Manifold decompositions and indices of Schrödinger operators
aa r X i v : . [ m a t h . A P ] J a n MANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS
GRAHAM COX, CHRISTOPER K.R.T. JONES, AND JEREMY L. MARZUOLA
Abstract.
The Maslov index is used to compute the spectra of different boundary value problems forSchr¨odinger operators on compact manifolds. The main result is a spectral decomposition formula for amanifold M divided into components Ω and Ω by a separating hypersurface Σ. A homotopy argumentrelates the spectrum of a second-order elliptic operator on M to its Dirichlet and Neumann spectra on Ω and Ω , with the difference given by the Maslov index of a path of Lagrangian subspaces. This Maslovindex can be expressed in terms of the Morse indices of the Dirichlet-to-Neumann maps on Σ. Applicationsare given to doubling constructions, periodic boundary conditions and the counting of nodal domains. Inparticular, a new proof of Courant’s nodal domain theorem is given, with an explicit formula for the nodaldeficiency. Introduction
Suppose M is a compact, orientable manifold, and L a selfadjoint, elliptic operator on M . It is of greatinterest to compute the spectrum of L , given boundary conditions on ∂M , and relate it to the underlyinggeometry of M and L . Of particular importance for many applications is the Morse index , or numberof negative eigenvalues. One approach to computing the Morse index is to deform M through a one-parameter family of domains { Ω t } and keep track of eigenvalues passing through 0 as t varies. For instance,if f : M → [0 ,
1] is a Morse function on M with f − (1) = ∂M , one can consider the sublevel sets Ω t = f − [0 , t )for t ∈ (0 , t → Vol(Ω t ) = 0 , it is easy to compute the Morse index of L on Ω t once t is sufficiently small. It thus remains to describe the“spectral flow” of the boundary value problems as t varies.This was done by Smale in [26] for the Dirichlet problem, assuming the Ω t remain diffeomorphic forall t , which is the case when f has no critical values in (0 , t to change but still assuming Dirichlet boundary conditions. In [12] Deng andJones used the Maslov index, a symplectic invariant, to generalize Smale’s result to more general boundaryconditions, but with the additional requirement that the domain be star-shaped. The star-shaped restrictionwas subsequently removed in [10], where the Maslov index was used to compute the spectral flow for anysmooth one-parameter family of domains, with quite general boundary conditions.To complete the picture, we must describe what happens when t passes through a critical value of f andthe topology of Ω t changes. More generally, we consider a decomposition of M into disjoint componentsalong a separating hypersurface Σ, as in Figure 1, and ask how the spectrum on M relates to the spectrumon each component. This question is answered in Theorem 1, which says the Morse index of L on M equals the sum of the Morse indices on each component (with appropriate boundary conditions on Σ) plusa “topological contribution,” which is given by the Maslov index of a path of Lagrangian subspaces in thesymplectic Hilbert space H / (Σ) ⊕ H − / (Σ) ⊕ H / (Σ) ⊕ H − / (Σ).By considering the limit in which either Ω or Ω is small in some appropriate sense, this can be relatedto classic results on eigenvalues of the Laplacian with respect to singular perturbations of the underlyingspatial domain. In [7] Chavel and Feldman studied the effect of removing a tubular neighborhood of a closedsubmanifold N ⊂ M , replacing M by the domain M ǫ = { x ∈ M : dist( x, N ) > ǫ } . Mathematics Subject Classification.
Primary: 35J10, 35P15, 58J50, 35J25; Secondary: 53D12, 35B05, 35B35.
Key words and phrases.
Schr¨odinger operator, manifold decomposition, Morse index, Maslov index, Dirichlet-to-Neumannmap, nodal domain. Ω Σ Ω ∂M Figure 1.
A manifold M with nontrivial boundary ∂M , separated into components Ω and Ω by an orientable hypersurface Σ.Assuming the codimension of N is at least 2, they proved convergence of the Dirichlet spectrum on M ǫ tothe spectrum on M as ǫ →
0. A similar analysis was carried out in [8] for manifolds to which a small handlehas been attached, with a sufficient condition given, in terms of an isoperimetric constant, for convergenceof the spectrum as the size of the handle decreases to zero. In [23] Rauch and Taylor considered a ratherweak notion of convergence for Euclidean domains and described the behavior of the Laplacian when oneremoves a small neighborhood of a polar set. (The definition of a polar set can be found in [23]; note inparticular that a submanifold of codimension at least 2 is a polar set, whereas a hypersurface is not.) In [15]Jimbo considered the case of two disjoint bounded domains connected by a small tube, and gave asymptoticformulas for the eigenvalues and eigenfunctions in the singular limit as the tube shrinks to a line.Several other authors have considered the reduction of spectral flows (and other analytic invariants)through similar manifold decompositions [6, 20, 29]. These results are all for first-order, Dirac-type ellipticoperators which have a particular form in a collar neighborhood of the separating hypersurface Σ.Our symplectic approach to this problem has many applications, which we explore in the last section ofthe paper. The first is a new proof of Courant’s nodal domain theorem, with an explicit formula for the nodaldeficiency. Then we compute the Morse indices of operators on “almost-doubled” manifolds, which consistof two identical (or almost identical) components glued together along a common boundary. We also usethe Maslov index to give a new proof of a well-known theorem relating the Dirichlet and Neumann countingfunctions to the spectrum of the Dirichlet-to-Neumann map. Finally, we relate the spectra of Schr¨odingeroperators on the torus—viewed as a cube with opposing faces identified—to the spectra on the cube withDirichlet boundary conditions, and find that the periodic and Dirichlet Morse indices are related by a kindof symmetrized Dirichlet-to-Neumann map.
