′ ∈ AC ([0 , R ]) , (1.4) γ θ ,θ R ( u ) = (cid:20) c c R (cid:21) ∈ C , (1.5)has a unique solution denoted by u ( z, · ) = u ( z, · ; ( θ , c ) , ( θ R , c R )) for each c , c R ∈ C . To each boundary value problem (1.4), (1.5), we now associate a family of general boundary data maps , Λ θ ′ ,θ ′ R θ ,θ R ( z ) : C → C , for θ , θ R , θ ′ , θ ′ R ∈ S π , whereΛ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:20) c c R (cid:21) = Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:0) γ θ ,θ R ( u ( z, · ; ( θ , c ) , ( θ R , c R ))) (cid:1) = γ θ ′ ,θ ′ R ( u ( z, · ; ( θ , c ) , ( θ R , c R ))) . (1.6)With u ( z, · ) = u ( z, · ; ( θ , c ) , ( θ R , c R )), then Λ θ ′ ,θ ′ R θ ,θ R ( z ) can be represented as a 2 × θ ′ ,θ ′ R θ ,θ R ( z ) (cid:20) c c R (cid:21) = Λ θ ′ ,θ ′ R θ ,θ R ( z ) " cos( θ ) u ( z,
0) + sin( θ ) u ′ ( z, θ R ) u ( z, R ) − sin( θ R ) u ′ ( z, R ) = " cos( θ ′ ) u ( z,
0) + sin( θ ′ ) u ′ ( z, θ ′ R ) u ( z, R ) − sin( θ ′ R ) u ′ ( z, R ) . (1.7)The map Λ θ ′ ,θ ′ R θ ,θ R ( z ) represents the principal object studied in this paper.We prove in Section 2 that Λ θ ′ ,θ ′ R θ ,θ R ( z ) is well-defined for z ∈ C \ σ ( H θ ,θ R ), that is,it is invariant with respect to a change of basis of solutions of (1.4), derive its basicproperties (cf. Corollary 2.4), and derive the explicit representation (2.50), (2.51)in terms of a distinguished basis of solutions (2.33).In Section 3, we relate a special case of Λ θ ′ ,θ ′ R θ ,θ R ( z ), given by the generalizedDirichlet-to-Neumann maps Λ θ ,θ R ( z ) = Λ ( θ + π ) mod(2 π ) , ( θ R + π ) mod(2 π ) θ ,θ R ( z ), to Weyl–Titchmarsh m -functions, derive its asymptotic behavior as z tends to infinity, andmost importantly, derive an explicit representation of the boundary data mapsΛ θ ′ ,θ ′ R θ ,θ R ( z ) in terms of the resolvent of the underlying Schr¨odinger operator H θ ,θ R and the associated boundary trace maps,Λ θ ′ ,θ ′ R θ ,θ R ( z ) S θ ′ − θ ,θ ′ R − θ R = γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ ,θ , θ R , θ ′ , θ ′ R ∈ S π , z ∈ C \ σ ( H θ ,θ R ) , (1.8)with S α,β denoting the 2 × S α,β = diag (cid:0) sin( α ) , sin( β ) (cid:1) .Theorem 4.1, the principal result in Section 4, then centers around the followinglinear fractional transformation relating the boundary data maps Λ θ ′ ,θ ′ R θ ,θ R ( z ) andΛ δ ′ ,δ ′ R δ ,δ R ( z ),Λ θ ′ ,θ ′ R θ ,θ R ( z ) = (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − (cid:2) S δ ′ − θ ′ ,δ ′ R − θ ′ R + S θ ′ − δ ,θ ′ R − δ R Λ δ ′ ,δ ′ R δ ,δ R ( z ) (cid:3) × (cid:2) S δ ′ − θ ,δ ′ R − θ R + S θ − δ ,θ R − δ R Λ δ ′ ,δ ′ R δ ,δ R ( z ) (cid:3) − S δ ′ − δ ,δ ′ R − δ R , (1.9)assuming θ , θ R , θ ′ , θ ′ R , δ , δ R , δ ′ , δ ′ R ∈ S π , δ ′ − δ = 0 mod( π ), δ ′ R − δ R = 0 mod( π ),and z ∈ C \ (cid:0) σ ( H θ ,θ R ) ∪ σ ( H δ ,δ R ) (cid:1) . The linear fractional transformation (1.9) then OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 3 is a major ingredient in our proof that Λ θ ′ ,θ ′ R θ ,θ R ( · ) S θ ′ − θ ,θ ′ R − θ R is a 2 × C + , the open complex upper half-plane, with anonnegative imaginary part) in the special case where H θ ,θ R is self-adjoint. (In thiscase, one necessarily assumes that θ , θ R , θ ′ , θ ′ R ∈ [0 , π ) and that V is real-valued.)In addition, we derive the Herglotz representation of Λ θ ′ ,θ ′ R θ ,θ R ( · ) S θ ′ − θ ,θ ′ R − θ R in termsof a 2 × R .The principal result proved in Section 5 then concerns Krein-type resolvent for-mulas explicitely relating the resolvents of H θ ,θ R and H θ ′ ,θ ′ R . A typical result tobe proved in Theorrem 5.3 is of the form( H θ ′ ,θ ′ R − zI ) − = ( H θ ,θ R − zI ) − − (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ S − θ ′ − θ ′ ,θ ′ R − θ R × h Λ θ ′ ,θ ′ R θ ,θ R ( z ) i − (cid:2) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:3) ,θ , θ R , θ ′ , θ ′ R ∈ S π , θ = θ ′ , θ R = θ ′ R , z ∈ C (cid:15)(cid:0) σ ( H θ ,θ R ) ∪ σ ( H θ ′ ,θ ′ R ) (cid:1) . (1.10)Formula (1.10) demonstrates why Λ θ ′ ,θ ′ R θ ,θ R is the ideal object for Krein-type resolventformulas.Finally, in Section 6, we describe some additional connections between Λ θ ,θ R ( z )and the Green’s function G θ ,θ R ( z, x, x ′ ) of H θ ,θ R , and then point out some interest-ing differences compared to the standard 2 × H θ ,θ R .For classical as well as recent fundamental literature on Weyl–Titchmarsh op-erators (i.e., spectral parameter dependent Dirichlet-to-Neumann maps, or moregenerally, Robin-to-Robin maps, resp., Poincar´e–Steklov operators), relevant in thecontext of boundary value spaces (boundary triples, etc.), we refer, for instance,to [2], [3], [4], [11], [12], [13]–[19], [24]– [28], [33]–[39], [42], [43, Ch. 3], [44], [45,Ch. 13], [48]–[50], [54], [57], [60], [64], [65], [70], [71], [72], [74]–[76], [81] and thereferences cited therein.Finally, we briefly summarize some of the notation used in this paper: Let H be a separable complex Hilbert space, ( · , · ) H the scalar product in H (linear inthe second argument), and I H the identity operator in H . Next, let T be a linearoperator mapping (a subspace of) a Banach space into another, with dom( T ) andker( T ) denoting the domain and kernel (i.e., null space) of T . The spectrum of aclosed linear operator in H will be denoted by σ ( · ). The Banach space of boundedlinear operators on H is denoted by B ( H ), the analogous notation B ( X , X ), willbe used for bounded operators between two Banach spaces X and X . Moreover, X ֒ → X denotes the continuous embedding of X into X .2. General Boundary Value Problems and Boundary Data Maps
This section is devoted to boundary data maps and their basic properties.Taking
R >
0, and fixing θ , θ R ∈ S π , with S π the strip S π = { z ∈ C | ≤ Re( z ) < π } , (2.1) S. CLARK, F. GESZTESY, AND MARIUS MITREA we introduce the linear map γ θ ,θ R , the trace map associated with the boundary { , R } of (0 , R ) and the parameters θ , θ R , by γ θ ,θ R : C ([0 , R ]) → C ,u " cos( θ ) u (0) + sin( θ ) u ′ (0)cos( θ R ) u ( R ) − sin( θ R ) u ′ ( R ) , θ , θ R ∈ S π , (2.2)where “prime” denotes d/dx . We note, in particular, that the Dirichlet trace γ D ,and the Neumnann trace γ N (in connection with the outward pointing unit normalvector at ∂ (0 , R ) = { , R } ), are given by γ D = γ , = − γ π,π , γ N = γ π/ , π/ = − γ π/ ,π/ . (2.3)Next, assuming V ∈ L ((0 , R ); dx ) , (2.4)we introduce the following family of densely defined closed linear operators H θ ,θ R in L ((0 , R ); dx ), H θ ,θ R f = − f ′′ + V f, θ , θ R ∈ S π ,f ∈ dom( H θ ,θ R ) = (cid:8) g ∈ L ((0 , R ); dx ) (cid:12)(cid:12) g, g ′ ∈ AC ([0 , R ]); γ θ ,θ R ( g ) = 0; (2.5)( − g ′′ + V g ) ∈ L ((0 , R ); dx ) (cid:9) . Here AC ([0 , R ]) denotes the set of absolutely continuous functions on [0 , R ]. Weemphasize that V is not assumed to be real-valued in the bulk of this paper.One notices that γ ( θ + π ) mod(2 π ) , ( θ R + π ) mod(2 π ) = − γ θ ,θ R , θ , θ R ∈ S π , (2.6)and, on the other hand, H ( θ + π ) mod(2 π ) , ( θ R + π ) mod(2 π ) = H θ ,θ R , θ , θ R ∈ S π , (2.7)hence it suffices to consider θ , θ R ∈ S π = { z ∈ C | ≤ Re( z ) < π } rather than θ , θ R ∈ S π in connection with H θ ,θ R , but for simplicity of notation we will keepusing the strip S π throughout this manuscript.That H θ ,θ R is indeed a closed operator follows, for instance, from [29, Sect.XII.4], especially, by combining Lemma 5 (c) and the first part of the proof ofLemma 26 and noting that g (0) , g ′ (0) (resp., g ( R ) , g ′ ( R )) are a complete set ofboundary values for the minimal operator H min associated with the differentialexpression − d /dx + V ( x ) in L ((0 , R ); dx ) at x = 0 (resp., at x = R ). Here H min f = − f ′′ + V f,f ∈ dom( H min ) = (cid:8) g ∈ L ((0 , R ); dx ) (cid:12)(cid:12) g, g ′ ∈ AC ([0 , R ]); (2.8) g (0) = g ′ (0) = g ( R ) = g ′ ( R ) = 0; ( − g ′′ + V g ) ∈ L ((0 , R ); dx ) (cid:9) . Morever, the adjoint of H θ ,θ R is given by( H θ ,θ R ) ∗ f = − f ′′ + V f, θ , θ R ∈ S π ,f ∈ dom (cid:0) ( H θ ,θ R ) ∗ (cid:1) = (cid:8) g ∈ L ((0 , R ); dx ) (cid:12)(cid:12) g, g ′ ∈ AC ([0 , R ]); γ θ ,θ R ( g ) = 0;( − g ′′ + V g ) ∈ L ((0 , R ); dx ) (cid:9) . (2.9)The fact that the spectrum of H θ ,θ R , σ ( H θ ,θ R ), is discrete is well-known, butdue to its importance in the context of this paper, we now briefly recall its prooffollowing an argument in Marchenko [59]: OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 5
Lemma 2.1 (See, [59], Sect. 1.3) . Suppose V ∈ L ((0 , R ); dx ) , assume θ , θ R ∈ S π , and let H θ ,θ R be defined as in (2.5) . Then σ ( H θ ,θ R ) is an infinite discretesubset of C ( i.e., a set without any finite limit point in C , but with a limit point atinfinity ) .Proof. Fix z ∈ C . Let θ ( z, · ) , θ ′ ( z, · ) , φ ( z, · ) , φ ′ ( z, · ) ∈ AC ([0 , R ]), and such that θ ( z, · ) and φ ( z, · ) are solutions of − ψ ′′ + V ψ = zψ uniquely determined by theinitial values at x = 0, θ ( z,
0) = φ ′ ( z,
0) = 1 , θ ′ ( z,
0) = φ ( z,
0) = 0 . (2.10)Consequently, θ ( z, · ) and φ ( z, · ) are entire with respect to z . Introducing ψ ( z, · ) = Aθ ( z, · ) + Bφ ( z, · ) , A, B ∈ C , (2.11)it follows that γ θ ,θ R ( ψ ) = " cos( θ ) ψ ( z,
0) + sin( θ ) ψ ′ ( z, θ R ) ψ ( z, R ) − sin( θ R ) ψ ′ ( z, R ) = 0 ∈ C . (2.12)Employing the initial conditions (2.10), one concludes that equation (2.12) is equiv-alent to0 = " cos( θ ) sin( θ )cos( θ R ) θ ( z, R ) − sin( θ R ) θ ′ ( z, R ) cos( θ R ) φ ( z, R ) − sin( θ R ) φ ′ ( z, R ) AB = U ( z, R, θ , θ R ) (cid:20) AB (cid:21) . (2.13)Consequently, z is an eigenvalue of H θ ,θ R if and only if z is a zero of the deter-minant ∆ defined as ∆( z, R, θ , θ R ) = det (cid:0) U ( z, R, θ , θ R ) (cid:1) . (2.14)Thus, ∆ is an entire function with respect to z , and an explicit computation revealsthat ∆( z, R, θ , θ R ) = cos( θ ) cos( θ R ) φ ( z, R ) − cos( θ ) sin( θ R ) φ ′ ( z, R ) − sin( θ ) cos( θ R ) θ ( z, R ) + sin( θ ) sin( θ R ) θ ′ ( z, R ) . (2.15)The standard Volterra integral equations θ ( z, x ) = cos( z / x ) + Z x dx ′ sin( z / ( x − x ′ )) z / V ( x ′ ) θ ( z, x ′ ) , (2.16) φ ( z, x ) = sin( z / x ) z / + Z x dx ′ sin( z / ( x − x ′ )) z / V ( x ′ ) φ ( z, x ′ ) , (2.17) z ∈ C , Im( z / ) ≥ , x ∈ [0 , R ] , then imply that θ ( z, x ) = | z |→∞ cos( z / x ) + O (cid:16) | z | − / e Im( z / ) x (cid:17) ,θ ′ ( z, x ) = | z |→∞ − z / sin( z / x ) + O (cid:16) e Im( z / ) x (cid:17) ,φ ( z, x ) = | z |→∞ sin( z / x ) z / + O (cid:16) | z | − e Im( z / ) x (cid:17) ,φ ′ ( z, x ) = | z |→∞ cos( z / x ) + O (cid:16) | z | − / e Im( z / ) x (cid:17) . (2.18) S. CLARK, F. GESZTESY, AND MARIUS MITREA
A comparison of (2.15) as | z | → ∞ and (2.18) demonstrates that ∆ does not vanishidentically. Thus the set of zeros of ∆, and hence the set of eigenvalues of H θ ,θ R ,constitutes a discrete set. Again, the asymptotic behavior of ∆ near infinity impliesthat ∆ is an entire function of order 1 / (cid:3) In addition (cf. also (3.37), (3.38)), the resolvent of H θ ,θ R is clearly a Hilbert–Schmidt operator in L ((0 , R ); dx ). In fact, it is even a trace class operator sincethe eigenvalues E θ ,θ R ,n of H θ ,θ R in the case of the separated boundary conditionsat hand are of the form E θ ,θ R ,n = [( nπ/R ) + ( a n /n )] with a n ∈ ℓ ∞ ( N ) as n → ∞ ,as shown in [59, Lemma 1.3.3].Having described the operator H θ ,θ R is some detail, still assuming (2.4), we nowbriefly recall the corresponding closed, sectorial, and densely defined sequilinearform, denoted by Q H θ ,θR , associated with H θ ,θ R (cf. [46, p. 312, 321, 327–328]): Q H θ ,θR ( f, g ) = Z R dx (cid:2) f ′ ( x ) g ′ ( x ) + V ( x ) f ( x ) g ( x ) (cid:3) − cot( θ ) f (0) g (0) − cot( θ R ) f ( R ) g ( R ) , (2.19) f, g ∈ dom( Q H θ ,θR ) = dom (cid:0) | H θ ,θ R | / (cid:1) = H ((0 , R ))= (cid:8) h ∈ L ((0 , R ); dx ) | h ∈ AC ([0 , R ]); h ′ ∈ L ((0 , R ); dx ) (cid:9) ,θ , θ R ∈ S π \{ , π } ,Q H ,θR ( f, g ) = Z R dx (cid:2) f ′ ( x ) g ′ ( x ) + V ( x ) f ( x ) g ( x ) (cid:3) − cot( θ R ) f ( R ) g ( R ) , (2.20) f, g ∈ dom( Q H ,θR ) = dom (cid:0) | H ,θ R | / (cid:1) = (cid:8) h ∈ L ((0 , R ); dx ) | h ∈ AC ([0 , R ]); h (0) = 0; h ′ ∈ L ((0 , R ); dx ) (cid:9) ,θ R ∈ S π \{ , π } ,Q H θ , ( f, g ) = Z R dx (cid:2) f ′ ( x ) g ′ ( x ) + V ( x ) f ( x ) g ( x ) (cid:3) − cot( θ ) f (0) g (0) , (2.21) f, g ∈ dom( Q H θ , ) = dom (cid:0) | H θ , | / (cid:1) = (cid:8) h ∈ L ((0 , R ); dx ) | h ∈ AC ([0 , R ]); h ( R ) = 0; h ′ ∈ L ((0 , R ); dx ) (cid:9) ,θ ∈ S π \{ , π } ,Q H , ( f, g ) = Z R dx (cid:2) f ′ ( x ) g ′ ( x ) + V ( x ) f ( x ) g ( x ) (cid:3) , (2.22) f, g ∈ dom( Q H , ) = dom (cid:0) | H , | / (cid:1) = H ((0 , R ))= (cid:8) h ∈ L ((0 , R ); dx ) | h ∈ AC ([0 , R ]); h (0) = 0 , h ( R ) = 0; h ′ ∈ L ((0 , R ); dx ) (cid:9) . Equations (2.19)–(2.22) follow from the fact that for any ε >
0, there exists η ( ε ) > h ∈ H ((0 , R )), | h ( x ) | ≤ ε k h ′ k L ((0 ,R ); dx ) + η ( ε ) k h k L ((0 ,R ); dx ) , x ∈ [0 , R ] , (2.23) k| V | / h k L ((0 ,R ); dx ) ≤ ε k h ′ k L ((0 ,R ); dx ) + η ( ε ) k h k L ((0 ,R ); dx ) (2.24)(cf. [46, p. 193, 345–346]). OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 7
Next, we recall the following elementary, yet fundamental, fact:
Lemma 2.2.