Structure of the paper.
In Section 2 we define the relevant operators and domains, and state the mainresults of the paper. The fundamental relation between Morse and Maslov indices is proved in Section 3. InSection 4 we study the Maslov index in more detail, and relate it to the Dirichlet-to-Neumann maps of themanifold decomposition. Finally, in Section 5 these results are applied to a variety of geometric scenarios.
Acknowledgments
The authors wish to thank Chris Judge, Rafe Mazzeo, Michael Taylor and Gunther Uhlmann for veryhelpful conversations during the preparation of this manuscript. JLM was supported in part by U.S. NSFDMS-1312874 and NSF CAREER Grant DMS-1352353. GC and CKRTJ were supported by U.S. NSF GrantDMS-1312906. 2.
Definitions and results
Throughout we assume that M is a compact, orientable manifold with Lipschitz boundary ∂M , andΣ ⊂ M is an embedded Lipschitz hypersurface that separates M into two disjoint (but not necessarilyconnected) components: M \ Σ = Ω ∪ Ω . We further assume that Σ ∩ ∂M = ∅ . A typical situation isshown in Figure 1.Let g be a Riemannian metric on M and V a real-valued function, both of class L ∞ . We define the formaldifferential operator L = − ∆ g + V, (1) ANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS 3 where ∆ g is the Laplace–Beltrami operator of g . (This is a formal operator in the sense that its domain hasnot been specified; we will allow L to act on functions on Ω , Ω and M .)Fix i ∈ { , } and suppose u ∈ H (Ω i ) and ∆ g u ∈ L (Ω i ). It follows from Theorem 3.37 and Lemma 4.3in [19] that u | ∂ Ω i ∈ H / ( ∂ Ω i ) , ∂u∂ν i (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ω i ∈ H − / ( ∂ Ω i ) (2)and so the following weak version of Green’s first identity Z Ω i h∇ u, ∇ v i = − Z Ω i (∆ g u ) v + Z ∂ Ω i v ∂u∂ν i (3)holds for any v ∈ H (Ω i ), where ν i denotes the outward unit normal to Ω i . Note that ∂ Ω i is the disjointunion Σ ∪ ( ∂ Ω i ∩ ∂M ) and ν = − ν on Σ.We suppose that either Dirichlet or Neumann boundary conditions are prescribed on each connectedcomponent of ∂M , and correspondingly write ∂M = Σ D ∪ Σ N . (Thus Σ D and Σ N are closed, disjointLipschitz hypersurfaces which need not be connected.)For i ∈ { , } we let L Di and L Ni denote the Dirichlet and Neumann realizations of L on Ω i , respectively.These are unbounded, selfadjoint operators on L (Ω i ), with domains D ( L Di ) = ( u ∈ H (Ω i ) : ∆ g u ∈ L (Ω i ) , u | Σ ∪ (Σ D ∩ ∂ Ω i ) = 0 and ∂u∂ν i (cid:12)(cid:12)(cid:12)(cid:12) Σ N ∩ ∂ Ω i = 0 ) (4) D ( L Ni ) = ( u ∈ H (Ω i ) : ∆ g u ∈ L (Ω i ) , u | Σ D ∩ ∂ Ω i = 0 and ∂u∂ν i (cid:12)(cid:12)(cid:12)(cid:12) Σ ∪ (Σ N ∩ ∂ Ω i ) = 0 ) . (5)The operators L Di and L Ni have the same boundary conditions on the “outer boundary” ∂M ∩ ∂ Ω i ; thesuperscript refers only to the conditions imposed on the “inner boundary” Σ. We let L G denote the “global”realization of L on M . This is an unbounded, selfadjoint operator on L ( M ), with domain D ( L G ) = ( u ∈ H ( M ) : ∆ g u ∈ L ( M ) , u | Σ D ∩ ∂ Ω i = 0 and ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ N ∩ ∂ Ω i = 0 ) . (6)Each of the operators L G , L Di and L Ni is bounded below and selfadjoint with compact resolvent, andtherefore has a well-defined Morse index (number of negative eigenvalues, counting multiplicity), which wedenote Mor( · ). We additionally let Mor = Mor + dim ker (7)denote the number of nonpositive eigenvalues.Our main result relates the Morse index of L G to the Morse indices of L Di and L Ni . We compare thesequantities by encoding the boundary conditions on Σ in a Lagrangian subspace, which is then rotated betweenglobal boundary conditions, corresponding to L G , and decoupled boundary conditions, corresponding to L N and L D ; see (11) below. The difference in Morse indices is equated to a symplectic winding number—theMaslov index—for the rotating path of boundary conditions. The relevant technical properties of the Maslovindex are summarized in Appendix B of [10]; a more complete presentation can be found in [4] or [14]. Someapplications of the Maslov index to boundary value problems for PDE can be found in [10, 11, 12, 17, 22, 27].Consider the Hilbert space H = H / (Σ) ⊕ H − / (Σ), with the symplectic form ω induced by the bilinearpairing of H / (Σ) with ( H / (Σ)) ∗ = H − / (Σ), that is ω (( x, φ ) , ( y, ψ )) = ψ ( x ) − φ ( y )for x, y ∈ H / (Σ) and φ, ψ ∈ H − / (Σ). To study the decomposition of M by Σ we use the doubled space H ⊞ := H ⊕ H , (8)with the symplectic form ω ⊞ := ω ⊕ ( − ω ). The negative sign on the second component is chosen so thediagonal subspace { ( x, φ, x, φ ) : x ∈ H / (Σ) , φ ∈ H − / (Σ) } is Lagrangian. GRAHAM COX, CHRISTOPER K.R.T. JONES, AND JEREMY L. MARZUOLA
For each i ∈ { , } and λ ∈ R we define K λi = ( u ∈ H (Ω i ) : Lu = λu, u | Σ D ∩ ∂ Ω i = 0 and ∂u∂ν i (cid:12)(cid:12)(cid:12)(cid:12) Σ N ∩ Ω i = 0 ) , (9)with the equality Lu = λu meant in the distributional sense. Thus K λi is the space of weak H (Ω i ) solutionsto Lu = λu that satisfy the given boundary conditions on ∂M ∩ ∂ Ω i but have no conditions imposed on Σ.We then define the space of two-sided Cauchy data on the separating hypersurface Σ by µ ( λ ) = (cid:26) (cid:18) u , ∂u ∂ν , u , − ∂u ∂ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Σ : u i ∈ K λi (cid:27) . (10)We also consider the one-parameter family of boundary conditions on Σ, given by β ( t ) = { ( x, tφ, tx, φ ) : x ∈ H / (Σ) , φ ∈ H − / (Σ) } (11)for t ∈ [0 , µ ( λ ) and β ( t ) comprise smooth families of Lagrangian subspaces in H ⊞ , and form a Fredholm pair for every ( λ, t ) ∈ R × [0 , Maslov index of β ( t ) withrespect to µ ( λ ) for any fixed λ . This is a homotopy invariant quantity that counts the intersections of thesubspaces β ( t ) and µ ( λ ), with sign and multiplicity, as t increases from 0 to 1.We are now ready to state the main result of the paper. Theorem 1.
Let ( M, g ) be a Riemannian manifold with Lipschitz boundary ∂M and a Lipschitz separatinghypersurface Σ . The operators L Di , L Ni and L G defined in (4) , (5) and (6) satisfy Mor( L G ) = Mor( L N ) + Mor( L D ) + Mas( β ( t ); µ (0)) . (12)In practice we can view this as a tool for computing the spectrum of L G by decomposing M into two simplerpieces, Ω and Ω . The relation between these spectral problems is given by the index Mas( β ( t ); µ (0)), so itremains to understand this term. We take several approaches to this. The first is to use spectral propertiesof the Dirichlet-to-Neumann maps for Ω and Ω , denoted Λ and Λ , respectively, to compute the numberof positive and negative intersections the path β ( t ) has with the fixed subspace µ (0). Theorem 2.
If the hypotheses of Theorem 1 are satisfied and / ∈ σ ( L D ) ∪ σ ( L D ) , then Mas( β ( t ); µ (0)) = Mor (Λ + Λ ) − Mor (Λ ) . This is particularly useful when combined with the following well-known result on the Dirichlet-to-Neumann map.
Theorem 3 (Friedlander [13]) . Fix i ∈ { , } . If / ∈ σ ( L Di ) , then Mor( L Ni ) − Mor( L Di ) = Mor (Λ i ) . This result was first proved by Friedlander in [13], where it was used to establish an inequality betweenthe Dirichlet and Neumann eigenvalues of a Euclidean domain. In [18] Mazzeo gave a more geometric proofand considered the possibility of generalizing Friedlander’s approach to non-Euclidean geometries. In Section5.3 we discuss Mazzeo’s proof in a symplectic framework.From Theorems 1, 2 and 3 we obtain the following.
Corollary 1.
Assume the hypotheses of Theorem 2. Then
Mor( L G ) = Mor( L D ) + Mor( L D ) + Mor (Λ + Λ ) , (13) hence Mor( L D ) + Mor( L D ) ≤ Mor( L G ) ≤ Mor( L N ) + Mor( L N ) . (14)It is well known that (14) can be derived by a min-max argument—see Proposition XIII.15.4 of [24], whereit is observed that the operator L increases when one either adds a hypersurface with Dirichlet boundaryconditions or removes a hypersurface with Neumann boundary conditions. Theorem 1 is a quantitativeimprovement of this result, since it provides the additional information that the inequalityMor( L G ) ≥ Mor( L D ) + Mor( L D ) ANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS 5 is strict when Mor(Λ + Λ ) >
0, and the inequalityMor( L G ) ≤ Mor( L N ) + Mor( L N )is strict when Mor (Λ + Λ ) < Mor (Λ ) + Mor (Λ ).The n -sphere yields a simple example in which both parts of (14) are strict. Define L = − ∆ g − c for c ∈ R , where ∆ g is the Laplace–Beltrami operator on S n , and let L G denote the global realizationof L , and L D , L N the Dirichlet and Neumann realizations of L on the upper hemisphere S n + . It followsfrom a reflection argument (cf. Theorem 4) that Mor( L G ) = Mor( L D ) + Mor( L N ), whereas (14) yields2 Mor( L D ) ≤ Mor( L G ) ≤ L N ). Therefore both inequalities are strict when Mor( L D ) < Mor( L N ).This can be achieved by choosing c > − ∆ g on S n + .In general the question of when Mor( L D ) < Mor( L N ) is quite subtle, and is intimately related to inequal-ities between the Dirichlet and Neumann eigenvalues—see [13, 18] and references therein.An immediate application of Corollary 1 is to the study of nodal domains. Suppose φ k is the k th eigenfunc-tion of L , with eigenvalue λ k . The nodal domains of φ k are the connected components of the set { φ k = 0 } .We denote the total number of nodal domains by n ( φ k ), and define the nodal deficiency δ ( φ k ) = k − n ( φ k ) . (15) Corollary 2.