Suppose that V ∈ L ((0 , R ); dx ) , fix θ , θ R ∈ S π , and assume that z ∈ C \ σ ( H θ ,θ R ) . Then the boundary value problem given by − u ′′ + V u = zu, u, u ′ ∈ AC ([0 , R ]) , (2.25) γ θ ,θ R ( u ) = (cid:20) c c R (cid:21) ∈ C , (2.26) has a unique solution u ( z, · ) = u ( z, · ; ( θ , c ) , ( θ R , c R )) for each c , c R ∈ C .Proof. This is well-known, but for the sake of completeness, we briefly recall theargument: Let ψ j ( z, · ), j = 1 ,
2, be a basis for the solutions of (2.25) and let ψ ( z, · ) = Aψ ( z, · ) + Bψ ( z, · ), A, B ∈ C , be the general solution of (2.25). Then γ θ ,θ R ( ψ ( z, · )) = (cid:2) γ θ ,θ R ( ψ ( z, · )) γ θ ,θ R ( ψ ( z, · )) (cid:3) (cid:20) AB (cid:21) = (cid:20) M , ( z ) M , ( z ) M , ( z ) M , ( z ) (cid:21) (cid:20) AB (cid:21) , (2.27)where M , ( z ) = cos( θ ) ψ ( z,
0) + sin( θ ) ψ ′ ( z, ,M , ( z ) = cos( θ ) ψ ( z,
0) + sin( θ ) ψ ′ ( z, ,M , ( z ) = cos( θ R ) ψ ( z, R ) − sin( θ R ) ψ ′ ( z, R ) ,M , ( z ) = cos( θ R ) ψ ( z, R ) − sin( θ R ) ψ ′ ( z, R ) . (2.28)Thus, prescribing c , c R ∈ C , the equation γ θ ,θ R ( ψ ( z, · )) = (cid:20) c c R (cid:21) (2.29)is uniquely solvable in terms of some A, B ∈ C if and only ifdet (cid:0) (cid:2) γ θ ,θ R ( ψ ( z, · )) γ θ ,θ R ( ψ ( z, · )) (cid:3) (cid:1) = det (cid:18)(cid:20) M , ( z ) M , ( z ) M , ( z ) M , ( z ) (cid:21)(cid:19) = 0 . (2.30)On the other hand, this determinant equals zero for some z ∈ C , if and only ifthere is a nonzero vector (cid:2) A B (cid:3) ⊤ ∈ C such that (cid:2) γ θ ,θ R ( ψ ( z , · )) γ θ ,θ R ( ψ ( z , · )) (cid:3) (cid:20) A B (cid:21) = 0 (2.31)which is equivalent to the existence of a nonzero solution ψ ( z , · ) = A ψ ( z , · ) + B ψ ( z , · ) of the corresponding boundary value problem given by (2.25) and (2.26)with z = z and homogeneous boundary conditions (i.e., with c = c R = 0).Equivalently, ψ ( z , · ) satisfies H θ ,θ R ψ ( z , · ) = z ψ ( z , · ) , ψ ( z , · ) ∈ dom( H θ ,θ R ) , (2.32)which in turn is equivalent to z ∈ σ ( H θ ,θ R ). (cid:3) Assuming z ∈ C \ σ ( H θ ,θ R ), a basis for the solutions of (2.25) is given by u − ,θ ( z, · ) = u ( z, · ; ( θ , , (0 , ,u + ,θ R ( z, · ) = u ( z, · ; (0 , , ( θ R , . (2.33) S. CLARK, F. GESZTESY, AND MARIUS MITREA
Explicitly, one then has u − ,θ ( z, R ) = 1 , cos( θ ) u − ,θ ( z,
0) + sin( θ ) u ′− ,θ ( z,
0) = 0 , (2.34) u + ,θ R ( z,
0) = 1 , cos( θ R ) u + ,θ R ( z, R ) − sin( θ R ) u ′ + ,θ R ( z, R ) = 0 . (2.35)Recalling the Wronskian of two functions f and g , W ( f, g )( x ) = f ( x ) g ′ ( x ) − f ′ ( x ) g ( x ) , f, g ∈ C ([0 , R ]) , (2.36)one then computes W ( u + ,θ R ( z, · ) , u − ,θ ( z, · ))= u + ,θ R ( z, x ) u ′− ,θ ( z, x ) − u ′ + ,θ R ( z, x ) u − ,θ ( z, x ) = 0 , x ∈ [0 , R ] , = u ′− ,θ ( z, − u ′ + ,θ R ( z, , u − ,θ ( z,
0) (2.37)= u + ,θ R ( z, R ) u ′− ,θ ( z, R ) − u ′ + ,θ R ( z, R ) . (2.38)To each boundary value problem (2.25), (2.26), we now associate a family of general boundary data maps , Λ θ ′ ,θ ′ R θ ,θ R ( z ) : C → C , for θ , θ R , θ ′ , θ ′ R ∈ S π , whereΛ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:20) c c R (cid:21) = Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:0) γ θ ,θ R ( u ( z, · ; ( θ , c ) , ( θ R , c R ))) (cid:1) = γ θ ′ ,θ ′ R ( u ( z, · ; ( θ , c ) , ( θ R , c R ))) . (2.39)With u ( z, · ) = u ( z, · ; ( θ , c ) , ( θ R , c R )), then Λ θ ′ ,θ ′ R θ ,θ R ( z ) can be represented as a 2 × θ ′ ,θ ′ R θ ,θ R ( z ) (cid:20) c c R (cid:21) = Λ θ ′ ,θ ′ R θ ,θ R ( z ) " cos( θ ) u ( z,
0) + sin( θ ) u ′ ( z, θ R ) u ( z, R ) − sin( θ R ) u ′ ( z, R ) = " cos( θ ′ ) u ( z,
0) + sin( θ ′ ) u ′ ( z, θ ′ R ) u ( z, R ) − sin( θ ′ R ) u ′ ( z, R ) . (2.40)The following result shows that Λ θ ′ ,θ ′ R θ ,θ R is well-defined for z ∈ C \ σ ( H θ ,θ R ), thatis, it is invariant with respect to a change of basis of solutions of (2.25). Theorem 2.3.
Let θ , θ R , θ ′ , θ ′ R ∈ S π and z ∈ C \ σ ( H θ ,θ R ) . In addition, denoteby ψ j ( z, · ) , j = 1 , , a basis for the solutions of (2.25) . Then, Λ θ ′ ,θ ′ R θ ,θ R ( z ) (2.41)= " cos( θ ′ ) ψ ( z,
0) + sin( θ ′ ) ψ ′ ( z,
0) cos( θ ′ ) ψ ( z,
0) + sin( θ ′ ) ψ ′ ( z, θ ′ R ) ψ ( z, R ) − sin( θ ′ R ) ψ ′ ( z, R ) cos( θ ′ R ) ψ ( z, R ) − sin( θ ′ R ) ψ ′ ( z, R ) × " cos( θ ) ψ ( z,
0) + sin( θ ) ψ ′ ( z,
0) cos( θ ) ψ ( z,
0) + sin( θ ) ψ ′ ( z, θ R ) ψ ( z, R ) − sin( θ R ) ψ ′ ( z, R ) cos( θ R ) ψ ( z, R ) − sin( θ R ) ψ ′ ( z, R ) − . Moreover, Λ θ ′ ,θ ′ R θ ,θ R ( z ) is invariant with respect to a change of basis for the solutionsof (2.25) .Proof. Letting ψ ( z, · ) = Aψ ( z, · ) + Bψ ( z, · ), A, B ∈ C , be an arbitrary solutionof (2.25), one observes, by (2.27) and (2.40), that the equation Λ θ ′ ,θ ′ R θ ,θ R ( γ θ ,θ R ( ψ )) = γ θ ′ ,θ ′ R ( ψ ) becomesΛ θ ′ ,θ ′ R θ ,θ R ( z ) (2.42) OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 9 × " cos( θ ) ψ ( z,
0) + sin( θ ) ψ ′ ( z,
0) cos( θ ) ψ ( z,
0) + sin( θ ) ψ ′ ( z, θ R ) ψ ( z, R ) − sin( θ R ) ψ ′ ( z, R ) cos( θ R ) ψ ( z, R ) − sin( θ R ) ψ ′ ( z, R ) AB (cid:21) = " cos( θ ′ ) ψ ( z,
0) + sin( θ ′ ) ψ ′ ( z,
0) cos( θ ′ ) ψ ( z,
0) + sin( θ ′ ) ψ ′ ( z, θ ′ R ) ψ ( z, R ) − sin( θ ′ R ) ψ ′ ( z, R ) cos( θ ′ R ) ψ ( z, R ) − sin( θ ′ R ) ψ ′ ( z, R ) AB (cid:21) for every (cid:2) A B (cid:3) ⊤ ∈ C . Equation (2.41) then follows by the invertibility of (cid:2) γ θ ,θ R ( ψ ) γ θ ,θ R ( ψ ) (cid:3) noted in (2.30).Let φ j ( z, · ), j = 1 ,
2, denote a second basis for the solutions of (2.25). Then,there is a nonsingular matrix K ∈ C × such that (cid:2) ψ ψ (cid:3) = (cid:2) φ φ (cid:3) K . Next,we introduce for each pair, θ , θ R ∈ S π , the following matrices C θ ,θ R = (cid:20) cos( θ ) 00 cos( θ R ) (cid:21) , S θ ,θ R = (cid:20) sin( θ ) 00 sin( θ R ) (cid:21) . (2.43)Introducing θ j to denote θ , and θ k to denote θ R , respectively; or, using θ j to denote θ ′ and θ k to denote θ ′ R , one computes, (cid:2) γ θ j ,θ k ( ψ ( z, · )) γ θ j ,θ k ( ψ ( z, · )) (cid:3) = C θ j ,θ k (cid:20) ψ ( z, ψ ( z, ψ ( z, R ) ψ ( z, R ) (cid:21) + S θ j ,θ k (cid:20) ψ ′ ( z, ψ ′ ( z, − ψ ′ ( z, R ) − ψ ′ ( z, R ) (cid:21) = (cid:18) C θ j ,θ k (cid:20) φ ( z, φ ( z, φ ( z, R ) φ ( z, R ) (cid:21) + S θ j ,θ k (cid:20) φ ′ ( z, φ ′ ( z, − φ ′ ( z, R ) − φ ′ ( z, R ) (cid:21) (cid:19) K = (cid:2) γ θ j ,θ k ( φ ( z, · )) γ θ j ,θ k ( φ ( z, · )) (cid:3) K. (2.44)As defined in (2.41), Λ θ ′ ,θ ′ R θ ,θ R = (cid:2) γ θ ′ ,θ ′ R ( ψ ) γ θ ′ ,θ ′ R ( ψ ) (cid:3) (cid:2) γ θ ,θ R ( ψ ) γ θ ,θ R ( ψ ) (cid:3) − ,and by (2.44),Λ θ ′ ,θ ′ R θ ,θ R = (cid:2) γ θ ′ ,θ ′ R ( ψ ) γ θ ′ ,θ ′ R ( ψ ) (cid:3) (cid:2) γ θ ,θ R ( ψ ) γ θ ,θ R ( ψ ) (cid:3) − = (cid:2) γ θ ′ ,θ ′ R ( φ ) γ θ ′ ,θ ′ R ( φ ) (cid:3) (cid:2) [ γ θ ,θ R ( φ ) γ θ ,θ R ( φ ) (cid:3) − , (2.45)completing the proof. (cid:3) Theorem 2.3 then readily implies the following result:
Corollary 2.4.
Let θ , θ R , θ ′ , θ ′ R , θ ′′ , θ ′′ R ∈ S π . Then, with I denoting the identitymatrix in C , Λ θ ,θ R θ ,θ R ( z ) = I , z ∈ C \ σ ( H θ ,θ R ) , (2.46)Λ θ ′′ ,θ ′′ R θ ′ ,θ ′ R ( z )Λ θ ′ ,θ ′ R θ ,θ R ( z ) = Λ θ ′′ ,θ ′′ R θ ,θ R ( z ) , z ∈ C \ (cid:0) σ ( H θ ,θ R ) ∪ σ ( H θ ′ ,θ ′ R ) (cid:1) , (2.47)Λ θ ,θ R θ ′ ,θ ′ R ( z ) = h Λ θ ′ ,θ ′ R θ ,θ R ( z ) i − , z ∈ C \ (cid:0) σ ( H θ ,θ R ) ∪ σ ( H θ ′ ,θ ′ R ) (cid:1) . (2.48) Remark 2.5.
By Theorem 2.3, Λ θ ′ ,θ ′ R θ ,θ R is invariant with respect to a change ofbasis for the solutions of (2.25). However, the representation of Λ θ ′ ,θ ′ R θ ,θ R with respectto a specific basis can be simplified considerably with an appropriate choice ofbasis. For example, by choosing the basis given in (2.33), and by letting ψ ( z, · ) = u + ,θ R ( z, · ) = u ( z, · ; (0 , , ( θ R , ψ ( z, · ) = u − ,θ ( z, · ) = u ( z, · ; ( θ , , (0 , (2.45) implies that, entrywise, (cid:2) γ θ ,θ R ( u + ,θ R ( z, · )) γ θ ,θ R ( u − ,θ ( z, · )) (cid:3) , = cos( θ ) + sin( θ ) u ′ + ,θ R ( z, , (cid:2) γ θ ,θ R ( u + ,θ R ( z, · )) γ θ ,θ R ( u − ,θ ( z, · )) (cid:3) , = cos( θ ) u − ,θ ( z,
0) + sin( θ ) u ′− ,θ ( z, , (cid:2) γ θ ,θ R ( u + ,θ R ( z, · )) γ θ ,θ R ( u − ,θ ( z, · )) (cid:3) , = cos( θ R ) u + ,θ R ( z, R ) − sin( θ R ) u ′ + ,θ R ( z, R ) , (cid:2) γ θ ,θ R ( u + ,θ R ( z, · )) γ θ ,θ R ( u − ,θ ( z, · )) (cid:3) , = cos( θ R ) − sin( θ R ) u ′− ,θ ( z, R ) . (2.49)Hence, for this basis,Λ θ ′ ,θ ′ R θ ,θ R ( z ) = h(cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:17) j,k i ≤ j,k ≤ , z ∈ C \ σ ( H θ ,θ R ) , (2.50) (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:17) , = cos( θ ′ ) + sin( θ ′ ) u ′ + ,θ R ( z, θ ) + sin( θ ) u ′ + ,θ R ( z, , (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:17) , = cos( θ ′ ) u − ,θ ( z,
0) + sin( θ ′ ) u ′− ,θ ( z, θ R ) − sin( θ R ) u ′− ,θ ( z, R ) , (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:17) , = cos( θ ′ R ) u + ,θ R ( z, R ) − sin( θ ′ R ) u ′ + ,θ R ( z, R )cos( θ ) + sin( θ ) u ′ + ,θ R ( z, , (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:17) , = cos( θ ′ R ) − sin( θ ′ R ) u ′− ,θ ( z, R )cos( θ R ) − sin( θ R ) u ′− ,θ ( z, R ) . (2.51)In particular, by (2.34) and (2.35), (cid:16) Λ θ ,θ ′ R θ ,θ R ( z ) (cid:17) , = 0 , (cid:16) Λ θ ′ ,θ R θ ,θ R ( z ) (cid:17) , = 0 . (2.52) Remark 2.6.
We note that Λ π , π , ( z ) represents the Dirichlet-to-Neumann map ,Λ D,N ( z ), for the boundary value problem (2.25), (2.26); that is, when θ = θ R = 0, θ ′ = θ ′ R = π/
2, then (2.40) becomesΛ
D,N ( z ) (cid:20) u ( z, u ( z, R ) (cid:21) = Λ π , π , ( z ) (cid:20) u ( z, u ( z, R ) (cid:21) = (cid:20) u ′ ( z, − u ′ ( z, R ) (cid:21) , z ∈ C \ σ ( H θ ,θ R ) , (2.53)with u ( z, · ) = u ( z, · ; (0 , c ) , (0 , c R )), u ( z,
0) = c , u ( z, R ) = c R . The Dirichlet-to-Neumann map in the case V = 0 has recently been considered in [71, Example5.1]. The Neumann-to-Dirichlet map Λ N,D ( z ) = Λ π,ππ/ ,π/ ( z ) = − [Λ D,N ( z )] − (cf.(4.70)) in the case V = 0 has earlier been computed in [27, Example 4.1]. We alsorefer to [12], [19], [26], [30] in the intimately related context of Q and M -functions.It would be interesting to establish precise connections between Λ θ ,θ R θ ,θ R ( z ) andthe dynamical response operator discussed, for instance, in [7], [8], [9], [10] inconnection with the problem of regularity and controllability of the wave equationon a compact interval.We conclude this section with an elementary result needed in the proof of Lemma3.4 and Theorem 4.6: OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 11
Lemma 2.7.
Let V ∈ L ((0 , R ); dx ) , fix θ , θ R ∈ S π , c , c R ∈ C , and assume that z ∈ C \ σ ( H θ ,θ R ) . Then the unique solution u ( z, · ) = u ( z, · ; ( θ , c ) , ( θ R , c R )) of (2.25) , (2.26) can be expressed in terms of the fundamental system θ ( z, · ) , φ ( z, · ) , asintroduced in the proof of Lemma , and satisfying the initial conditions (2.10) ,as follows: u ( z, · ) = u ( z, · ; ( θ , c ) , ( θ R , c R )) = α ( z ) θ ( z, · ) + β ( z ) φ ( z, · ) , (2.54) " α ( z ) β ( z ) = 1∆( z, R, θ , θ R ) " [cos( θ R ) φ ( z, R ) − sin( θ R ) φ ′ ( z, R )] c − sin( θ ) c R − [cos( θ R ) θ ( z, R ) − sin( θ R ) θ ′ ( z, R )] c + cos( θ ) c R , with ∆ given by (2.15) . In particular, one has u + ,θ R ( z, · ) = u ( z, · ; ((0 , , ( θ R , α ( z ) θ ( z, · ) + β ( z ) φ ( z, · ) , " α ( z ) β ( z ) = 1∆( z, R, , θ R ) " cos( θ R ) φ ( z, R ) − sin( θ R ) φ ′ ( z, R ) − cos( θ R ) θ ( z, R ) + sin( θ R ) θ ′ ( z, R ) , (2.55) and u − ,θ ( z, · ) = u ( z, · ; ( θ , , (0 , z, R, θ , (cid:0) − sin( θ ) θ ( z, · ) + cos( θ ) φ ( z, · ) (cid:1) . Proof.
With u ( z, · ) = u ( z, · ; ( θ , c ) , ( θ R , c R )) = αθ ( z, · ) + βφ ( z, · ) one observesfrom (2.26) that (cid:20) c c R (cid:21) = γ θ ,θ R ( u ( z, · )) = γ θ ,θ R ( αθ ( z, · ) + βφ ( z, · ))= (cid:2) γ θ ,θ R ( θ ( z, · )) γ θ ,θ R ( φ ( z, · )) (cid:3) (cid:20) αβ (cid:21) = " cos( θ ) sin( θ )cos( θ R ) θ ( z, R ) − sin( θ R ) θ ′ ( z, R ) cos( θ R ) φ ( z, R ) − sin( θ R ) φ ′ ( z, R ) αβ = U ( z, R, θ , θ R ) (cid:20) αβ (cid:21) , (2.57)with U introduced in (2.13). Solving (2.57) for α, β (and observing that det( U ) =∆ according to (2.14)) then yields (2.54) and hence the special cases (2.55) and(2.56). (cid:3) Resolvent Formulas
In the principal result of this section, we will derive a formula for Λ θ ′ ,θ ′ R θ ,θ R in termsof the resolvent of H θ ,θ R and the boundary traces γ θ ′ ,θ ′ R . But first we focus on thespecial boundary data map given byΛ θ ,θ R ( z ) = Λ ( θ + π ) mod(2 π ) , ( θ R + π ) mod(2 π ) θ ,θ R ( z ) , θ , θ R ∈ S π , z ∈ C \ σ ( H θ ,θ R ) , (3.1)a generalization of the Dirichlet-to-Neumann map Λ D,N ( z ) = Λ , ( z ). Using thebasis for solutions of (2.25) given in (2.33) by u + ,θ R ( z, · ) , u − ,θ ( z, · ), and with θ ′ =( θ + π/
2) mod(2 π ), θ ′ R = ( θ R + π/
2) mod(2 π ), equation (2.50) becomesΛ θ ,θ R ( z ) = (cid:2)(cid:0) Λ θ ,θ R ( z ) (cid:1) j,k (cid:3) ≤ j,k ≤ , z ∈ C \ σ ( H θ ,θ R ) , (3.2) (cid:0) Λ θ ,θ R ( z ) (cid:1) , = − sin( θ ) + cos( θ ) u ′ + ,θ R ( z, θ ) + sin( θ ) u ′ + ,θ R ( z, , (cid:0) Λ θ ,θ R ( z ) (cid:1) , = − sin( θ ) u − ,θ ( z,
0) + cos( θ ) u ′− ,θ ( z, θ R ) − sin( θ R ) u ′− ,θ ( z, R ) , (cid:0) Λ θ ,θ R ( z ) (cid:1) , = − sin( θ R ) u + ,θ R ( z, R ) − cos( θ R ) u ′ + ,θ R ( z, R )cos( θ ) + sin( θ ) u ′ + ,θ R ( z, , (cid:0) Λ θ ,θ R ( z ) (cid:1) , = − sin( θ R ) − cos( θ R ) u ′− ,θ ( z, R )cos( θ R ) − sin( θ R ) u ′− ,θ ( z, R ) . (3.3)Next, consider α ( ξ ) = (cid:2) cos( ξ ) sin( ξ ) (cid:3) , ξ ∈ S π , (3.4)and let Ψ( z, · ; x , α ( ξ )) denote the fundamental matrix of solutions of (2.25) givenby Ψ( z, · ; x , α ( ξ )) = (cid:2) Θ( z, · ; x , α ( ξ )) Φ( z, · ; x , α ( ξ )) (cid:3) = (cid:20) ϑ ( z, · ; x , α ( ξ )) ϕ ( z, · ; x , α ( ξ )) ϑ ′ ( z, · ; x α ( ξ )) ϕ ′ ( z, · ; x , α ( ξ )) (cid:21) , (3.5)in particular,Ψ( z, x ; x , α ( ξ )) = (cid:20) cos( ξ ) − sin( ξ )sin( ξ ) cos( ξ ) (cid:21) , x ∈ [0 , R ] . In addition, set β ( η ) = (cid:2) cos( η ) − sin( η ) (cid:3) = α ( − η ) , η ∈ S π . (3.6)As shown in [21, Section 2] and [22], one can then introduce for all x , y ∈ [0 , R ], x = y , the Weyl–Titchmarsh function , m ( z ; x , α ( ξ ); y , β ( η )), given by m ( z ; x , α ( ξ ); y , β ( η ))= − [ β ( η )Φ( z, y ; x , α ( ξ ))] − [ β ( η )Θ( z, y ; x , α ( ξ ))]= − cos( η ) ϑ ( z, y ; x , α ( ξ )) − sin( η ) ϑ ′ ( z, y ; x , α ( ξ ))cos( η ) ϕ ( z, y ; x , α ( ξ )) − sin( η ) ϕ ′ ( z, y ; x , α ( ξ )) , (3.7) ξ, η ∈ S π , z ∈ C \ σ ( H ξ,η ) . Then, with ψ ( z, · ; x , α ( ξ ); y , β ( η )) defined by ψ ( z, · ; x , α ( ξ ); y , β ( η ))= ϑ ( z, · ; x , α ( ξ )) + ϕ ( z, · ; x , α ( ξ )) m ( z ; x , α ( ξ ); y , β ( η )) , (3.8) ϕ ( z, · ; x , α ( ξ )) and ψ ( z, · ; x , α ( ξ ); y , β ( η )) are linearly independent solutions of(2.25) satisfying cos( ξ ) ϕ ( z, x ; x , α ( ξ )) + sin( ξ ) ϕ ′ ( z, x ; x , α ( ξ )) = 0 , (3.9)cos( η ) ψ ( z, y ; x , α ( ξ ); y , β ( η )) − sin( η ) ψ ′ ( z, y ; x , α ( ξ ); y , β ( η )) = 0 . (3.10)The following result has been proved in [21, Lemma 2.10] in a self-adjoint con-text, but self-adjointness is of no relevance for the result (3.11) below. For theconvenience of the reader we reproduce its proof here. OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 13
Lemma 3.1 ([21], Lemma 2.10) . Suppose ξ, η ∈ S π , z ∈ C (cid:15)(cid:0) σ ( H ξ,η ) ∪ σ ( H ,η ) (cid:1) ,let α ( ξ ) and β ( η ) be defined as in (3.4) and (3.6) , and suppose that x , y ∈ [0 , R ] .Then, the following linear fractional transformation holds, m ( z ; x , α ( ξ ); y , β ( η )) = − sin( ξ ) + cos( ξ ) m ( z ; x , α (0); y , β ( η ))cos( ξ ) + sin( ξ ) m ( z ; x , α (0); y , β ( η )) , (3.11) with m defined as in (3.7) .Proof. With ψ ( z, · ; x , α ( ξ ); y , β ( η )) defined in (3.8), by (3.10) one obtains β ( η ) (cid:20) ψ ( z, y ; x , α (0); y , β ( η )) ψ ′ ( z, y ; x , α (0); y , β ( η )) (cid:21) = β ( η ) (cid:20) ψ ( z, y ; x , α ( ξ ); y , β ( η )) ψ ′ ( z, y ; x , α ( ξ ); y , β ( η )) (cid:21) = 0 , (3.12)and as a consequence, for some c ( z ) ∈ C \{ } , (cid:20) ψ ( z, y ; x , α ( ξ ); y , β ( η )) ψ ′ ( z, y ; x , α ( ξ ); y , β ( η )) (cid:21) = c ( z ) (cid:20) ψ ( z, y ; x , α (0); y , β ( η )) ψ ′ ( z, y ; x , α (0); y , β ( η )) (cid:21) . (3.13)By the uniqueness of solutions for the initial value problem associated with (2.25), (cid:20) ψ ( z, x ; x , α ( ξ ); y , β ( η )) ψ ′ ( z, x ; x , α ( ξ ); y , β ( η )) (cid:21) = c ( z ) (cid:20) ψ ( z, x ; x , α (0); y , β ( η )) ψ ′ ( z, x ; x , α (0); y , β ( η )) (cid:21) , x ∈ [0 , R ] . (3.14)Next, one notes that (cid:20) ψ ( z, x ; x , α ( ξ ); y , β ( η )) ψ ′ ( z, x ; x , α ( ξ ); y , β ( η )) (cid:21) = Ψ( z, x ; x , α ( ξ )) (cid:20) m ( z ; x , α ( ξ ); y , β ( η )) (cid:21) = (cid:20) cos( ξ ) − sin( ξ )sin( ξ ) cos( ξ ) (cid:21) (cid:20) m ( z ; x , α ( ξ ); y , β ( η )) (cid:21) , (3.15)and similarly that (cid:20) ψ ( z, x ; x , α (0); y , β ( η )) ψ ′ ( z, x ; x , α (0); y , β ( η )) (cid:21) = Ψ( z, x ; x , α (0)) (cid:20) m ( z ; x , α (0); y , β ( η )) (cid:21) = (cid:20) m ( z ; x , α (0); y , β ( η )) (cid:21) . (3.16)By (3.14)–(3.16), one concludes that (cid:20) m ( z ; x , α ( ξ ); y , β ( η )) (cid:21) = c ( z ) (cid:20) cos( ξ ) sin( ξ ) − sin( ξ ) cos( ξ ) (cid:21) (cid:20) m ( z ; x , α (0); y , β ( η )) (cid:21) (3.17)implying (3.11). (cid:3) With α ( ξ ) and β ( η ) defined in (3.4) and (3.6), and with m ( z ; x , α ( ξ ); y , β ( η ))defined in (3.7), for the next result we let α = α ( θ ) = (cid:2) cos( θ ) sin( θ ) (cid:3) , β R = β ( θ R ) = (cid:2) cos( θ R ) − sin( θ R ) (cid:3) , (3.18) m + ,θ ( z, θ R ) = m ( z ; 0 , α ; R, β R ) , m − ,θ R ( z, θ ) = m ( z ; R, β R ; 0 , α ) . (3.19) Theorem 3.2.
Let θ , θ R ∈ S π and z ∈ C \ σ ( H θ ,θ R ) . Then, Λ θ ,θ R ( z ) = " m + ,θ ( z, θ R ) Λ θ ,θ R ( z ) , Λ θ ,θ R ( z ) , − m − ,θ R ( z, θ ) , (3.20) where Λ θ ,θ R ( z ) , = Λ θ ,θ R ( z ) ,
14 S. CLARK, F. GESZTESY, AND MARIUS MITREA = − sin( θ ) u − ,θ ( z,
0) + cos( θ ) u ′− ,θ ( z, θ R ) − sin( θ R ) u ′− ,θ ( z, R )= − sin( θ R ) u + ,θ R ( z, R ) − cos( θ R ) u ′ + ,θ R ( z, R )cos( θ ) + sin( θ ) u ′ + ,θ R ( z, . (3.21) Proof.
We temporarily assume in addition that z / ∈ (cid:0) σ ( H θ , ) ∪ σ ( H ,θ R ) (cid:1) . If x = 0, y = R , then with ϕ ( z, · ; 0 , α ) defined in (3.5), and with ψ ( z, · ; 0 , α ; R, β R )defined in (3.8), one notes, by (3.9) and (3.10), that ϕ ( z, · ; 0 , α ) = C − ( z ) u − ,θ ( z, · ) , (3.22) ψ ( z, · ; 0 , α ; R, β R ) = C + ( z ) u + ,θ R ( z, · ) , (3.23)for some C ± ( z ) ∈ C \{ } , where u + ,θ R ( z, · ) , u − ,θ ( z, · ) represents the basis for thesolutions of (2.25) described in (2.33). With m + ,θ ( z, θ R ) defined in (3.19), onenotes when θ = 0 that m + , ( z, θ R ) = ψ ′ ( z,
0; 0 , α ; R, β R ) ψ ( z,
0; 0 , α ; R, β R ) (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = u ′ + ,θ R ( z, m + ,θ ( z, θ R ) = − sin( θ ) + cos( θ ) m + , ( z, θ R )cos( θ ) + sin( θ ) m + , ( z, θ R )= − sin( θ ) + cos( θ ) u ′ + ,θ R ( z, θ ) + sin( θ ) u ′ + ,θ R ( z, . (3.25)By Theorem 2.3, Λ θ ,θ R is invariant with respect to a change of basis for (2.25);thus the (1 , θ ,θ R , provided in (3.3), equals m + ,θ ( z, θ R ).Next, if x = R , y = 0, then with ϕ ( z, · ; R, β R ) defined in (3.5), and with ψ ( z, · ; R, β R ; 0 , α ) defined in (3.8), one notes, by (3.9) and (3.10), that ϕ ( z, · ; R, β R ) = D + ( z ) u + ,θ R ( z, · ) , (3.26) ψ ( z, · ; R, β R ; 0 , α ) = D − ( z ) u − ,θ ( z, · ) , (3.27)for some D ± ( z ) ∈ C \{ } , where u + ,θ R ( z, · ) , u − ,θ ( z, · ) again denotes the basis forthe solutions of (2.25) described in (2.33). With m − ,θ R ( z, θ ) defined in (3.19), onenow obtains when θ R = 0 that, m − , ( z, θ ) = ψ ′ ( z, R ; R, β R ; 0 , α ) ψ ( z, R ; R, β R ; 0 , α ) (cid:12)(cid:12)(cid:12)(cid:12) θ R =0 = u ′− ,θ ( z, R ); (3.28)hence by (3.11) that m − ,θ R ( z, θ ) = sin( θ R ) + cos( θ R ) m − , ( z, θ )cos( θ R ) − sin( θ R ) m − , ( z, θ )= sin( θ R ) + cos( θ R ) u ′− ,θ ( z, R )cos( θ R ) − sin( θ R ) u ′− ,θ ( z, R ) . (3.29)Again, by the invariance of Λ θ ,θ R with respect to a change of basis for solutions of(2.25), the (2 , θ ,θ R , provided in (3.3), is given by − m − ,θ R ( z, θ ).To see that the off-diagonal elements of (3.20) are equal, we introduce W ( z ) = W ( u + ,θ R ( z, · ) , u − ,θ ( z, · )) , θ , θ R ∈ S π , z ∈ C \ σ ( H θ ,θ R ) , (3.30) OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 15 where u + ,θ R ( z, · ) , u − ,θ ( z, · ), is the basis for the solutions of (2.25) as described in(2.33). A straightforward computation, then yields[ − sin( θ ) u − ,θ ( z,
0) + cos( θ ) u ′− ,θ ( z, θ ) + sin( θ ) u ′ + ,θ R ( z, W ( z ) (3.31)= [cos( θ R ) − sin( θ R ) u ′− ,θ ( z, R )][ − sin( θ R ) u + ,θ R ( z, R ) − cos( θ R ) u ′ + ,θ R ( z, R )] . Indeed, the first equality in (3.31) follows by inserting the second term in (2.34)into (2.37); similarly, the second equality in (3.31) follows by inserting the secondterm in (2.35) into (2.38). Equation (3.31) immediately yields (3.21).Finally, a meromorphic continuation with respect to z then removes the addi-tional assumption z / ∈ (cid:0) σ ( H θ , ) ∪ σ ( H ,θ R ) (cid:1) , completing the proof. (cid:3) Remark 3.3.
Assume the special self-adjoint case where V is real-valued and θ , θ R ∈ [0 , π ). By Marchenko’s fundamental uniqueness result [58, Ch. 2], oneobserves the inverse spectral theory fact that each of the two diagonal terms ofΛ θ ,θ R ( · ) already uniquely determines the potential coefficient V ( x ) for a.e. x ∈ [0 , R ]. In addition, the known leading asymptotic behavior of m + ,θ ( z, θ R ) as z → i ∞ , m + ,θ ( z, θ R ) −→ z → i ∞ cot( θ ) + o (1) , θ ∈ [0 , π ) \{ , π } ,m + , ( z, θ R ) −→ z → i ∞ iz / + o (cid:0) z / (cid:1) , θ ∈ { , π } , (3.32)determines the boundary condition parameter θ , and similarly, that of m − ,θ R ( z, θ )determines θ R . These facts are obviously not shared by the usual 2 × × θ ,θ R ( · ) as a special case of the principal result in Theorem 4.6 (cf.(4.77)).Next we turn to the asymptotic behavior of Λ θ ,θ R ( z ) as | z | → ∞ , Im( z / ) > Lemma 3.4.
Suppose that V ∈ L ((0 , R ); dx ) and let θ , θ R ∈ S π . Then, Λ θ ,θ R ( z ) = | z |→∞ Im( z / ) > cot( θ ) iz − / e iz / R sin( θ ) sin( θ R )2 iz − / e iz / R sin( θ ) sin( θ R ) − cot( θ ) + " O ( | z | − / ) O (cid:0) | z | − e − Im( z / ) R (cid:1) O (cid:0) | z | − e − Im( z / ) R (cid:1) O ( | z | − / ) , θ = 0 , θ R = 0 , (3.33)Λ ,θ R ( z ) = | z |→∞ Im( z / ) > iz / − e iz / R sin( θ R ) − e iz / R sin( θ R ) − cot( θ R ) + " O (1) O (cid:0) | z | − / e − Im( z / ) R (cid:1) O (cid:0) | z | − / e − Im( z / ) R (cid:1) O ( | z | − / ) , θ = 0 , θ R = 0 , (3.34) Λ θ , ( z ) = | z |→∞ Im( z / ) > cot( θ ) − e iz / R sin( θ ) − e iz / R sin( θ ) − iz / + " O ( | z | − / ) O (cid:0) | z | − / e − Im( z / ) R (cid:1) O (cid:0) | z | − / e − Im( z / ) R (cid:1) O (1) , θ = 0 , θ R = 0 , (3.35)Λ , ( z ) = | z |→∞ Im( z / ) > " iz / − iz / e iz / R − iz / e iz / R − iz / + " O (1) O (cid:0) e − Im( z / ) R (cid:1) O (cid:0) e − Im( z / ) R (cid:1) O (1) , θ = 0 , θ R = 0 . (3.36) Proof.