Suppose λ k is a simple eigenvalue of L and is a regular value of φ k . Define L ( ǫ ) = L − ( λ k + ǫ ) and let Λ ± ( ǫ ) denote the corresponding Dirichlet-to-Neumann maps on Ω ± = {± φ k > } . If ǫ > issufficiently small, then δ ( φ k ) = Mor (Λ + ( ǫ ) + Λ − ( ǫ )) . Since the right-hand side is nonnegative, this implies n ( φ k ) ≤ k , which is Courant’s nodal domain theorem[9]. In [3] Berkolaiko, Kuchment and Smilinsky gave a different formula for the nodal deficiency as the Morseindex of a certain energy functional defined on the space of equipartitions of M .In some cases we can compute Mas( β ( t ); µ (0)) by finding conditions that ensure the index vanishes. Theeasiest case is when the Dirichlet-to-Neumann maps for Ω and Ω coincide. In this situation Theorem 2yields Mas( β ( t ); µ (0)) = Mor (Λ + Λ ) − Mor (Λ ) = 0because Λ + Λ = 2Λ has the same Morse index as Λ .The problem of determining when Λ = Λ is in general quite difficult, even for L = − ∆ g . If τ : M → M is an isometry with τ (Σ D ) = Σ D , τ (Σ N ) = τ (Σ N ) and τ | Σ = id, then Λ = Λ . In the real analytic case,Λ = Λ implies Ω and Ω are isometric [16], but the smooth case is still unresolved.However, we are able to prove that β ( t ) ∩ µ (0) = { } for all t , hence Mas( β ( t ); µ (0)) = 0, provided theDirichlet-to-Neumann maps are sufficiently close. To that end, it is convenient to view them as boundedoperators e Λ i : H / (Σ) −→ H − / (Σ) , which are well defined if 0 / ∈ σ ( L Di ). If 0 / ∈ σ ( L Ni ), by an abuse of notation we let e Λ − i : H − / (Σ) −→ H / (Σ)denote the corresponding Neumann-to-Dirichlet map. (If 0 / ∈ σ ( L Di ) ∪ σ ( L Ni ), the operators e Λ i and e Λ − i both exist and are mutually inverse.) Theorem 4.
Assume / ∈ σ ( L N ) ∪ σ ( L D ) . If there exists c such that (cid:13)(cid:13)(cid:13)e Λ − e Λ − cI (cid:13)(cid:13)(cid:13) B ( H / (Σ)) < c, then Mas( β ( t ); µ (0)) = 0 , hence Mor( L G ) = Mor( L N ) + Mor( L D ) . The corollary applies to the cylinder shown in Figure 2 if one prescribes the same boundary conditions(either Dirichlet or Neumann) on both ends of the cylinder. With mixed boundary conditions it is possiblethat the Maslov index is nonzero, as is demonstrated by a simple example in Section 5.2.Using similar methods, we can also describe the spectra of periodic eigenvalue problems and, more gen-erally, problems in which the boundary is divided into two components, ∂M = Γ ∪ Γ , which are identified GRAHAM COX, CHRISTOPER K.R.T. JONES, AND JEREMY L. MARZUOLA Ω Σ Ω ∂M ∂M Figure 2.
An example of the doubling construction in Section 5.2.Γ Γ Γ Γ M Figure 3.
A manifold M with boundary ∂M = Γ ∪ Γ , as in the statement of Theorem 5.by a map τ : Γ → Γ as shown in Figure 3. We let L D denote the Dirichlet realization of L , and L P the“periodic” realization, with domain D ( L P ) = ( u ∈ H ( M ) : Lu ∈ L ( M ) , u | Γ = u | Γ ◦ τ and ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Γ = − ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Γ ◦ τ ) . To state the result we must also define the “periodic Dirichlet-to-Neumann map” Λ τ , assuming 0 / ∈ σ ( L D ).For a function f on Γ we let u denote the unique solution to the boundary value problem Lu = 0 , u | Γ = f, u | Γ = f ◦ τ − . and define Λ τ f = ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Γ + ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Γ ◦ τ. (16)It is shown in Section 5.4 that Λ τ defines an unbounded, selfadjoint operator on L (Γ ), with domain D (Λ τ ) = ( f ∈ L (Γ ) : ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Γ + ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Γ ◦ τ ∈ L (Γ ) ) . Moreover, it is bounded from below and has compact resolvent, and hence has a well-defined Morse index.For the following theorem to hold, it is necessary that the boundary can be subdivided into pieces onwhich the map τ is Lipschitz. (In general τ will not be globally Lipschitz—for the cube with opposing facesidentified it fails to be continuous at the corners.) We let dµ and dµ denote the induced area forms on Γ and Γ , respectively. Theorem 5.