This follows from an elementary computation upon inserting (2.55) (resp.,(2.56)) into (3.25) (resp., (3.29)) and using the asymptotic expansions (2.18). (cid:3)
For the principal result of this section, an explicit formula for Λ θ ′ ,θ ′ R θ ,θ R ( z ) in termsof the resolvent ( H θ ,θ R − zI (cid:1) − of H θ ,θ R and the boundary traces γ θ ′ ,θ ′ R , we recallthe Green’s function associated with the operator H θ ,θ R in (2.5), G θ ,θ R ( z, x, x ′ ) = ( H θ ,θ R − zI ) − ( x, x ′ )= 1 W ( u + ,θ R ( z, · ) , u − ,θ ( z, · )) ( u − ,θ ( z, x ′ ) u + ,θ R ( z, x ) , x ′ x,u − ,θ ( z, x ) u + ,θ R ( z, x ′ ) , x x ′ , (3.37) z ∈ C \ σ ( H θ ,θ R ) , x, x ′ ∈ [0 , R ] . Here u + ,θ R ( z, · ) , u − ,θ ( z, · ) is a basis for solutions of (2.25) as described in (2.33)and I = I L ((0 ,R ); dx ) abbreviates the identity operator in L ((0 , R ); dx ). Thus, oneobtains (cid:0) ( H θ ,θ R − zI ) − g (cid:1) ( x ) = Z R dx ′ G θ ,θ R ( z, x, x ′ ) g ( x ′ ) ,g ∈ L ((0 , R ); dx ) , z ∈ C \ σ ( H θ ,θ R ) , x ∈ (0 , R ) . (3.38)Equation (3.37) is a crucial input in the proof of the following fundamental result: Theorem 3.5.
Assume that θ , θ R , θ ′ , θ ′ R ∈ S π , let H θ ,θ R be defined as in (2.5) ,suppose that z ∈ C \ σ ( H θ ,θ R ) , and let S θ ′ − θ ,θ ′ R − θ R be defined according to (2.43) .Then Λ θ ′ ,θ ′ R θ ,θ R ( z ) S θ ′ − θ ,θ ′ R − θ R = γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ . (3.39) In particular, with θ ′ = ( θ + π/
2) mod(2 π ) , θ ′ R = ( θ R + π/
2) mod(2 π ) , one obtains Λ θ ,θ R ( z ) = b γ θ ,θ R (cid:2)b γ θ ,θ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ , (3.40) where b γ θ ,θ R = γ ( θ + π ) mod(2 π ) , ( θ R + π ) mod(2 π ) . (3.41) As a consequence, Λ ( θ + π ) mod(2 π ) , ( θ R + π ) mod(2 π ) ( z ) = Λ θ ,θ R ( z ) , θ , θ R ∈ S π . (3.42) OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 17
Proof.
We start by noting that the Green’s function G ∗ θ ,θ R ( z, x, x ′ ) of ( H θ ,θ R ) ∗ isgiven by G ∗ θ ,θ R ( z, x, x ′ ) = (cid:0) ( H θ ,θ R ) ∗ − zI (cid:1) − ( x, x ′ )= 1 W ∗ ( z ) u − ,θ ( z, x ′ ) u + ,θ R ( z, x ) , x ′ x,u − ,θ ( z, x ) u + ,θ R ( z, x ′ ) , x x ′ ,z ∈ C \ σ (cid:0) ( H θ ,θ R ) ∗ (cid:1) , x, x ′ ∈ [0 , R ] , where u + ,θ R ( z, · ) , u − ,θ ( z, · ) is a basis for solutions of (2.25) as described in (2.33),and W ∗ ( z ) is defined by W ∗ ( z ) = W (cid:0) u + ,θ R ( z, · ) , u − ,θ ( z, · ) (cid:1) , z ∈ C \ σ ( H θ ,θ R ) . (3.43)In particular, we note the fact that W ∗ ( z ) = W ( z ) , (3.44)with W ( z ) the Wronskian defined in (3.30). Thus, one obtains (cid:0) (( H θ ,θ R ) ∗ − zI ) − g (cid:1) ( x ) = Z R dx ′ G ∗ θ ,θ R ( z, x, x ′ ) g ( x ′ ) ,g ∈ L ((0 , R ); dx ) , z ∈ C \ σ ( H θ ,θ R ) , x ∈ (0 , R ) . (3.45)Next, for g ∈ L ((0 , R ); dx ) let b g ( z, x ) = (cid:0) (( H θ ,θ R ) ∗ − zI ) − g (cid:1) ( x )= 1 W ( z ) (cid:18) u + ,θ R ( z, x ) Z x dx ′ u − ,θ ( z, x ′ ) g ( x ′ ) (3.46)+ u − ,θ ( z, x ) Z Rx dx ′ u + ,θ R ( z, x ′ ) g ( x ′ ) (cid:19) . Then one notes that b g ′ ( z, x ) = 1 W ( z ) (cid:18) u ′ + ,θ R ( z, x ) Z x dx ′ u − ,θ ( z, x ′ ) g ( x ′ )+ u ′− ,θ ( z, x ) Z Rx dx ′ u + ,θ R ( z, x ′ ) g ( x ′ ) (cid:19) , (3.47)and, as a consequence, that b g ( z,
0) = 1 W ( z ) u − ,θ ( z, Z R dx ′ u + ,θ R ( z, x ′ ) g ( x ′ ) , (3.48) b g ′ ( z,
0) = 1 W ( z ) u ′− ,θ ( z, Z R dx ′ u + ,θ R ( z, x ′ ) g ( x ′ ) , (3.49) b g ( z, R ) = 1 W ( z ) u + ,θ R ( z, R ) Z R dx ′ u − ,θ ( z, x ′ ) g ( x ′ ) , (3.50) b g ′ ( z, R ) = 1 W ( z ) u ′ + ,θ R ( z, R ) Z R dx ′ u − ,θ ( z, x ′ ) g ( x ′ ) . (3.51) In turn, this permits one to compute γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − g = γ θ ′ ,θ ′ R b g (¯ z, · )= " cos( θ ′ ) b g ( z,
0) + sin( θ ′ ) b g ′ ( z, θ ′ R ) b g ( z, R ) − sin( θ ′ R ) b g ′ ( z, R ) . (3.52)Using (3.48)–(3.52) one infers, with [ a a R ] ⊤ ∈ C and ( · , · ) C denoting the scalarproduct in C , that (cid:0) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − g, [ a a R ] ⊤ (cid:1) C = " cos( θ ′ ) b g ( z,
0) + sin( θ ′ ) b g ′ ( z, θ ′ R ) b g ( z, R ) − sin( θ ′ R ) b g ′ ( z, R ) , " a a R C = 1 W ( z ) Z R dx ′ g ( x ′ ) (cid:0) cos( θ ′ ) a u − ,θ ( z, u + ,θ R ( z, x ′ )+ sin( θ ′ ) a u ′− ,θ ( z, u + ,θ R ( z, x ′ ) + cos( θ ′ R ) a R u + ,θ R ( z, R ) u − ,θ ( z, x ′ ) − sin( θ ′ R ) a R u ′ + ,θ R ( z, R ) u − ,θ ( z, x ′ ) (cid:1) . (3.53)Hence one concludes that (cid:16)(cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [ a a R ] ⊤ (cid:17) ( x )= 1 W ( z ) (cid:16)(cid:2) cos( θ ′ ) u − ,θ ( z,
0) + sin( θ ′ ) u ′− ,θ ( z, (cid:3) a u + ,θ R ( z, x )+ (cid:2) cos( θ ′ R ) u + ,θ R ( z, R ) − sin( θ ′ R ) u ′ + ,θ R ( z, R ) (cid:3) a R u − ,θ ( z, x ) (cid:17) . (3.54)Consequently, writing γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [ a a R ] ⊤ = "(cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [ a a R ] ⊤ (cid:1) (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [ a a R ] ⊤ (cid:1) , (3.55)it follows that (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [ a a R ] ⊤ (cid:1) = 1 W ( z ) (cid:16) cos( θ ′ ) (cid:2) cos( θ ′ ) u − ,θ ( z,
0) + sin( θ ′ ) u ′− ,θ ( z, (cid:3) a + cos( θ ′ ) (cid:2) cos( θ ′ R ) u + ,θ R ( z, R ) − sin( θ ′ R ) u ′ + ,θ R ( z, R ) (cid:3) a R u − ,θ ( z, θ ′ ) (cid:2) cos( θ ′ ) u − ,θ ( z,
0) + sin( θ ′ ) u ′− ,θ ( z, (cid:3) a u ′ + ,θ R ( z, θ ′ ) (cid:2) cos( θ ′ R ) u + ,θ R ( z, R ) − sin( θ ′ R ) u ′ + ,θ R ( z, R ) (cid:3) a R u ′− ,θ ( z, (cid:17) , (3.56)and (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [ a a R ] ⊤ (cid:1) = 1 W ( z ) (cid:16) cos( θ ′ R ) (cid:2) cos( θ ′ ) u − ,θ ( z,
0) + sin( θ ′ ) u ′− ,θ ( z, (cid:3) a u + ,θ R ( z, R )+ cos( θ ′ R ) (cid:2) cos( θ ′ R ) u + ,θ R ( z, R ) − sin( θ ′ R ) u ′ + ,θ R ( z, R ) (cid:3) a R − sin( θ ′ R ) (cid:2) cos( θ ′ ) u − ,θ ( z,
0) + sin( θ ′ ) u ′− ,θ ( z, (cid:3) a u ′ + ,θ R ( z, R ) OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 19 − sin( θ ′ R ) (cid:2) cos( θ ′ R ) u + ,θ R ( z, R ) − sin( θ ′ R ) u ′ + ,θ R ( z, R ) (cid:3) a R u ′− ,θ ( z, R ) (cid:17) . (3.57)Hence, writing γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ = h(cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ (cid:1) j,k i j,k =1 , , (3.58)one obtains (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ (cid:1) , = 1 W ( z ) (cid:2) cos( θ ′ ) u − ,θ ( z,
0) + sin( θ ′ ) u ′− ,θ ( z, (cid:3) × (cid:2) cos( θ ′ ) + sin( θ ′ ) u ′ + ,θ R ( z, (cid:3) , (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI (cid:1) − (cid:3) ∗ (cid:1) , = 1 W ( z ) (cid:2) cos( θ ′ R ) u + ,θ R ( z, R ) − sin( θ ′ R ) u ′ + ,θ R ( z, R ) (cid:3) × (cid:2) cos( θ ′ ) u − ,θ ( z,
0) + sin( θ ′ ) u ′− ,θ ( z, (cid:3) , (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ (cid:1) , = 1 W ( z ) (cid:2) cos( θ ′ ) u − ,θ ( z,
0) + sin( θ ′ ) u ′− ,θ ( z, (cid:3) × (cid:2) cos( θ ′ R ) u + ,θ R ( z, R ) − sin( θ ′ R ) u ′ + ,θ R ( z, R ) (cid:3) , (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ (cid:1) , = 1 W ( z ) (cid:2) cos( θ ′ R ) u + ,θ R ( z, R ) − sin( θ ′ R ) u ′ + ,θ R ( z, R ) (cid:3) × (cid:2) cos( θ ′ R ) − sin( θ ′ R ) u ′− ,θ ( z, R ) (cid:3) . (3.59)Employing (2.37) and (2.38) one finally concludes from (2.51) that the expressionsin (3.59) are equivalent to the following ones (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ (cid:1) , = sin( θ ′ − θ ) (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:17) , , (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ (cid:1) , = sin( θ ′ R − θ R ) (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:17) , , (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ (cid:1) , = sin( θ ′ − θ ) (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:17) , , (cid:0) γ θ ′ ,θ ′ R (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ (cid:1) , = sin( θ ′ R − θ R ) (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:17) , , (3.60)proving (3.39).Finally, equation (3.42) is a consequence of (2.6) and (3.40). (cid:3) Remark 3.6.
A formula of the type (3.40) for Dirichlet-to-Neumann maps asso-ciated with multi-dimensional Schr¨odinger operators was published by Amrein andPearson [4] in 2004. It has recently been extended in various directions in [35], [36],[38], [39]. Formula (3.40) for the Dirichlet-to-Neumann map Λ π/ ,π/ in the specialcase V = 0 has also been derived by Posilicano [71, Example 5.1]. Remark 3.7.
While it is tempting to view γ θ ,θ R as an unbounded but denselydefined operator on L ((0 , R ); dx ) whose domain contains the space C ∞ ((0 , R )),one should note, in this case, that its adjoint γ ∗ θ ,θ R is not densely defined. Indeed,the adjoint γ ∗ θ ,θ R of γ θ ,θ R would have to be an unbounded operator from C to L ((0 , R ); dx ) such that( γ θ ,θ R f, g ) C = ( f, γ ∗ θ ,θ R g ) L ((0 ,R ); dx ) for all f ∈ dom( γ θ ,θ R ) , g ∈ dom( γ ∗ θ ,θ R ) . (3.61)In particular, choosing f ∈ C ∞ ((0 , R )), in which case γ θ ,θ R f = 0, one concludesthat ( f, γ ∗ θ ,θ R g ) L ((0 ,R ); dx ) = 0 for all f ∈ C ∞ ((0 , R )) . (3.62)Thus, one obtains γ ∗ θ ,θ R g = 0 for all g ∈ dom( γ ∗ θ ,θ R ). Since obviously γ θ ,θ R = 0,(3.61) implies dom( γ ∗ θ ,θ R ) = { } and hence γ θ ,θ R is not a closable linear operatorin L ((0 , R ); dx ). This is the reason for our careful choice of notation in (3.39) and(3.40).We conclude this section by providing an explicit example in the special case V = 0 a.e. on [0 , R ]. For notational purposes we add the superscript (0) to thecorresponding quantities below: Example 3.8.
Let V = 0 a.e. on [0 , R ], x ∈ (0 , R ), θ , θ R , θ ∈ S π , x, x ′ , s ∈ [0 , R ],and let f ( z, s, α, β ) = f ( z, s, β, α ) (3.63)= z sin( α ) sin( β ) sin( √ zs ) + √ z sin( α + β ) cos( √ zs ) − cos( α ) cos( β ) sin( √ zs ) ,g ( z, s, α, β ) = f ( z, s, α + π/ , β ) (3.64)= z cos( α ) sin( β ) sin( √ zs ) + √ z cos( α + β ) cos( √ zs ) + sin( α ) cos( β ) sin( √ zs ) . Then, u (0) ( z, x, ( θ , c ) , ( θ R , c R )) = c f ( z, R − x, , θ R ) + c R f ( z, x, θ , f ( z, R, θ , θ R ) , (3.65) u (0)+ ,θ R ( z, x ) = u ( z, · ; (0 , , ( θ R , f ( z, R − x, , θ R ) f ( z, R, , θ R ) , (3.66) u (0) ′ + ,θ R ( z, x ) = f ′ ( z, R − x, , θ R ) f ( z, R, , θ R ) = f ( z, R − x, π/ , θ R ) f ( z, R, , θ R ) , (3.67) u (0) − ,θ ( z, x ) = u ( z, · ; ( θ , , (0 , f ( z, x, θ , f ( z, R, θ , , (3.68) u (0) ′ − ,θ ( z, x ) = f ′ ( z, x, θ , f ( z, R, θ ,
0) = − f ( z, x, θ , π/ f ( z, R, θ , , (3.69) W ( u (0)+ ,θ R ( z, · ) , u (0) − ,θ ( z, · )) = −√ zf ( z, R, θ , θ R ) f ( z, R, θ , f ( z, R, , θ R ) , (3.70) m (0)+ ,θ ( z, θ R ) = g ( z, R, θ , θ R ) f ( z, R, θ , θ R ) , (3.71) m (0) − ,θ R ( z, θ ) = − g ( z, R, θ R , θ ) f ( z, R, θ R , θ ) , (3.72)Λ (0) θ ,θ R ( z ) = g ( z,R,θ ,θ R ) f ( z,R,θ ,θ R ) −√ zf ( z,R,θ ,θ R ) −√ zf ( z,R,θ ,θ R ) g ( z,R,θ R ,θ ) f ( z,R,θ ,θ R ) ! , OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 21 m (0)+ , ( z, x , θ R ) = u (0) ′ + ,θ R ( z, x ) u (0)+ ,θ R ( z, x ) = g ( z, R − x , , θ R ) f ( z, R − x , , θ R ) , (3.73) m (0) − , ( z, x , θ ) = u (0) ′ − ,θ ( z, x ) u (0) − ,θ ( z, x ) = − g ( z, x , , θ ) f ( z, x , , θ ) , (3.74) G (0) θ ,θ R ( z, x, x ′ ) = (cid:0) H (0) θ ,θ R − zI (cid:1) − ( x, x ′ )= 1 W ( u (0)+ ,θ R ( z, · ) , u (0) − ,θ ( z, · )) ( u (0) − ,θ ( z, x ′ ) u (0)+ ,θ R ( z, x ) , x ′ x,u (0) − ,θ ( z, x ) u (0)+ ,θ R ( z, x ′ ) , x x ′ , = f ( z, x ′ , θ , f ( z, R − x, , θ R ) −√ zf ( z, R, θ , θ R ) , x ′ x,f ( z, x, θ , f ( z, R − x ′ , , θ R ) −√ zf ( z, R, θ , θ R ) , x x ′ , z ∈ C \ σ (cid:0) H (0) θ ,θ R (cid:1) . (3.75)The special case of the Neumann-to-Dirichlet map, Λ (0) π/ ,π/ in (3.73), was com-puted in [27, Example 4.1], and more recently, the special case of the Dirichlet-to-Neumann map, Λ (0)0 , in (3.73), was computed in [71, Example 5.1].4. Linear Fractional Transformations and the Herglotz Property inthe Self-Adjoint Case
The principal purpose of this section is to prove that Λ θ ′ ,θ ′ R θ ,θ R ( z ) and Λ δ ′ ,δ ′ R δ ,δ R ( z )satisfy a linear fractional transformation. As a consequence we will show thatΛ θ ′ ,θ ′ R θ ,θ R ( z ) S θ ′ − θ ,θ ′ R − θ R is a 2 × V and θ , θ R , θ ′ , θ ′ R are real-valued.In the following we denote by C n × n , n ∈ N , the set of n × n matrices withcomplex-valued entries, and by I n the identity matrix in C n .Let A = (cid:2) A j,k (cid:3) ≤ j,k ≤ ∈ C × , with A j,k ∈ C × , 1 ≤ j, k ≤
2, and L ∈ C × ,chosen such that ker( A , + A , L ) = { } ; that is, ( A , + A , L ) is invertible in C . Define for such A (cf., e.g., [51]), M A ( L ) = ( A , + A , L )( A , + A , L ) − , (4.1)and observe that M I ( L ) = L, (4.2) M AB ( L ) = M A ( M B ( L )) , (4.3) M A − ( M A ( L )) = L = M A ( M A − ( L )) , A invertible, (4.4) M A ( L ) = M AB − ( M B ( L )) , (4.5)whenever the right-hand sides (and hence the left-hand sides) in (4.3)–(4.5) exist. Theorem 4.1.
Assume that θ , θ R , θ ′ , θ ′ R , δ , δ R , δ ′ , δ ′ R ∈ S π , δ ′ − δ = 0 mod( π ) , δ ′ R − δ R = 0 mod( π ) , and that z ∈ C \ (cid:0) σ ( H θ ,θ R ) ∪ σ ( H δ ,δ R ) (cid:1) . Then, with S θ ,θ R defined as in (2.43) , Λ θ ′ ,θ ′ R θ ,θ R ( z ) = (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − (cid:2) S δ ′ − θ ′ ,δ ′ R − θ ′ R + S θ ′ − δ ,θ ′ R − δ R Λ δ ′ ,δ ′ R δ ,δ R ( z ) (cid:3) × (cid:2) S δ ′ − θ ,δ ′ R − θ R + S θ − δ ,θ R − δ R Λ δ ′ ,δ ′ R δ ,δ R ( z ) (cid:3) − S δ ′ − δ ,δ ′ R − δ R . (4.6) Proof.
Assume θ , θ R , θ ′ , θ ′ R ∈ S π , z ∈ C \ σ ( H θ ,θ R ), and let ψ j ( z, · ), j = 1 , C θ ,θ R and S θ ,θ R as definedin (2.43), equation (2.42) yieldsΛ θ ′ ,θ ′ R θ ,θ R ( z ) (cid:18) C θ ,θ R (cid:20) ψ ( z, ψ ( z, ψ ( z, R ) ψ ( z, R ) (cid:21) + S θ ,θ R (cid:20) ψ ′ ( z, ψ ′ ( z, − ψ ′ ( z, R ) − ψ ′ ( z, R ) (cid:21) (cid:19) = (cid:18) C θ ′ ,θ ′ R (cid:20) ψ ( z, ψ ( z, ψ ( z, R ) ψ ( z, R ) (cid:21) + S θ ′ ,θ ′ R (cid:20) ψ ′ ( z, ψ ′ ( z, − ψ ′ ( z, R ) − ψ ′ ( z, R ) (cid:21) (cid:19) . (4.7)From Remark 2.6, recall, for z ∈ C \ σ ( H , ), that Λ D,N ( z ) = Λ π , π , ( z ) = Λ , ( z )and note that (2.45) then yieldsΛ , ( z ) = (cid:20) ψ ′ ( z, ψ ′ ( z, − ψ ′ ( z, R ) − ψ ′ ( z, R ) (cid:21) (cid:20) ψ ( z, ψ ( z, ψ ( z, R ) ψ ( z, R ) (cid:21) − . (4.8)Then, with C θ ,θ R defined as in (2.43), and with z ∈ C (cid:15)(cid:0) σ ( H θ ,θ R ) ∪ σ ( H , ) (cid:1) , (4.7)can be written asΛ θ ′ ,θ ′ R θ ,θ R ( z ) = (cid:2) C θ ′ ,θ ′ R + S θ ′ ,θ ′ R Λ , ( z ) (cid:3)(cid:2) C θ ,θ R + S θ ,θ R Λ , ( z ) (cid:3) − . (4.9)Next, assume that θ , θ R , θ ′ , θ ′ R , δ , δ R , δ ′ , δ ′ R ∈ S π , and let A, B ∈ C × bedefined by A = (cid:20) C θ ,θ R S θ ,θ R C θ ′ ,θ ′ R S θ ′ ,θ ′ R (cid:21) , B = (cid:20) C δ ,δ R S δ ,δ R C δ ′ ,δ ′ R S δ ′ ,δ ′ R (cid:21) . (4.10)Then, by (4.1), and (4.9),Λ θ ′ ,θ ′ R θ ,θ R ( z ) = M A (Λ , ( z )) , Λ δ ′ ,δ ′ R δ ,δ R ( z ) = M B (Λ , ( z )) (4.11)for z ∈ C (cid:15)(cid:0) σ ( H θ ,θ R ) ∪ σ ( H δ ,δ R ) ∪ σ ( H , ) (cid:1) . If additionally, one assumes that δ ′ − δ = 0 mod( π ), δ ′ R − δ R = 0 mod( π ), then AB − = "(cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − S δ ′ − θ ,δ ′ R − θ R (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − S θ − δ ,θ R − δ R (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − S δ ′ − θ ′ ,δ ′ R − θ ′ R (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − S θ ′ − δ ,θ ′ R − δ R , (4.12)and (4.6) then follows from (4.1), (4.3), (4.11) and (4.12), given thatΛ θ ′ ,θ ′ R θ ,θ R ( z ) = M A (Λ , ( z )) = M AB − ( M B (Λ , ( z )) = M AB − (Λ δ ′ ,δ ′ R δ ,δ R ( z )) . (4.13)By meromorphic continuation, (4.6) holds for z ∈ C (cid:15)(cid:0) σ ( H θ ,θ R ) ∪ σ ( H δ ,δ R ) (cid:1) . (cid:3) If H θ ,θ R and H δ ,δ R are self-adjoint, then (4.6) holds for z ∈ C \ R . Remark 4.2.
In the special case of (4.6) which relates two generalized Dirichlet-to-Neumann maps one obtainsΛ θ ,θ R ( z ) = (cid:2) − S θ − δ ,θ R − δ R + C θ − δ ,θ R − δ R Λ δ ,δ R ( z ) (cid:3) × (cid:2) C θ − δ ,θ R − δ R + S θ − δ ,θ R − δ R Λ δ ,δ R ( z ) (cid:3) − , (4.14) θ , θ R , δ , δ R ∈ S π , z ∈ C \ (cid:0) σ ( H θ ,θ R ) ∪ σ ( H δ ,δ R ) (cid:1) . The following reformulation of (4.6) (motivated by the form of (3.39) in Theorem3.5) will be crucial in the proof of Theorem 4.6.
OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 23
Corollary 4.3.
Assume that θ , θ R , θ ′ , θ ′ R , δ , δ R , δ ′ , δ ′ R ∈ S π , θ ′ − θ = 0 mod( π ) , θ ′ R − θ R = 0 mod( π ) , δ ′ − δ = 0 mod( π ) , δ ′ R − δ R = 0 mod( π ) , and that z ∈ C \ (cid:0) σ ( H θ ,θ R ) ∪ σ ( H δ ,δ R ) (cid:1) . Then Λ θ ′ ,θ ′ R θ ,θ R ( z ) S θ ′ − θ ,θ ′ R − θ R = h S δ ′ − θ ′ ,δ ′ R − θ ′ R + (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − S θ ′ − δ ,θ ′ R − δ R Λ δ ′ ,δ ′ R δ ,δ R ( z ) S δ ′ − δ ,δ ′ R − δ R i × h(cid:0) S θ ′ − θ ,θ ′ R − θ R (cid:1) − S δ ′ − θ ,δ ′ R − θ R + (cid:0) S θ ′ − θ ,θ ′ R − θ R (cid:1) − (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − × S θ − δ ,θ R − δ R Λ δ ′ ,δ ′ R δ ,δ R ( z ) S δ ′ − δ ,δ ′ R − δ R i − . (4.15) Proof.
This follows from multiplying (4.6) by S θ ′ − θ ,θ ′ R − θ R from the right, insertingthe terms S δ ′ − δ ,δ ′ R − δ R (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − to the right of Λ δ ′ ,δ ′ R δ ,δ R ( z ) twice, and thenalgebraically manipulate the various terms in the equation resulting from theseinsertions into (4.6) (repeatedly using ( ST ) − = T − S − , etc.). (cid:3) Since the self-adjoint case is the principal focus for the remainder of this section,we now recall some additional pertinent facts from [51] (see also [42, Sect. 6]) onlinear fractional transformations of matrices. Defining J = (cid:20) − I I (cid:21) , (4.16)and A = { A ∈ C × | A ∗ J A = J } , (4.17)representing A ∈ C × by A = (cid:20) A , A , A , A , (cid:21) , A p,q ∈ C × , ≤ p, q ≤ , (4.18)the condition A ∗ J A = J in (4.17) is equivalent to A ∗ , A , = A ∗ , A , , A ∗ , A , = A ∗ , A , ,A ∗ , A , − A ∗ , A , = I = A ∗ , A , − A ∗ , A , , (4.19)or equivalently, to (cid:20) A ∗ , − A ∗ , − A ∗ , A ∗ , (cid:21) (cid:20) A , A , A , A , (cid:21) = I . (4.20)Since left inverses in C × are also right inverses, (4.20) implies (cid:20) A , A , A , A , (cid:21) (cid:20) A ∗ , − A ∗ , − A ∗ , A ∗ , (cid:21) = I , (4.21)that is, A , A ∗ , = A , A ∗ , , A , A ∗ , = A , A ∗ , ,A , A ∗ , − A , A ∗ , = I = A , A ∗ , − A , A ∗ , , (4.22)or equivalently, AJ A ∗ = J . (4.23)In particular, A ∈ A if and only if A − ∈ A . (4.24)At this point we turn to the particularly important special self-adjoint casewhere V and θ , θ R , θ ′ , θ ′ R are real-valued. In this case we will now prove that Λ θ ′ ,θ ′ R θ ,θ R ( z ) S θ ′ − θ ,θ ′ R − θ R is a 2 × Lemma 4.4.
Assume that θ , θ R , θ ′ , θ ′ R , δ , δ R , δ ′ , δ ′ R ∈ [0 , π ) , θ ′ − θ = 0 mod( π ) , θ ′ R − θ R = 0 mod( π ) , δ ′ − δ = 0 mod( π ) , δ ′ R − δ R = 0 mod( π ) , and introduce inaccordance with (4.15) , A ( θ, δ ) = (cid:2) A ( θ, δ ) j,k (cid:3) ≤ j,k ≤ ∈ C × ,A ( θ, δ ) , = (cid:0) S θ ′ − θ ,θ ′ R − θ R (cid:1) − S δ ′ − θ ,δ ′ R − θ R ,A ( θ, δ ) , = (cid:0) S θ ′ − θ ,θ ′ R − θ R (cid:1) − (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − S θ − δ ,θ R − δ R ,A ( θ, δ ) , = S δ ′ − θ ′ ,δ ′ R − θ ′ R ,A ( θ, δ ) , = (cid:0) S δ ′ − δ ,δ ′ R − δ R (cid:1) − S θ ′ − δ ,θ ′ R − δ R . (4.25) Then A ( θ, δ ) ∈ A . (4.26) Proof.
Since according to (2.43) S α,β are 2 × A θ,δ in (4.25) satisfy the relations in (4.19). (cid:3) We denote by C + the open complex upper half-plane and abbreviate Im( L ) =( L − L ∗ ) / (2 i ) for L ∈ C n × n , n ∈ N . In addition, d k Σ k C will denote the totalvariation of the 2 × d Σ below in (4.32).We recall that M ( · ) is called an n × n matrix-valued Herglotz function if it isanalytic on C + and Im( M ( z )) ≥ z ∈ C + . In this context we also recall thefollowing result: Lemma 4.5.
Assume that A = (cid:2) A j,k (cid:3) ≤ j,k ≤ ∈ A and L ∈ C × . Then Im( L ) > implies ker( A , + A , L ) = { } (4.27) and M A ( L ) = ( A , + A , L )( A , + A , L ) − ( defined according to (4.1)) satisfies Im( M A ) = (cid:0) ( A , + A , L ) − (cid:1) ∗ Im( L )( A , + A , L ) − > . (4.28) In particular, if M ( · ) is a × matrix-valued Herglotz function satisfying Im( M ( z )) > , z ∈ C + , (4.29) then M A ( · ) ( in obvious notation defined according to (4.1) with L replaced by M ( · )) is a a × matrix-valued Herglotz function satisfying Im( M A ( z )) > , z ∈ C + . (4.30) Proof.
This is the special finite-dimensional case of [42, Theorem 6.4]. (cid:3)
Now we are in position to prove the fundamental Herglotz property of the matrixΛ θ ′ ,θ ′ R θ ,θ R ( · ) S θ ′ − θ ,θ ′ R − θ R in the case where H θ ,θ R is self-adjoint. Theorem 4.6.
Let θ , θ R , θ ′ , θ ′ R ∈ [0 , π ) , θ ′ − θ = 0 mod( π ) , θ ′ R − θ R = 0 mod( π ) , z ∈ C \ σ ( H θ ,θ R ) , and H θ ,θ R be defined as in (2.5) . In addition, suppose that V is real-valued ( and hence H θ ,θ R is self-adjoint ) . Then Λ θ ′ ,θ ′ R θ ,θ R ( · ) S θ ′ − θ ,θ ′ R − θ R is a × matrix-valued Herglotz function admitting the representation Λ θ ′ ,θ ′ R θ ,θ R ( z ) S θ ′ − θ ,θ ′ R − θ R = Ξ θ ′ ,θ ′ R θ ,θ R + Z R d Σ θ ′ ,θ ′ R θ ,θ R ( λ ) (cid:18) λ − z − λ λ (cid:19) , (4.31) OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 25 z ∈ C \ σ ( H θ ,θ R ) , Ξ θ ′ ,θ ′ R θ ,θ R = (cid:16) Ξ θ ′ ,θ ′ R θ ,θ R (cid:17) ∗ ∈ C × , Z R d (cid:13)(cid:13) Σ θ ′ ,θ ′ R θ ,θ R ( λ ) (cid:13)(cid:13) C λ < ∞ , (4.32) where Σ θ ′ ,θ ′ R θ ,θ R (( λ , λ ]) = 1 π lim δ ↓ lim ε ↓ Z λ + δλ + δ dλ Im (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( λ + iε ) (cid:17) ,λ , λ ∈ R , λ < λ . (4.33) In addition, Im (cid:16) Λ θ ′ ,θ ′ R θ ,θ R ( z ) S θ ′ − θ ,θ ′ R − θ R (cid:17) > , z ∈ C + . (4.34) Proof.