Suppose Γ can be decomposed as Γ ∪ · · · ∪ Γ N , where each Γ i is an open subset of ∂M withLipschitz boundary, and the restrictions τ | Γ i : Γ i → τ (Γ i ) are Lipschitz. If / ∈ σ ( L D ) and τ ∗ dµ = dµ ,then Mor( L P ) = Mor( L D ) + Mor (Λ τ ) . Thus the periodic problem on M is related to the Dirichlet problem, and the difference in Morse indicesis quantified by the periodic Dirichlet-to-Neumann map Λ τ . This is useful because separated boundaryconditions are often easier to work with than periodic boundary conditions and more techniques are availablefor their study. ANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS 7 Proof of the main theorem
In this section we prove Theorem 1. We first describe how the subspaces µ ( λ ) and β ( t ), defined in (10)and (11), contain spectral data for the operators L G , L Di and L Ni . Next we prove that µ ( λ ) and β ( t ) aresmooth families of Lagrangian subspaces in H ⊞ (as defined in (8)) and comprise a Fredholm pair, so theirMaslov index is well defined. Finally, we use the homotopy invariance of the Maslov index to prove thetheorem.Recall that µ ( λ ) encodes the boundary data of weak solutions to the equation Lu = λu , with no boundaryconditions imposed on Σ, whereas β ( t ) defines a one-parameter family of boundary conditions that does notdepend on L . The significance of the endpoints t = 0 , Lemma 1.
Let λ ∈ R . Then dim [ µ ( λ ) ∩ β (0)] = dim ker( L N − λ ) + dim ker( L D − λ )dim [ µ ( λ ) ∩ β (1)] = dim ker( L G − λ ) . The proof relies on a version of the unique continuation principle (Proposition 2.5 of [2]) which says thatthere is a one-to-one correspondence between weak solutions in K λi and their Cauchy data in H / (Σ) ⊕ H − / (Σ); cf. Proposition 2.2 in [10]. Proof.
For the first claim observe that β (0) = { ( x, , , φ ) : x ∈ H / (Σ) , φ ∈ H − / (Σ) } , and so µ ( λ ) ∩ β (0)is nontrivial when there exist functions u i ∈ K λi , not both zero, such that ∂u ∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ = 0 , u | Σ = 0 . Therefore u ∈ D ( L N ) and u ∈ D ( L D ), with ( L N − λ ) u = 0 and ( L D − λ ) u = 0. Since (at least) one of u and u is nonzero, we conclude that λ ∈ σ ( L N ) ∪ σ ( L D ). On the other hand, if λ ∈ σ ( L N ), the correspondingeigenfunction satisfies u ∈ K λ , hence ( u | Σ , , , ∈ µ ( λ ) ∩ β (0) , and similarly when λ ∈ σ ( L D ). This completes the proof of the first equality.For the second equality we observe that β (1) = { ( x, φ, x, φ ) : x ∈ H / (Σ) , φ ∈ H − / (Σ) } , and so µ ( λ ) ∩ β (1) is nontrivial if and only if there exist functions u i ∈ K λi such that u | Σ = u | Σ , ∂u ∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ = − ∂u ∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ . But this is true precisely when there exists a weak solution u ∈ H ( M ) to Lu = λu (with u | Ω i = u i for i ∈ { , } ) such that u | Σ D = 0 , ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ N = 0 , which is equivalent to λ ∈ σ ( L G ). (cid:3) We next show that the set of λ for which µ ( λ ) and β ( t ) intersect nontrivially is bounded below uniformlyin t . Lemma 2.
There exists λ ∞ < such that µ ( λ ) ∩ β ( t ) = { } for all λ ≤ λ ∞ and t ∈ [0 , .Proof. Suppose µ ( λ ) ∩ β ( t ) = { } . By definition, there exist functions u i ∈ K λi such that u | Σ = t u | Σ , ∂u ∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ = − t ∂u ∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ , hence u ∂u ∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ = − u ∂u ∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ . GRAHAM COX, CHRISTOPER K.R.T. JONES, AND JEREMY L. MARZUOLA
Integrating by parts, we obtain Z Ω (cid:2) |∇ u | + ( V − λ ) u (cid:3) = Z Σ u ∂u ∂ν = − Z Σ u ∂u ∂ν = − Z Ω (cid:2) |∇ u | + ( V − λ ) u (cid:3) which implies λ (cid:18)Z Ω u + Z Ω u (cid:19) ≥ Z Ω V u + Z Ω V u and so it suffices to choose λ ∞ < inf x ∈ M V ( x ) . (cid:3) To define the Maslov index of µ with respect to β (and vice versa), we need to prove that µ ( λ ) and β ( t )are continuous curves in the Lagrangian Grassmannian of H ⊞ and comprise a Fredholm pair for each λ and t . We recall that a curve γ : I → Λ( H ⊞ ) is said to be C k if the corresponding curve of orthogonal projections, t P γ ( t ) , is contained in C k ( I, B ( H ⊞ )). Lemma 3.