Without loss of generality, we take z ∈ C + . An analytic continuation withrespect to z then extends the result (4.31) to z ∈ C \ σ ( H θ ,θ R ).We will first prove the Herglotz property for Λ π/ ,π/ and then use a special caseof the linear fractional transformation (4.15) to conclude that Λ θ ′ ,θ ′ R θ ,θ R ( · ) S θ ′ − θ ,θ ′ R − θ R is a 2 × θ , θ R , θ ′ , θ ′ R ∈ [0 , π ). Our pointof departure is formula (3.40) in the special case θ = θ R = π/
2, that is,Λ π/ ,π/ ( z ) = γ , (cid:2) γ , ( H π/ ,π/ − z ) − ] ∗ , z ∈ C + , (4.35)noticing that b γ π/ ,π/ = γ π,π = − γ , (4.36)by (2.6).First, we slightly change the definition of γ , by introducing e γ , : H ((0 , R )) → C ,u " u (0) u ( R ) , e γ , ∈ B (cid:0) H ((0 , R )) , C (cid:1) , (4.37)instead. One notes that e γ , is well-defined in the following sense: Any u ∈ H ((0 , R )) has a representative in its equivalence class of Lebesgue measurable andsquare integrable elements, again denoted by u for simplicity, that is absolutelycontinuous on [0 , R ]. In fact, by a standard Sobolev embedding result, one has H (0 , R )) ֒ → C / ((0 , R )) = C / ([0 , R ]). In particular, the limits lim x ↓ u ( x ) = u (0) and lim x ↑ R u ( x ) = u ( R ) are well-defined for this representative u .Next, using H ((0 , R )) ֒ → L ((0 , R ); dx )) ֒ → H ((0 , R )) ∗ , (4.38)one infers that (cf. also Remark 3.7)( e γ , ) ∗ : C → H ((0 , R )) ∗ , " v v v δ + v δ R , ( e γ , ) ∗ ∈ B (cid:0) C , H ((0 , R )) ∗ (cid:1) , (4.39)where (cf. also [47, Example 2]) δ , δ R ∈ H ((0 , R )) ∗ ,δ ( u ) = H ((0 ,R )) (cid:10) u, δ (cid:11) H ((0 ,R )) ∗ = u (0) , (4.40) δ R ( u ) = H ((0 ,R )) (cid:10) u, δ R (cid:11) H ((0 ,R )) ∗ = u ( R ) , u ∈ H ((0 , R )) , with H ((0 ,R )) h · , · i H ((0 ,R )) ∗ denoting the duality pairing between H ((0 , R )) and H ((0 , R )) ∗ (linear in the second argument and antilinear in the first, see, e.g., [37,Sect. 2] for more details). Indeed, (4.40) follows from u (0) v + u ( R ) v = ( e γ , u, v ) C = H ((0 ,R )) (cid:10) u, ( e γ , ) ∗ v (cid:11) H ((0 ,R )) ∗ , (4.41) v = (cid:20) v v (cid:21) ∈ C , u ∈ H ((0 , R )) . (4.42)Next, still following the material discussed in [37, Sect. 2], we extend the operator H π/ ,π/ in L ((0 , R ); dx ),( H π/ ,π/ − zI ) : dom (cid:0) H π/ ,π/ (cid:1) → L ((0 , R ); dx ) , z ∈ z ∈ C + , (4.43)where dom (cid:0) H π/ ,π/ (cid:1) ֒ → L ((0 , R ); dx ), to its extension ^ H π/ ,π/ , which maps H ((0 , R )) boundedly into H ((0 , R )) ∗ , ^ H π/ ,π/ ∈ B (cid:0) H ((0 , R )) , H ((0 , R )) ∗ (cid:1) , (4.44)such that (with e I : H ((0 , R )) ֒ → H ((0 , R )) ∗ the continuous embedding operator) (cid:0) ^ H π/ ,π/ + e I (cid:1) ∈ B (cid:0) H ((0 , R )) , H ((0 , R )) ∗ (cid:1) (4.45)and (cid:0) ^ H π/ ,π/ + e I (cid:1) : H ((0 , R )) → H ((0 , R )) ∗ is unitary. (4.46)In addition (cf. (2.19)),( H π/ ,π/ + I ) / ∈ B (cid:0) H ((0 , R )) , L ((0 , R ); dx ) (cid:1) (4.47)and ( H π/ ,π/ + I ) / : H ((0 , R )) → L ((0 , R ); dx ) is unitary (4.48)(cf. [37, Sect. 2]). Moreover, ^ H π/ ,π/ is self-adjoint, (cid:16) ^ H π/ ,π/ (cid:17) ∗ = ^ H π/ ,π/ , (4.49)in the sense that H ((0 ,R )) (cid:10) w , ^ H π/ ,π/ w (cid:11) H ((0 ,R )) ∗ = H ((0 ,R )) (cid:10) w , ^ H π/ ,π/ w (cid:11) H ((0 ,R )) ∗ ,w , w ∈ H ((0 , R )) (4.50)(again we refer to [37, Sect. 2] for more details).In addition, (cid:16) ^ H π/ ,π/ − z e I (cid:17) − : H ((0 , R )) ∗ → H ((0 , R )) , z ∈ C + , (4.51)and hence, (cid:16)(cid:16) ^ H π/ ,π/ − z e I (cid:17) − w (cid:17) ( x ) = H ((0 ,R )) (cid:10) G π/ ,π/ ( z, x, · ) , w (cid:11) H ((0 ,R )) ∗ ,w ∈ H ((0 , R )) ∗ , z ∈ C + , (4.52)using the fact that G π/ ,π/ ( z, x, · ) ∈ H ((0 , R )) , x ∈ R , z ∈ C \ σ ( H π/ ,π/ ) . (4.53)By (2.34) and (2.35), the Wronskian W is of the form W ( z ) = W ( u + ,π/ ( z, · ) , u − ,π/ ( z, · ))= − u ′ + ,π/ ( z, u − ,π/ ( z,
0) = u + ,π/ ( z, R ) u ′− ,π/ ( z, R ) . (4.54) OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 27
Using (3.37), one computes (cid:16)(cid:16) ^ H π/ ,π/ − z e I (cid:17) − ( e γ , ) ∗ v (cid:17) ( x )= 1 W ( z ) H ((0 ,R )) (cid:10) G π/ ,π/ ( z, x, · ) , [ v δ + v δ R ] (cid:11) H ((0 ,R )) ∗ = 1 W ( z ) (cid:0) u − ,π/ ( z, x ) u + ,π/ ( z, R ) v + u + ,π/ ( z, x ) u − ,π/ ( z, v (cid:1) , (4.55) v = [ v v ] ⊤ ∈ C , and hence e γ , (cid:16) ^ H π/ ,π/ − z e I (cid:17) − ( e γ , ) ∗ v = 1 W ( z ) " u − ,π/ ( z, u + ,π/ ( z, R ) v + u − ,π/ ( z, v u + ,π/ ( z, R ) v + u + ,π/ ( z, R ) u − ,π/ ( z, v = 1 W ( z ) " u − ,π/ ( z, u − ,π/ ( z, u + ,π/ ( z, R ) u − ,π/ ( z, u + ,π/ ( z, R ) u + ,π/ ( z, R ) v v = Λ π/ ,π/ ( z ) v, v = [ v v ] ⊤ ∈ C , (4.56)inserting (4.54) for W ( · ). Consequently, one hasΛ π/ ,π/ ( z ) = e γ , (cid:16) ^ H π/ ,π/ − z e I (cid:17) − ( e γ , ) ∗ , z ∈ C + , (4.57)and also Λ π/ ,π/ ( z ) ∗ = Λ π/ ,π/ ( z ) , z ∈ C + . (4.58)In particular, (cid:0) v, Im (cid:0) Λ π/ ,π/ ( z ) (cid:1) v (cid:1) C = 12 i (cid:0) v, (cid:0) Λ π/ ,π/ ( z ) − Λ π/ ,π/ ( z ) ∗ (cid:1) v (cid:1) C = 12 i (cid:16) v, e γ , (cid:16)(cid:16) ^ H π/ ,π/ − z e I (cid:17) − − (cid:16) ^ H π/ ,π/ − z e I (cid:17) − (cid:17) ( e γ , ) ∗ v (cid:17) C = Im( z ) (cid:16) v, e γ , h(cid:16) ^ H π/ ,π/ − z e I (cid:17) − e I (cid:16) ^ H π/ ,π/ − z e I (cid:17) − i ( e γ , ) ∗ v (cid:17) C = Im( z ) (cid:16) v, he γ , (cid:16) ^ H π/ ,π/ − z e I (cid:17) − ie I he γ , (cid:16) ^ H π/ ,π/ − z e I (cid:17) − i ∗ v (cid:17) C = Im( z ) H ((0 ,R )) Dhe γ , (cid:16) ^ H π/ ,π/ − z e I (cid:17) − i ∗ v, e I × he γ , (cid:16) ^ H π/ ,π/ − z e I (cid:17) − i ∗ v E H ((0 ,R )) ∗ = Im( z ) (cid:0)(cid:2) γ , ( H π/ ,π/ − zI ) − (cid:3) ∗ v, (cid:2) γ , ( H π/ ,π/ − zI ) − (cid:3) ∗ v (cid:1) L ((0 ,R ); dx ) = Im( z ) (cid:13)(cid:13)(cid:2) γ , ( H π/ ,π/ − zI ) − (cid:3) ∗ v (cid:13)(cid:13) L ((0 ,R ); dx ) ≥ , v ∈ C , z ∈ C + , (4.59)since the duality pairing H ((0 ,R )) h · , · i H ((0 ,R )) ∗ between the spaces H ((0 , R )) and H ((0 , R )) ∗ is compatible with the scalar product in L ((0 , R ); dx ), that is, H ((0 ,R )) h w , e Iw i H ((0 ,R )) ∗ = ( w , w ) L ((0 ,R ); dx ) , w , w ∈ H ((0 , R )) . (4.60) In particular,Im (cid:0) Λ π/ ,π/ ( z ) (cid:1) = (cid:2) γ , ( H π/ ,π/ − zI ) − (cid:3)(cid:2) γ , ( H π/ ,π/ − zI ) − (cid:3) ∗ ≥ ,z ∈ C + , (4.61)and thus, Λ π/ ,π/ ( · ) is a 2 × (cid:0) Λ π/ ,π/ ( z ) (cid:1) > , z ∈ C + , (4.62)since ker (cid:0)(cid:2) γ , ( H π/ ,π/ − zI ) − (cid:3) ∗ (cid:1) = { } , z ∈ C \ σ ( H π/ ,π/ ) . (4.63)To prove (4.63), one can argue as follows: Suppose that[ a a R ] ⊤ ∈ ker (cid:0)(cid:2) γ , ( H π/ ,π/ − zI ) − (cid:3) ∗ (cid:1) , (4.64)then by (3.54), (cid:0)(cid:2) γ , ( H π/ ,π/ − zI ) − (cid:3) ∗ [ a a R ] ⊤ (cid:1) ( x ) = 1 W ( u + ,π/ ( z, · ) , u − ,π/ ( z, · )) × (cid:2) a u − ,π/ ( z, u + ,π/ ( z, x ) + a R u + ,π/ ( z, R ) u − ,π/ ( z, x ) (cid:3) = 0 , (4.65) z ∈ C \ σ ( H π/ ,π/ ) . Since by definition (cf. (2.34), (2.35)), u ′− ,π/ ( z,
0) = u ′ + ,π/ ( z, R ) = 0, one con-cludes that u − ,π/ ( z, = 0 , u + ,π/ ( z, R ) = 0 . (4.66)Moreover, since W ( u + ,π/ ( z, · ) , u − ,π/ ( z, · )) = 0 for all z ∈ C \ σ ( H π/ ,π/ ) (other-wise, z would be an eigenvalue of H π/ ,π/ ), u + ,π/ ( z, · ) and u − ,π/ ( z, · )) are linearlyindependent, implying a = a R = 0 , (4.67)and hence (4.63).Next, using the notation introduced in (3.1), and applying the linear fractionaltransformation (4.6) one can show that (with z ∈ C + )Λ π/ ,π/ ( z ) (cid:2) C π/ ,π/ + S π/ ,π/ Λ , ( z ) (cid:3) = (cid:2) C π,π + S π,π Λ , ( z ) (cid:3) , (4.68)or equivalently, that Λ π/ ,π/ ( z )Λ , ( z ) = − I , (4.69)and hence,Λ π/ ,π/ , ( z ) = Λ , ( z ) = − (cid:2) Λ π/ ,π/ ( z ) (cid:3) − = − (cid:2) Λ π,ππ/ ,π/ ( z ) (cid:3) − , (4.70)is a 2 × (cid:0) Λ π/ ,π/ , ( z ) (cid:1) > , z ∈ C + . (4.71)The Herglotz property of Λ θ ′ ,θ ′ R θ ,θ R ( · ) S θ ′ − θ ,θ ′ R − θ R , θ , θ R , θ ′ , θ ′ R ∈ [0 , π ) thenfollows again from the linear fractional transformation (4.15) and from (4.30) inLemma 4.5 upon identifying the 2 × A = (cid:2) A j,k (cid:3) ≤ j,k ≤ in Lemma4.5 with A ( θ, δ ) ∈ A in the special case where θ , θ R , θ ′ , θ ′ R ∈ [0 , π ) , θ ′ − θ = 0 mod( π ) , θ ′ R − θ R = 0 mod( π ) ,δ = δ R = 0 , δ ′ = δ ′ R = π/ . (4.72) OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 29
Given the Herglotz property of Λ θ ′ ,θ ′ R θ ,θ R ( · ) S θ ′ − θ ,θ ′ R − θ R , one obtains as in [42, Theo-rem 5.4] the representationΛ θ ′ ,θ ′ R θ ,θ R ( z ) S θ ′ − θ ,θ ′ R − θ R = Ξ θ ′ ,θ ′ R θ ,θ R + Υ θ ′ ,θ ′ R θ ,θ R z + Z R d Σ θ ′ ,θ ′ R θ ,θ R ( λ ) (cid:18) λ − z − λ λ (cid:19) ,z ∈ C \ σ ( H θ ,θ R ) , (4.73)with Ξ θ ′ ,θ ′ R θ ,θ R and Σ θ ′ ,θ ′ R θ ,θ R ( · ) as in (4.32) and (4.33), and with Υ θ ′ ,θ ′ R θ ,θ R satisfying0 ≤ Υ θ ′ ,θ ′ R θ ,θ R ∈ C × . (4.74)Thus, to conclude the proof of (4.31), it remains to prove that actually, Υ θ ′ ,θ ′ R θ ,θ R = 0.The latter fact is clear sinceΛ θ ′ ,θ ′ R θ ,θ R ( z ) = | z |→∞ Im( z / ) > O ( | z | / ) , (4.75)using the fact that by (4.6),Λ θ ′ ,θ ′ R θ ,θ R ( z ) = C θ ′ − θ ,θ ′ R − θ R + S θ ′ − θ ,θ ′ R − θ R Λ θ ,θ R ( z ) , (4.76)and applying Lemma 3.4.Finally, (4.34) is a consequence of (4.71) and Lemma 4.5 with A = A ( θ, δ ) chosenas in (4.72). (cid:3) As a particular case of Theorem 4.6 where θ ′ = ( θ + ( π/ π ), θ ′ R =( θ R + ( π/ π ), one concludes thatΛ θ ,θ R ( z ) = Λ ( θ + π ) mod(2 π ) , ( θ R + π ) mod(2 π ) θ ,θ R ( z ) , θ , θ R ∈ [0 , π ) , z ∈ C \ σ ( H θ ,θ R ) , (4.77)is a 2 × Krein-Type Resolvent Formulas
Krein-type resolvent formulas have been studied in a great variety of contexts,far too numerous to account for all in this paper. For instance, they are of funda-mental importance in connection with the spectral and inverse spectral theory ofordinary and partial differential operators. Abstract versions of Krein-type resol-vent formulas (see also the brief discussion at the end of our introduction), con-nected to boundary value spaces (boundary triples) and self-adjoint extensions ofclosed symmetric operators with equal (possibly infinite) deficiency spaces, havereceived enormous attention in the literature. In particular, we note that Robin-to-Dirichlet maps in the context of ordinary differential operators reduce to thecelebrated (possibly, matrix-valued) Weyl–Titchmarsh function, the basic object ofspectral analysis in this context. Since it is impossible to cover the literature inthis paper, we refer the reader to the rather extensive recent bibliography in [35],[36], and [38]. Here we just mention, for instance, [1, Sect. 84], [2], [3], [4], [5], [6],[11], [12], [14], [15], [16], [17], [18], [19], [25], [26], [28], [33], [34], [39], [42], [43, Ch.3], [44], [45, Ch. 13],[48], [49], [50], [52], [53], [54], [57], [60], [62], [63], [64], [65],[66], [68], [69], [70], [71], [72], [74], [73], [76], [77], [78], [81], and the references citedtherein.
We start by explicitly computing operators of the type γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − which play a role at various places in this manuscript (cf. Theorem 3.5, Lemma 5.2,and Theorem 5.3): Lemma 5.1.
Assume that θ , θ R , θ ′ , θ ′ R ∈ S π , let H θ ,θ R be defined as in (2.5) ,and suppose that z ∈ C \ σ ( H θ ,θ R ) . Then, assuming f ∈ L ((0 , R ); dx ) , and writing γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − f = "(cid:0) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:1) f (cid:0) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:1) f ∈ C , (5.1) one has (cid:0) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:1) f = sin( θ ′ − θ ) W ( u + ,θ R ( z, · ) , u − ,θ ( z, · )) (cid:0) u + ,θ R ( z, · ) , f ) L ((0 ,R ); dx ) × − u − ,θ ( z, θ ) , θ ∈ S π \{ , π } , u ′− ,θ ( z, θ ) , θ ∈ S π \{ π/ , π/ } , (5.2) (cid:0) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:1) f = sin( θ ′ R − θ R ) W ( u + ,θ R ( z, · ) , u − ,θ ( z, · )) (cid:0) u − ,θ ( z, · ) , f ) L ((0 ,R ); dx ) × − u + ,θR ( z,R )sin( θ R ) , θ R ∈ S π \{ , π } , u ′ + ,θR ( z,R )cos( θ R ) , θ R ∈ S π \{ π/ , π/ } , (5.3) in particular, | (cid:0) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:1) f | = θ ′ → θ O ( θ ′ − θ ) C ( z ) k f k L ((0 ,R ); dx ) , (5.4) | (cid:0) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:1) f | = θ ′ R → θ R O ( θ ′ R − θ R ) C ( z ) k f k L ((0 ,R ); dx ) , (5.5) for some constants C j ( z ) > , j = 1 , , and hence γ θ ,θ R ( H θ ,θ R − zI ) − = 0 in B (cid:0) L ((0 , R ); dx ) , C (cid:1) , (5.6) (cid:0) γ θ ,θ R ( H θ ,θ R − zI ) − (cid:1) k = 0 in B (cid:0) L ((0 , R ); dx ) , C (cid:1) , k = 1 , . (5.7) Proof.
Employing (3.37) and (3.38) one obtains γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − f = γ θ ′ ,θ ′ R (cid:18) Z R dx ′ G θ ,θ R ( z, · , x ′ ) f ( x ′ ) (cid:19) = (cid:18) γ θ ′ ,θ ′ R (cid:18) R R dx ′ G θ ,θ R ( z, · , x ′ ) f ( x ′ ) (cid:19)(cid:19) (cid:18) γ θ ′ ,θ ′ R (cid:18) R R dx ′ G θ ,θ R ( z, · , x ′ ) f ( x ′ ) (cid:19)(cid:19) (5.8)and hence (with W ( z ) = W ( u + ,θ R ( z, · ) , u − ,θ ( z, · ))) (cid:18) γ θ ′ ,θ ′ R (cid:18) Z R dx ′ G θ ,θ R ( z, · , x ′ ) f ( x ′ ) (cid:19)(cid:19) = 1 W ( z ) (cid:26) cos( θ ′ ) (cid:18) Z R dx ′ G θ ,θ R ( z, , x ′ ) f ( x ′ ) (cid:19) + sin( θ ′ ) (cid:18) Z R dx ′ (cid:18) ∂∂x G θ ,θ R ( z, x, x ′ ) (cid:12)(cid:12)(cid:12)(cid:12) x 0) + sin( θ ′ ) u ′ + ,θ ( z, (cid:3) Z R dx ′ u + ,θ R ( z, x ′ ) f ( x ′ ) (cid:27) = 1 W ( z ) (cid:0) u + ,θ R ( z, · ) , f ) L ((0 ,R ); dx ) × ( [cos( θ ′ ) − sin( θ ′ ) cot( θ )] u − ,θ ( z, , θ ∈ S π \{ , π } , [ − cos( θ ′ ) tan( θ ) + sin( θ ′ )] u ′− ,θ ( z, , θ ∈ S π \{ π/ , π/ } . (5.9)Equation (5.9) is easily seen to be equivalent to (5.2). Equation (5.3) is derivedanalogously. (cid:3) Introducing the orthogonal projections in C , P = (cid:20) (cid:21) , P = (cid:20) (cid:21) , (5.10)one obtains the following result, patterned after [61, Lemma 6] in the context ofSchr¨odinger operators with Dirichlet and Neumann boundary conditions on a cubein R n : Lemma 5.2. Assume that θ , θ R , θ ′ , θ ′ R ∈ S π , let H θ ,θ R and H θ ′ ,θ ′ R be defined asin (2.5) , and suppose that z ∈ C (cid:15)(cid:0) σ ( H θ ,θ R ) ∪ σ ( H θ ′ ,θ ′ R ) (cid:1) . Then ( H θ ′ ,θ ′ R − zI ) − = ( H θ ,θ R − zI ) − + (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ S − θ ′ − θ ,θ ′ R − θ R (cid:2) γ θ ,θ R ( H θ ′ ,θ ′ R − zI ) − (cid:3) , (5.11) θ ′ = θ , θ ′ R = θ R , ( H θ ,θ ′ R − zI ) − = ( H θ ,θ R − zI ) − + (cid:2) γ θ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [sin( θ ′ R − θ R )] − P (cid:2) γ θ ,θ R ( H θ ,θ ′ R − zI ) − (cid:3) ,θ ′ R = θ R , (5.12)( H θ ′ ,θ R − zI ) − = ( H θ ,θ R − zI ) − + (cid:2) γ θ ′ ,θ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [sin( θ ′ − θ )] − P (cid:2) γ θ ,θ R ( H θ ′ ,θ R − zI ) − (cid:3) ,θ ′ = θ . (5.13) Proof. We first consider the case θ , θ R , θ ′ , θ ′ R ∈ S π \{ , π } , θ ′ = θ , θ ′ R = θ R , (5.14)which illustrates the principal idea of the proof. To get started, we pick f, g ∈ L ((0 , R ); dx ) and introduce φ = (( H θ ,θ R ) ∗ − zI ) − f ∈ dom(( H θ ,θ R ) ∗ ) ,ψ = ( H θ ′ ,θ ′ R − zI ) − g ∈ dom( H θ ′ ,θ ′ R ) . (5.15)Then one computes(( H θ ,θ R ) ∗ − zI ) φ, ψ ) L ((0 ,R ); dx ) − ( φ, ( H θ ′ ,θ ′ R − zI ) ψ ) L ((0 ,R ); dx ) = − Z R dx φ ′′ ( x ) ψ ( x ) + Z R dx φ ( x ) ψ ′′ ( x )= − φ ′ ( R ) ψ ( R ) + φ ′ (0) ψ (0) + φ ( R ) ψ ′ ( R ) − φ (0) ψ ′ (0)= [cot( θ ′ R ) − cot( θ R )] φ ( R ) ψ ( R ) + [cot( θ ′ ) − cot( θ )] φ (0) ψ (0) = − sin( θ ′ R − θ R )sin( θ ′ R ) sin( θ R ) φ ( R ) ψ ( R ) − sin( θ ′ − θ )sin( θ ′ ) sin( θ ) φ (0) ψ (0) , (5.16)using the fact that (5.15) impliescos( θ ) φ (0) + sin( θ ) φ ′ (0) = 0 , cos( θ R ) φ ( R ) − sin( θ R ) φ ′ ( R ) = 0 , cos( θ ′ ) ψ (0) + sin( θ ′ ) ψ ′ (0) = 0 , cos( θ ′ R ) ψ ( R ) − sin( θ ′ R ) ψ ′ ( R ) = 0 . (5.17)Using (5.17) once again, one also computes( γ θ ′ ,θ ′ R φ, γ θ ,θ R ψ ) C = − sin ( θ ′ R − θ R )sin( θ ′ R ) sin( θ R ) φ ( R ) ψ ( R ) − sin ( θ ′ − θ )sin( θ ′ ) sin( θ ) φ (0) ψ (0) . (5.18)A comparison of (5.16) and (5.18) then yields(( H θ ,θ R ) ∗ − zI ) φ, ψ ) L ((0 ,R ); dx ) − ( φ, ( H θ ′ ,θ ′ R − zI ) ψ ) L ((0 ,R ); dx ) = (cid:0) γ θ ′ ,θ ′ R φ, S − θ ′ − θ ,θ ′ R − θ R γ θ ,θ R ψ (cid:1) C , (5.19)or equivalently, (cid:0) f, ( H θ ′ ,θ ′ R − zI ) − g (cid:1) L ((0 ,R ); dx ) = (cid:0) f, ( H θ ,θ R − zI ) − g (cid:1) L ((0 ,R ); dx ) + (cid:0) f, (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ S − θ ′ − θ ,θ ′ R − θ R (5.20) × (cid:2) γ θ ,θ R ( H θ ′ ,θ ′ R − zI ) − (cid:3) g ) L ((0 ,R ); dx ) , and hence (5.11) since f, g ∈ L ((0 , R ); dx ) are arbitrary, under the additionalassumptions in (5.14).Employing Lemma 5.1 (and particularly, (5.4)–(5.7)) then shows that (5.11) iscontinuous in θ , θ R , θ ′ , θ ′ R ∈ S π with respect to the norm in B (cid:0) L ((0 , R ); dx ) (cid:1) ,removing the restrictions θ , θ R , θ ′ , θ ′ R ∈ S π \{ , π } in (5.20), and thus proving(5.11).Analogous considerations imply (5.12) and (5.13). (cid:3) The principal result of this section, Krein’s formula for the difference of resolventsof H θ ′ ,θ ′ R and H θ ,θ R , then reads as follows: Theorem 5.3. Assume that θ , θ R , θ ′ , θ ′ R ∈ S π , let H θ ,θ R and H θ ′ ,θ ′ R be definedas in (2.5) , and suppose that z ∈ C (cid:15)(cid:0) σ ( H θ ,θ R ) ∪ σ ( H θ ′ ,θ ′ R ) (cid:1) . Then, with Λ θ ′ ,θ ′ R θ ,θ R ( z ) introduced in (2.39) , ( H θ ′ ,θ ′ R − zI ) − = ( H θ ,θ R − zI ) − − (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ S − θ ′ − θ ′ ,θ ′ R − θ R (5.21) × h Λ θ ′ ,θ ′ R θ ,θ R ( z ) i − (cid:2) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:3) , θ = θ ′ , θ R = θ ′ R . ( H θ ,θ ′ R − zI ) − = ( H θ ,θ R − zI ) − − (cid:2) γ θ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [sin( θ ′ R − θ R )] − P (5.22) × h Λ θ ,θ ′ R θ ,θ R ( z ) i − P (cid:2) γ θ ,θ ′ R ( H θ ,θ R − zI ) − (cid:3) , θ R = θ ′ R , OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 33 ( H θ ′ ,θ R − zI ) − = ( H θ ,θ R − zI ) − − (cid:2) γ θ ′ ,θ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ [sin( θ ′ − θ )] − P (5.23) × h Λ θ ′ ,θ R θ ,θ R ( z ) i − P (cid:2) γ θ ′ ,θ R ( H θ ,θ R − zI ) − (cid:3) , θ = θ ′ . Proof. Taking adjoints on both sides of (5.11), and subsequently replacing z by z , θ , θ R by θ , θ R , and V by V , then yields( H θ ′ ,θ ′ R − zI ) − = ( H θ ,θ R − zI ) − + (cid:2) γ θ ,θ R (( H θ ′ ,θ ′ R ) ∗ − zI ) − (cid:3) ∗ S − θ ′ − θ ,θ ′ R − θ R (cid:2) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:3) , (5.24) θ ′ = θ , θ ′ R = θ R . Applying γ θ ,θ R to both sides of (5.24), and using the fact that γ θ ,θ R ( H θ ,θ R − zI ) − = 0 by (5.6), one obtains γ θ ,θ R ( H θ ′ ,θ ′ R − zI ) − = γ θ ,θ R (cid:2) γ θ ,θ R (( H θ ′ ,θ ′ R ) ∗ − zI ) − (cid:3) ∗ × S − θ ′ − θ ,θ ′ R − θ R (cid:2) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:3) , θ ′ = θ , θ ′ R = θ R . (5.25)An insertion of (5.25) into (5.11) implies( H θ ′ ,θ ′ R − zI ) − = ( H θ ,θ R − zI ) − + (cid:2) γ θ ′ ,θ ′ R (( H θ ,θ R ) ∗ − zI ) − (cid:3) ∗ S − θ ′ − θ ,θ ′ R − θ R γ θ ,θ R (cid:2) γ θ ,θ R (( H θ ′ ,θ ′ R ) ∗ − zI ) − (cid:3) ∗ × S − θ ′ − θ ,θ ′ R − θ R (cid:2) γ θ ′ ,θ ′ R ( H θ ,θ R − zI ) − (cid:3) , θ ′ = θ , θ ′ R = θ R . (5.26)Using (2.48) and (3.39) one obtains γ θ ,θ R (cid:2) γ θ ,θ R (( H θ ′ ,θ ′ R ) ∗ − zI ) − (cid:3) ∗ = − h Λ θ ′ ,θ ′ R θ ,θ R ( z ) i − S θ ′ − θ ,θ ′ R − θ R , z ∈ C + , (5.27)and inserting (5.27) into (5.26) yields (5.21).Since equations (5.22) and (5.23) are proved similarly, we just briefly sketch theproof of (5.22): First, in analogy to (5.24), one derives from (5.12) that( H θ ,θ ′ R − zI ) − = ( H θ ,θ R − zI ) − + (cid:2) γ θ ,θ R (( H θ ,θ ′ R ) ∗ − zI ) − (cid:3) ∗ [sin( θ ′ R − θ R )] − P (cid:2) γ θ ,θ ′ R ( H θ ,θ R − zI ) − (cid:3) ,θ ′ R = θ R . (5.28)Applying γ θ ,θ R to both sides of (5.28), and using again the fact that γ θ ,θ R ( H θ ,θ R − zI ) − = 0, one obtains γ θ ,θ R ( H θ ,θ ′ R − zI ) − = γ θ ,θ R (cid:2) γ θ ,θ R (( H θ ,θ ′ R ) ∗ − zI ) − (cid:3) ∗ × [sin( θ ′ R − θ R )] − P (cid:2) γ θ ,θ ′ R ( H θ ,θ R − zI ) − (cid:3) , θ ′ R = θ R . (5.29)Finally, an insertion of (5.29) into the right-hand side of (5.12), and using (5.27)in the special case θ ′ = θ , that is, γ θ ,θ R (cid:2) γ θ ,θ R (( H θ ,θ ′ R ) ∗ − zI ) − (cid:3) ∗ = − h Λ θ ,θ ′ R θ ,θ R ( z ) i − S ,θ ′ R − θ R = − h Λ θ ,θ ′ R θ ,θ R ( z ) i − [sin( θ ′ R − θ R )] − P , z ∈ C + , (5.30)yields (5.22). (cid:3) A Brief Outlook In this section we provide a brief comparison between Λ θ ,θ R ( z ) and the 2 × H θ ,θ R in the self-adjointcontext, that is, under the assumptions θ , θ R ∈ [0 , π ) and V is real-valued inaddition to (2.4). While both 2 × θ ,θ R ( z ) and various versions of theWeyl–Titchmarsh matrix we will link the entries in both matrices to the Green’sfunction G θ ,θ R ( z, x, x ′ ) of H θ ,θ R and provide some formulas which may well be ofindependent interest.We start by linking Λ θ ,θ R ( z ) and G θ ,θ R ( z, x, x ′ ) and list a variety of pertinentformulas (choosing z ∈ C \ R for notational simplicity):Λ θ ,θ R ( z ) , = m + ,θ ( z, θ R )= − sin( θ ) + cos( θ ) m + , ( z, θ R )cos( θ ) + sin( θ ) m + , ( z, θ R )= − sin( θ ) + cos( θ ) u ′ + ,θ R ( z, θ ) + sin( θ ) u ′ + ,θ R ( z, ( θ ) (cid:2) G θ ,θ R ( z, , 0) + sin( θ ) cos( θ ) (cid:3) , (6.1) θ ∈ [0 , π ) \{ , π } , θ R ∈ [0 , π ) , Λ , ( z ) , = m + , ( z, 0) = − sin( θ )cos( θ ) + sin( θ ) m + , ( z, θ R )= sin( θ ) (cid:2) − cos( θ ) + sin( θ ) m + ,θ ( z, θ R ) (cid:3) . (6.3)In the same manner one obtainsΛ θ ,θ R ( z ) , = − m − ,θ R ( z, θ )= − sin( θ R ) + cos( θ R ) m − , ( z, θ )cos( θ R ) − sin( θ R ) m − , ( z, θ )= − sin( θ R ) + cos( θ R ) u ′− ,θ ( z, R )cos( θ R ) − sin( θ R ) u ′− ,θ ( z, R )= 1sin ( θ R ) (cid:2) G θ ,θ R ( z, R, R ) + sin( θ R ) cos( θ R ) (cid:3) , (6.4) θ ∈ [0 , π ) , θ R ∈ [0 , π ) \{ , π } , Λ , ( z ) , = − m − , ( z, Here we used (3.29) and (cf. also (3.28)) G θ ,θ R ( z, R, R ) = − sin( θ R )cos( θ R ) − sin( θ R ) m − , ( z, θ )= sin( θ R ) (cid:2) − cos( θ R ) − sin( θ R ) m − ,θ R ( z, θ ) (cid:3) . (6.6)Similarly, the off-diagonal terms of Λ θ ,θ R ( z ) in (3.21) can be written asΛ θ ,θ R ( z ) , = Λ θ ,θ R ( z ) , = 1sin( θ R ) G θ ,θ R ( z, R, R ) − u ′− ,θ ( z, θ ) , θ ∈ [0 , π ) \{ π/ , π/ } , u − ,θ ( z, θ ) , θ ∈ [0 , π ) \{ , π } , (6.7) θ R ∈ [0 , π ) \{ , π } , = 1sin( θ ) G θ ,θ R ( z, , u ′ + ,θR ( z,R )cos( θ R ) , θ R ∈ [0 , π ) \{ π/ , π/ } , u + ,θR ( z,R )sin( θ R ) , θ R ∈ [0 , π ) \{ , π } , (6.8) θ ∈ [0 , π ) \{ , π } . Next we turn to (variants of) the 2 × H θ ,θ R with respect to an interior reference point x ∈ (0 , R ). We start byintroducing (again, choosing z ∈ C \ R for notational simplicity) m + , ( z, x , θ R ) = u ′ + ,θ R ( z, x ) u + ,θ R ( z, x ) , x ∈ (0 , R ) , (6.9) m − , ( z, x , θ ) = u ′− ,θ ( z, x ) u − ,θ ( z, x ) , x ∈ (0 , R ) , (6.10)and more generally, m + ,α ( z, x , θ R ) = − sin( α ) + cos( α ) m + , ( z, x , θ R )cos( α ) + sin( α ) m + , ( z, x , θ R ) , α ∈ [0 , π ) , (6.11) m − ,α ( z, x , θ ) = − sin( α ) + cos( α ) m − , ( z, x , θ )cos( α ) + sin( α ) m − , ( z, x , θ ) , α ∈ [0 , π ) . (6.12)We note that m + ,α ( · , x , θ R ) and − m − ,α ( · , x , θ ) are known to be Herglotz func-tions (cf., e.g., [23, Sect. 9.5]), [56, Sect. II.8], [67, Sect. 6.5], [79, Ch. III]).Associated with (6.9)–(6.12) one then defines the 2 × H θ ,θ R ) by, M ( z, x , θ , θ R ) = (cid:2) M ,j,k ( z, x , θ , θ R ) (cid:3) j,k =1 , = (cid:2) m − , ( z, x , θ ) − m + , ( z, x , θ R ) (cid:3) − (6.13) × (cid:20) [ m − , ( z, x , θ ) + m + , ( z, x , θ R )] [ m − , ( z, x , θ ) + m + , ( z, x , θ R )] m − , ( z, x , θ ) m + , ( z, x , θ R ) (cid:21) , and more generally, by M α ( z, x , θ , θ R ) = (cid:2) M α,j,k ( z, x , θ , θ R ) (cid:3) j,k =1 , = (cid:2) m − ,α ( z, x , θ ) − m + ,α ( z, x , θ R ) (cid:3) − (6.14) × (cid:20) [ m − ,α ( z, x , θ ) + m + ,α ( z, x , θ R )] [ m − ,α ( z, x , θ ) + m + ,α ( z, x , θ R )] m − ,α ( z, x , θ ) m + ,α ( z, x , θ R ) (cid:21) , α ∈ [0 , π ) . By inspection,det( M α ( z, x , θ , θ R )) = − / , α ∈ [0 , π ) , z ∈ C \ R , (6.15)Im( M α ( z, x , θ , θ R )) > , α ∈ [0 , π ) , z ∈ C + , (6.16)and hence M α ( · , x , θ , θ R ) is a 2 × M α,j,j ( · , x , θ , θ R ) are Herglotz functions for j = 1 , 2. In particular, the matrices M α ( z, x , θ , θ R ) can be shown to have Herglotz representations of the type (4.31).For the connection of M ( z, x , θ , θ R ) and M α ( z, x , θ , θ R ) with the Green’sfunction G θ ,θ R ( z, x, x ′ ) of H θ ,θ R we first introduce a bit of notation: ∂ G θ ,θ R ( z, x , x ′ ) = ∂ x G θ ,θ R ( z, x , x ′ ) (cid:12)(cid:12) x = x ,∂ G θ ,θ R ( z, x, x ) = ∂ x G θ ,θ R ( z, x, x ) (cid:12)(cid:12) x = x , (6.17) ∂ ∂ G θ ,θ R ( z, x , x ) = ∂ x ∂ x G θ ,θ R ( z, x , x ) (cid:12)(cid:12) x = x ,x = x , etc.The expressions (6.13) and (6.14) for M ( z, x , θ , θ R ) and M α ( z, x , θ , θ R ) thencan be rewritten as follows: M , , ( z, x , θ , θ R ) = G θ ,θ R ( z, x , x ) . (6.18) M , , ( z, x , θ , θ R ) = M , , ( z, x , θ , θ R )= (1 / ∂ + ∂ ) G θ ,θ R ( z, x ± , x ∓ , (6.19) M , , ( z, x , θ , θ R ) = ∂ ∂ G θ ,θ R ( z, x , x ) , (6.20)and M α, , ( z, x , θ , θ R )= (cid:0) cos( α ) + sin( α ) ∂ (cid:1)(cid:0) cos( α ) + sin( α ) ∂ (cid:1) G θ ,θ R ( z, x , x ) . (6.21) M α, , ( z, x , θ , θ R ) = M α, , ( z, x , θ , θ R )= (1 / (cid:0) (cos( α ) + sin( α ) ∂ )( − sin( α ) + cos( α ) ∂ )+ ( − sin( α ) + cos( α ) ∂ )(cos( α ) + sin( α ) ∂ ) (cid:1) G θ ,θ R ( z, x ± , x ∓ , (6.22) M α, , ( z, x , θ , θ R )= (cid:0) − sin( α ) + cos( α ) ∂ (cid:1)(cid:0) − sin( α ) + cos( α ) ∂ (cid:1) G θ ,θ R ( z, x , x ) . (6.23)For relevant references in the context of (6.9)–(6.23), we refer, for instance, to[20, Ch. III], [23, Ch. 9], [31, App. J], [32], [40], [41], [55, Ch. 2], [56, Ch. 2], [67,Ch. 6], [79, Chs. II, III], and the references cited therein.A comparison of equations (6.1), (6.2), (6.4), (6.5), (6.7), (6.8) with equations(6.18)–(6.20) and (6.21)–(6.23), respectively, clearly exhibits the different characterof Λ θ ,θ R ( z ) and M α ( z, x , θ , θ R ), α ∈ [0 , π ), despite the fact that both are 2 × H θ ,θ R . Additional differences are highlighted in Remark3.3, and we feel that Λ θ ,θ R ( z ) (and more generally, Λ θ ′ ,θ ′ R θ ,θ R ( z )) is deserving of a moredetailed study.We conclude with a final observation: OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 37 Remark 6.1. With only minor modifications, all results in this paper extend togeneral, regular three-coefficient differential expressions of the type1 r (cid:18) − ddx p ddx + q (cid:19) , x ∈ [0 , R ] , (6.24)where p > , r > , R ) , p , q, r ∈ L ((0 , R ); dx ) , (6.25)generating Sturm–Liouville operator realizations in L ((0 , R ); rdx ). One just needsto consistently replace the derivative f ′ for elements in operator domains and Wron-skians by the first quasi-derivative ( pf ′ ). Acknowledgments. We are indebted to Sergei Avdonin, Pavel Kurasov, andMark Malamud for helpful discussions on this topic, and to Dorina Mitrea for acritical reading of this manuscript. References [1] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Volume II ,Pitman, Boston, 1981.