For ( λ, t ) ∈ R × [0 , the subspaces µ ( λ ) and β ( t ) are Lagrangian, and the curves λ µ ( λ ) and t β ( t ) are smooth.Proof. The space µ ( λ ) of two-sided Cauchy data can be decomposed as µ ( λ ) = µ ( λ ) ⊕ µ ( λ ), where µ and µ are the spaces of Cauchy data for weak solutions to Lu = λu on Ω and Ω , respectively. It was shownin Proposition 3.5 of [10] that λ µ ( λ ) and λ µ ( λ ) are smooth curves in Λ( H ), hence their sum µ ( λ )is a smooth curve in Λ( H ⊞ ).The fact that β ( t ) is Lagrangian follows from a direct computation, and the regularity of t β ( t ) isimmediate from the definition. (cid:3) Lemma 4.
For ( λ, t ) ∈ R × [0 , , µ ( λ ) and β ( t ) comprise a Fredholm pair.Proof. For convenience we abbreviate µ = µ ( λ ) and β = β ( t ). Let P β : H ⊞ → H ⊞ denote the orthogonalprojection onto β , and P ⊥ β = I − P β the complementary projection. By Proposition 2.27 of [14], µ and β comprise a Fredholm pair if and only if the restriction P ⊥ β (cid:12)(cid:12) µ : µ → H ⊞ is a Fredholm operator.By Peetre’s lemma (Lemma 3 of [21]), it suffices to find a compact embedding ι : µ → Y and a positiveconstant C such that k z k H ⊞ ≤ C (cid:0) k P ⊥ β z k H ⊞ + k ιz k Y (cid:1) for all z ∈ µ . Since the operator Tr : K λ ⊕ K λ → µ defined byTr( u , u ) = (cid:18) u , ∂u ∂ν , u , − ∂u ∂ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Σ is boundedly invertible by Lemmas 3.2 and 3.3 of [10], it suffices to prove k ( u , u ) k H ≤ C (cid:0) k P ⊥ β Tr( u , u ) k H ⊞ + k ι Tr( u , u ) k Y (cid:1) for u i ∈ K λi , where we have defined k ( u , u ) k H = k u k H (Ω ) + k u k H (Ω ) . Defining Y = L (Ω ) ⊕ L (Ω )and letting ι : µ → Y denote the composition µ Tr − −−−→ H (Ω ) ⊕ H (Ω ) ֒ −→ L (Ω ) ⊕ L (Ω ) , we need to show that k ( u , u ) k H ≤ C (cid:0) k P ⊥ β Tr( u , u ) k H ⊞ + k ( u , u ) k L (cid:1) (17)for u i ∈ K λi .To prove (17), we first observe that there is a constant C > k u i k H (Ω i ) ≤ C k u i k L (Ω i ) + Z Σ u i ∂u i ∂ν i (18) ANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS 9 for i ∈ { , } and u i ∈ K λi . Using the definition of β = β ( t ) we can write P β Tr( u , u ) = ( x, tφ, tx, φ ) , P ⊥ β Tr( u , u ) = ( − ty, ψ, y, − tψ )for some x, y ∈ H / (Σ) and φ, ψ ∈ H − / (Σ). Therefore Z Σ u ∂u ∂ν + u ∂u ∂ν = ( tφ + ψ )( x − ty ) − ( φ − tψ )( tx + y )= (1 + t ) [ ψ ( x ) − φ ( y )] ≤ t (cid:0) ǫ k x k H / (Σ) + ǫ − k ψ k H − / (Σ) + ǫ − k y k H / (Σ) + ǫ k φ k H − / (Σ) (cid:1) = ǫ k P β Tr( u , u ) k H ⊞ + 12 ǫ k P ⊥ β Tr( u , u ) k H ⊞ for any ǫ >
0, hence (18) implies k ( u , u ) k H ≤ C k ( u , u ) k L + ǫ k P β Tr( u , u ) k H ⊞ + 12 ǫ k P ⊥ β Tr( u , u ) k H ⊞ . Choosing ǫ small enough that ǫ k P β Tr( u , u ) k H ⊞ ≤ k ( u , u ) k H , which is possible by Lemma 3.2 of [10],the desired estimate (17) follows. (cid:3) Remark 1.
The proof of Lemma 3.8 in [10] , in which certain pairs of Lagrangian subspaces are shown tobe Fredholm, can be greatly simplified by an application Peetre’s lemma as above.
The Maslov index counts signed intersections of Lagrangian subspaces and so, in light of Lemma 1, it isnot surprising that the Maslov indices of µ ( λ ) with respect to β (0) and β (1) are related to the Morse indicesof the corresponding boundary value problems. This is a consequence of the fact that µ ( λ ) always passesthrough β ( t ) in the same direction. This is proved by computing the crossing form—a symmetric bilinearform associated to a nontrivial intersection—and showing that it is sign definite (c.f. the proof of Lemma4.2 in [10]). The necessary properties of crossing forms can be found in Appendix B of [10]; see also [14] fora more thorough treatment. Proposition 1.
For | λ ∞ | sufficiently large we have Mas( µ ( λ ); β (0)) = − Mor( L N ) − Mor( L D ) and Mas( µ ( λ ); β (1)) = − Mor( L G ) , where the Maslov index is computed over the interval [ λ ∞ , .Proof. We use a crossing form computation to show that, for fixed t ∈ [0 , µ ( λ ) with β ( t ) is negative definite. This impliesMas( µ ( λ ); β ( t )) = − X λ ∞ ≤ λ< dim ( µ ( λ ) ∩ β ( t ))= − X λ< dim ( µ ( λ ) ∩ β ( t )) , where in the second equality we have used Lemma 2, and the result then follows from Lemma 1.To prove monotonicity, we assume there is a crossing at some λ ∗ ∈ R . Then there exist differentiablepaths of functions λ u i ( λ ) ∈ H (Ω i ) such that u i ( λ ) ∈ K λi for | λ − λ ∗ | ≪
1, hence z ( λ ) = (cid:18) u ( λ ) , ∂u ∂ν ( λ ) , u ( λ ) , − ∂u ∂ν ( λ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Σ defines a differentiable curve in H ⊞ , with z ( λ ) ∈ µ ( λ ) and z ( λ ∗ ) ∈ µ ( λ ∗ ) ∩ β ( t ). Since ω ⊞ = ω ⊕ ( − ω ), thecrossing form is given by ω ⊞ (cid:18) z, dzdλ (cid:19) = ω (cid:18) z , dz dλ (cid:19) − ω (cid:18) z , dz dλ (cid:19) . (19) To compute the first term on the right-hand side of (19), we define the quadratic formΦ( u, v ) = Z Ω [ g ( ∇ u, ∇ v ) + V uv ]for u, v ∈ H (Ω ). Since u ( λ ) ∈ K λ , Green’s first identity (3) impliesΦ( u ( λ ) , v ) = λ h u ( λ ) , v i L (Ω ) + Z Σ v ∂u ∂ν (20)for any v ∈ H (Ω ) with v | Σ D ∩ ∂ Ω = 0. Choosing v = du /dλ , we obtainΦ (cid:18) u , du dλ (cid:19) = λ (cid:28) u , du dλ (cid:29) L (Ω ) + Z Σ du dλ ∂u ∂ν . On the other hand, differentiating (20) with respect to λ and then choosing v = u yieldsΦ (cid:18) du dλ , u (cid:19) = k u k L (Ω ) + λ (cid:28) du dλ , u (cid:29) L (Ω ) + Z Σ u ddλ ∂u ∂ν . Using the symmetry of Φ and recalling the definition of ω , it follows that ω (cid:18) z , dz dλ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ = λ ∗ = Z Σ (cid:20) u ddλ ∂u ∂ν − dudλ ∂u ∂ν (cid:21) = −k u ( λ ∗ ) k L (Ω ) . The second term on the right-hand side of (19) is computed similarly, and we obtain for the crossing form ω ⊞ (cid:18) z, dzdλ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ = λ ∗ = −k u ( λ ∗ ) k L (Ω ) − k u ( λ ∗ ) k L (Ω ) , which is strictly negative. (cid:3) We are now ready to prove Theorem 1.
Proof of Theorem 1.
The unitary group acts transitively on Λ( H ⊞ ) and, by Theorem 2.14 of [14], gives it thestructure of a principal fiber bundle, so there exists a continuous family of unitary operators U ( t ) : H ⊞ → H ⊞ such that β ( t ) = U ( t ) β (0). We define a homotopy [ λ ∞ , × [0 , → F Λ β (0) ( H ⊞ ) by ( λ, t ) U ( t ) − µ ( λ ).The invariance of the Maslov index under unitary transformations impliesMas (cid:0) U ( t ) − µ ( λ ); β (0) (cid:1) = Mas( µ ( λ ); β ( t ))and Mas (cid:0) U ( t ) − µ ( λ ); β (0) (cid:1) = − Mas( β ( t ); µ ( λ ))for any fixed t and λ . The image of the boundary of [ λ ∞ , × [0 ,
1] is null homotopic in F Λ β (0) ( H ⊞ ) andhence has zero Maslov index. This impliesMas( µ ( λ ); β (1)) = Mas( µ ( λ ); β (0)) + Mas( β ( t ); µ ( λ ∞ )) − Mas( β ( t ); µ (0)) . The proof follows immediately from the above formula, Lemma 2 (which implies Mas( β ( t ); µ ( λ ∞ )) = 0) andProposition 1. (cid:3) The Maslov index of β ( t )In this section we prove Theorem 2: if 0 / ∈ σ ( L D ) ∪ σ ( L D ), thenMas( β ( t ); µ (0)) = Mor (Λ + Λ ) − Mor (Λ ) . Instead of directly analyzing the crossings of β ( t ) with µ (0), which may be degenerate, we deform β ( t ) to anondegenerate path for which the Maslov index can be easily computed, and then appeal to the homotopyinvariance of the index.We first define the (unbounded) Dirichlet-to-Neumann map Λ i for i ∈ { , } . This is an unboundedoperator on L (Σ) with domain D (Λ i ) = (cid:26) f ∈ L (Σ) : ∃ u ∈ K i such that u | Σ = f and ∂u∂ν i (cid:12)(cid:12)(cid:12)(cid:12) Σ ∈ L (Σ) (cid:27) , ANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS 11 defined by Λ i f = ∂u∂ν i (cid:12)(cid:12)(cid:12) Σ . Recall from (9) that u ∈ K i means that u | Σ D ∩ ∂ Ω i = 0 , ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ N ∩ ∂ Ω i = 0and u is a weak solution to the equation Lu = 0 in Ω i .We can also view the Dirichlet-to-Neumann maps as bounded operators e Λ , e Λ : H / (Σ) −→ H − / (Σ) . We relate the spectrum of the unbounded operator Λ i to its bounded counterpart e Λ i . Let J : H / (Σ) ֒ → H − / (Σ) denote the compact inclusion. Lemma 5.