[2] S. Albeverio, J. F. Brasche, M. M. Malamud, and H. Neidhardt, Inverse spectral theory forsymmetric operators with several gaps: scalar-type Weyl functions , J. Funct. Anal. ,144–188 (2005).[3] D. Alpay and J. Behrndt, Generalized Q -functions and Dirichlet-to-Neumann maps for el-liptic differential operators , J. Funct. Anal. , 1666–1694 (2009).[4] W. O. Amrein and D. B. Pearson, M operators: a generalisation of Weyl–Titchmarsh theory ,J. Comp. Appl. Math. , 1–26 (2004).[5] Yu. M. Arlinski˘ı and E. R. Tsekanovski˘ı, On von Neumann’s problem in extension theory ofnonnegative operators , Proc. Amer. Math. Soc. , 3143–3154 (2003).[6] Yu. M. Arlinski˘ı and E. R. Tsekanovski˘ı, The von Neumann problem for nonnegative sym-metric operators , Integr. Eq. Operator Th. , 319–356 (2005).[7] S. A. Avdonin, M. I. Belishev, and S. A. Ivanov, Boundary control and a matrix inverseproblem for the equation u tt − u xx + V ( x ) u = 0, Math. USSR Sbornik , 287–310 (1992).[8] S. Avdonin and P. Kurasov, Inverse problems for quantum trees , Inverse Probl. Imaging ,1–21 (2008).[9] S. Avdonin, S. Lenhart, and V. Protopopescu, Solving the dynamical inverse problem for theSchr¨odinger equation by the boundary control method , Inverse Probl. , 349–361 (2002).[10] S. Avdonin, S. Lenhart, and V. Protopopescu, Determining the potential in the Schr¨odingerequation from the Dirichlet to Neumann map by the boundary control method , J. Inv. Ill-Posed Probl., , 1–14 (2005).[11] J. Behrndt and M. Langer, Boundary value problems for partial differential operators onbounded domains , J. Funct. Anal. , 536–565 (2007).[12] J. Behrndt, M. M. Malamud and H. Neidhardt, Scattering matrices and Weyl functions ,Proc. London Math. Soc. (3) , 568–598 (2008).[13] J. F. Brasche, M. M. Malamud, and H. Neidhardt, Weyl functions and singular continuousspectra of self-adjoint extensions , in Stochastic Processes, Physics and Geometry: New Inter-plays. II. A Volume in Honor of Sergio Albeverio , F. Gesztesy, H. Holden, J. Jost, S. Paycha,M. R¨ockner, and S. Scarlatti (eds.), Canadian Mathematical Society Conference Proceedings,Vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 75–84.[14] J. F. Brasche, M. M. Malamud, and H. Neidhardt, Weyl function and spectral properties ofself-adjoint extensions , Integral Eq. Operator Th. , 264–289 (2002).[15] B. M. Brown, G. Grubb, and I. G. Wood, M -functions for closed extensions of adjoint pairsof operators with applications to elliptic boundary problems , Math. Nachr. , 314–347(2009). [16] M. Brown, J. Hinchcliffe, M. Marletta, S. Naboko, and I. Wood, The abstract Titchmarsh–Weyl M -function for adjoint operator pairs and its relation to the spectrum , Integral Equ.Operator Th. , 297–320 (2009).[17] B. M. Brown and M. Marletta, Spectral inclusion and spectral exactness for PDE’s on exteriordomains , IMA J. Numer. Anal. , 21–43 (2004).[18] B. M. Brown, M. Marletta, S. Naboko, and I. Wood, Boundary triplets and M -functions fornon-selfadjoint operators, with applications to elliptic PDEs and block operator matrices , J.London Math. Soc. (2) , 700–718 (2008).[19] J. Br¨uning, V. Geyler, and K. Pankrashkin, Spectra of self-adjoint extensions and applicationsto solvable Schr¨odinger operators , Rev. Math. Phys. , 1–70 (2008).[20] R. Carmona and J. Lacroix, Spectral Theory of Random Schr¨odinger Operators , Birkh¨auser,Basel, 1990.[21] S. Clark and F. Gesztesy, Weyl–Titchmarsh M -function asymptotics and Borg-type theoremsfor Dirac operators , Trans. Amer. Math. Soc. , 3475–3534 (2002).[22] S. Clark and F. Gesztesy, On self-adjoint and J -self-adjoint Dirac-type operators: A casestudy , in Recent Advances in Differential Equations and Mathematical Physics , N. Chernov,Y. Karpeshina, I. W. Knowles, R. T. Lewis, and R. Weikard (eds.), Contemp. Math. Vol.412, Amer. Math. Soc., Providence, RI, 2006, pp. 103–140.[23] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations , Krieger,Malabar, 1985.[24] V. A. Derkach, S. Hassi, M. M. Malamud, and H. S. V. de Snoo, Generalized resolvents ofsymmetric operators and admissibility , Meth. Funct. Anal. Top. , No. 3, 24–55 (2000).[25] V. Derkach, S. Hassi, M. Malamud, and H. de Snoo, Boundary relations and their Weylfamilies , Trans. Amer. Math. Soc. , 5351–5400 (2006).[26] V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problemsfor Hermitian operators with gaps , J. Funct. Anal. , 1–95 (1991).[27] V. A. Derkach and M. M. Malamud, Characteristic functions of almost solvable extensionsof Hermitian operators , Ukrain. Math. J. , 379–401 (1992).[28] V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and themoment problem , J. Math. Sci. , 141–242 (1995).[29] N. Dunford and J. T. Schwartz, Linear Operators Part II: Spectral Theory , Interscience, NewYork, 1988.[30] C. Fox, V. Oleinik, and B. Pavlov, A Dirichlet-to-Neumann map approach to resonance gapsand bands of periodic networks , in Recent Advances in Differential Equations and Mathe-matical Physics , N. Chernov, Y. Karpeshina, I. W. Knowles, R. T. Lewis, and R. Weikard(eds.), Contemp. Math. Vol. 412, Amer. Math. Soc., Providence, RI, 2006, pp. 151–169.[31] F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Vol-ume I: (1 + 1) -Dimensional Continuous Models , Cambridge Studies in Advanced Mathemat-ics, Vol. 79, Cambridge University Press, Cambridge, 2003.[32] F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, Higher order trace relations for Schr¨odingeroperators , Rev. Math. Phys. , 893–922 (1995).[33] F. Gesztesy, N. J. Kalton, K. A. Makarov, and E. Tsekanovskii, Some Applications ofOperator-Valued Herglotz Functions , in Operator Theory, System Theory and Related Topics.The Moshe Livˇsic Anniversary Volume , D. Alpay and V. Vinnikov (eds.), Operator Theory:Advances and Applications, Vol. 123, Birkh¨auser, Basel, 2001, pp. 271–321.[34] F. Gesztesy, K. A. Makarov, and E. Tsekanovskii, An Addendum to Krein’s formula , J. Math.Anal. Appl. , 594–606 (1998).[35] F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps,and Krein-type resolvent formulas for Schr¨odinger operators on bounded Lipschitz domains ,in Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: AVolume in Honor of Vladimir G. Maz’ya’s 70th Birthday , D. Mitrea and M. Mitrea (eds.),Proceedings of Symposia in Pure Mathematics, Vol. 79, Amer. Math. Soc., Providence, RI,2008, pp. 105–173.[36] F. Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type resolvent formulas forSchr¨odinger operators on bounded Lipschitz domains , in Modern Analysis and Applications.The Mark Krein Centenary Conference , Vol. 2, V. Adamyan, Y. M. Berezansky, I. Gohberg,M. L. Gorbachuk, V. Gorbachuk, A. N. Kochubei, H. Langer, and G. Popov (eds.), OperatorTheory: Advances and Applications, Vol. 191, Birkh¨auser, Basel, 2009, pp. 81–113. OUNDARY DATA MAPS FOR SCHR ¨ODINGER OPERATORS ON AN INTERVAL 39 [37] F. Gesztesy, M. Mitrea. Nonlocal Robin Laplacians and some remarks on a paper by Filonovon eigenvalue inequalities . J. Diff. Eq. , 2871–2896 (2009).[38] F. Gesztesy and M. Mitrea, Self-adjoint extensions of the Laplacian and Krein-type resolventformulas in nonsmooth domains , preprint, 2009.[39] F. Gesztesy, M. Mitrea, and M. Zinchenko, Variations on a theme of Jost and Pais , J. Funct.Anal. , 399–448 (2007).[40] F. Gesztesy, R. Ratnaseelan, and G. Teschl, The KdV hierarchy and associated trace for-mulas , in Recent Developments in Operator Theory and its Applications , I. Gohberg, P.Lancaster, and P. N. Shivakumar (eds.), Operator Theory: Advances and Applications, Vol.87, Birkh¨auser, Basel, 1996, pp. 125–163.[41] F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schr¨odinger operators , Trans. Amer. Math. Soc. , 349–373 (1996).[42] F. Gesztesy and E. Tsekanovskii, On matrix-valued Herglotz functions , Math. Nachr. ,61–138 (2000).[43] V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator DifferentialEquations , Kluwer, Dordrecht, 1991.[44] G. Grubb, Krein resolvent formulas for elliptic boundary problems in nonsmooth domains ,Rend. Semin. Mat. Univ. Politec. Torino , 271–297 (2008).[45] G. Grubb, Distributions and Operators , Graduate Texts in Mathematics, Vol. 252, Springer,New York, 2009.[46] T. Kato, Perturbation Theory for Linear Operators , corr. printing of the 2nd ed., Springer,Berlin, 1980.[47] A. Kiselev and B. Simon, Rank one perturbations with infinitesimal coupling , J. Funct. Anal. , 345–356 (1995).[48] M. G. Krein and I. E. Ovcharenko, Q -functions and sc -resolvents of nondensely definedhermitian contractions , Sib. Math. J. , 728–746 (1977).[49] M. G. Krein and I. E. Ovˇcarenko, Inverse problems for Q -functions and resolvent matricesof positive hermitian operators , Sov. Math. Dokl. , 1131–1134 (1978).[50] M. G. Krein, S. N. Saakjan, Some new results in the theory of resolvents of hermitian oper-ators , Sov. Math. Dokl. , 1086–1089 (1966).[51] M. G. Krein and Ju. L. Smul’jan, On linear-fractional transformations with operator coeffi-cients , Amer. Math. Soc. Transl. (2) , 125–152 (1974).[52] P. Kurasov, Triplet extensions I: Semibounded operators in the scale of Hilbert spaces , J.Analyse Math. , 251–286 (2009).[53] P. Kurasov and S. T. Kuroda, Krein’s resolvent formula and perturbation theory , J. OperatorTh. , 321–334 (2004).[54] H. Langer, B. Textorius, On generalized resolvents and Q -functions of symmetric linearrelations (subspaces) in Hilbert space , Pacific J. Math. , 135–165 (1977).[55] B. M. Levitan, Inverse Sturm–Liouville Problems , VNU Science Press, Utrecht, 1987.[56] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory , Amer. Math. Soc., Provi-dence, RI, 1975.[57] M. M. Malamud and V. I. Mogilevskii, Krein type formula for canonical resolvents of dualpairs of linear relations , Methods Funct. Anal. Topology, , no. 4, 72–100 (2002).[58] V. A. Marchenko, Some questions in the theory of one-dimensional linear differential opera-tors of the second order, I , Trudy Moskov. Mat. Ob˘s˘c. , 327–420 (1952). (Russian.) Englishtransl. in Amer. Math. Soc. Transl., Ser. 2, , 1–104 (1973).[59] V. A. Marchenko, Sturm–Liouville Operators and Applications , Birkh¨auser, Basel, 1986.[60] M. Marletta, Eigenvalue problems on exterior domains and Dirichlet to Neumann maps , J.Comp. Appl. Math. , 367–391 (2004).[61] S. Nakamura, A remark on the Dirichlet–Neumann decoupling and the integrated density ofstates , J. Funct. Anal. , 136–152 (2001).[62] G. Nenciu, Applications of the Kre˘ın resolvent formula to the theory of self-adjoint extensionsof positive symmetric operators , J. Operator Th. , 209–218 (1983).[63] K. Pankrashkin, Resolvents of self-adjoint extensions with mixed boundary conditions , Rep.Math. Phys. , 207–221 (2006).[64] B. Pavlov, The theory of extensions and explicitly-soluble models , Russ. Math. Surv. ,127–168 (1987). [65] B. Pavlov, S -matrix and Dirichlet-to-Neumann operators , Ch. 6.1.6 in Scattering: Scatteringand Inverse Scattering in Pure and Applied Science , Vol. 2, R. Pike and P. Sabatier (eds.),Academic Press, San Diego, 2002, pp. 1678–1688.[66] B. Pavlov, Krein formula with compensated singularities for the ND-mapping and the gen-eralized Kirchhoff condition at the Neumann Schr¨odinger junction , Russ. J. Math. Phys. ,364–388 (2008).[67] D. B. Pearson, Quantum Scattering and Spectral Theory , Academic Press, London, 1988.[68] A. Posilicano, A Krein-like formula for singular perturbations of self-adjoint operators andapplications , J. Funct. Anal. , 109–147 (2001).[69] A. Posilicano, Self-adjoint extensions by additive perturbations , Ann. Scuola Norm. Sup. PisaCl. Sci. (5) , no. 1, 1–20 (2003).[70] A. Posilicano, Boundary triples and Weyl functions for singular perturbations of self-adjointoperators , Meth. Funct. Anal. Topology , no. 2, 57–63 (2004).[71] A. Posilicano, Self-adjoint extensions of restrictions , Operators and Matrices , 483–506(2008).[72] A. Posilicano and L. Raimondi, Krein’s resolvent formula for self-adjoint extensions of sym-metric second-order elliptic differential operators , J. Phys. A: Math. Theor. , 015204 (11pp)(2009).[73] A. Rybkin, On the boundary control approach to inverse spectral and scattering theory forSchr¨odinger operators , Inverse Probl. Imaging , 139–149 (2009).[74] V. Ryzhov, A general boundary value problem and its Weyl function , Opuscula Math. ,305–331(2007).[75] V. Ryzhov, Weyl–Titchmarsh function of an abstract boundary value problem, operator col-ligations, and linear systems with boundary control , Complex Anal. Operator Theory ,289–322 (2009).[76] V. Ryzhov, Spectral boundary value problems and their linear operators , preprint, 2009.[77] Sh. N. Saakjan, On the theory of the resolvents of a symmetric operator with infinite defi-ciency indices , Dokl. Akad. Nauk Arm. SSR , 193–198 (1965). (Russian.)[78] A. V. Straus, Extensions and generalized resolvents of a non-densely defined symmetricoperator , Math. USSR Izv. , 179–208 (1970).[79] E. C. Titchmarsh, Eigenfunction Expansions, Part I, The Theory of Functions , 2nd ed., Oxford University Press, Oxford, 1985.[81] E. R. Tsekanovskii and Yu. L. Shmul’yan, The theory of bi-extensions of operators on riggedHilbert spaces. Unbounded operator colligations and characteristic functions , Russ. Math.Surv. , 73–131 (1977). Department of Mathematics & Statistics, University of Missouri, Rolla, MO 65409,USA E-mail address : [email protected] URL : http://web.umr.edu/~sclark/index.html Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail address : [email protected] URL : Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail address : [email protected] URL ::