Let i ∈ { , } and suppose / ∈ σ ( L Di ) . Then s ∈ σ (Λ i ) if and only if there exists f ∈ H / (Σ) such that ( e Λ i − s J ) f = 0 .Proof. First suppose that s ∈ σ (Λ i ), so there exists u i ∈ K i with ∂u i ∂ν i (cid:12)(cid:12)(cid:12)(cid:12) Σ = s u i | Σ ∈ L (Σ) . Then f := u i | Σ is contained in H / (Σ) and satisfies e Λ i f = s J f , as required.On the other hand, suppose u i ∈ K i satisfies ∂u i ∂ν i (cid:12)(cid:12)(cid:12)(cid:12) Σ = s J ( u i | Σ ) ∈ H − / (Σ) . This implies ∂u i ∂ν i (cid:12)(cid:12)(cid:12) Σ ∈ H / (Σ), hence f := u i | Σ is contained in D (Λ i ) and satisfies Λ i f = sf . (cid:3) For s ≤ β ( s, t ) = n ( x, tφ + s J x, tx, φ ) : x ∈ H / (Σ) , φ ∈ H − / (Σ) o . (21)When s = 0 this is just β ( t ). It is easy to see that β ( s, t ) ⊥ = n ( − ty − s J ∗ ψ, ψ, y, − tψ ) : y ∈ H / (Σ) , ψ ∈ H − / (Σ) o . This modification of β yields a homotopy that relates the Maslov index of β ( t ) to the Morse indices of theDirchlet-to-Neumann maps. This relies crucially on a certain monotonicity with respect to s , which is shownin Proposition 2. Lemma 6. If ( s, t ) ∈ ( −∞ , × [0 , , then β ( s, t ) is a Lagrangian subspace of H ⊞ , and the map ( s, t ) β ( s, t ) is smooth.Proof. Since β (0 ,
0) is Lagrangian, it suffices to find a smooth family of selfadjoint operators A ( s, t ) : β (0 , → β (0 ,
0) such that β ( s, t ) is the graph of A ( s, t ), i.e. β ( s, t ) = { z + J ⊞ A ( s, t ) z : z ∈ β (0 , } , where J ⊞ ( x, φ, y, ψ ) = ( R − φ, − Rx, − R − ψ, Ry )and R : H / (Σ) → H − / (Σ) ∼ = ( H / (Σ)) ∗ is the Riesz duality isomorphism (cf. equation (17) in [10]).It suffices to choose A ( s, t )( x, , , ψ ) = − (cid:0) R − ( tψ + s J x ) , , , tRx (cid:1) , which is selfadjoint because the composition R − ◦ J : H / (Σ) → H / (Σ) is selfadjoint.To prove the selfadjointness of R − ◦ J we use the identity R − = R ∗ to compute (cid:10) R − J f, g (cid:11) H / (Σ) = hJ f, Rg i H − / (Σ) = ( J f ) g = Z Σ f g for any f, g ∈ H / (Σ), where ( J f ) g denotes the action of the functional J f ∈ H − / (Σ) on g ∈ H / (Σ).Since the right-hand side of the above equality is symmetric in f and g , we obtain (cid:10) R − J f, g (cid:11) H / (Σ) = (cid:10) R − J g, f (cid:11) H / (Σ) as required. (cid:3) Lemma 7. If ( s, t ) ∈ ( −∞ , × [0 , , then β ( s, t ) and µ (0) are a Fredholm pair.Proof. As in the proof of Lemma 4, it suffices to have an estimate of the form k ( u , u ) k H ≤ C (cid:0) k P ⊥ β Tr( u , u ) k H ⊞ + k ( u , u ) k L (cid:1) (22)for all u i ∈ K i .We first decompose an arbitrary element Tr( u , u ) ∈ µ (0) into P β Tr( u , u ) = ( x, tφ + s J x, tx, φ ) , P ⊥ β Tr( u , u ) = ( − ty − s J ∗ ψ, ψ, y, − tψ )for some x, y ∈ H / (Σ) and φ, ψ ∈ H − / (Σ), and observe that k P β Tr( u , u ) k H ⊞ ≥ k x k H / (Σ) + k φ k H − / (Σ) k P ⊥ β Tr( u , u ) k H ⊞ ≥ k y k H / (Σ) + k ψ k H − / (Σ) . We then compute Z Σ u ∂u ∂ν + u ∂u ∂ν = ( tφ + s J x + ψ )( x − ty − s J ∗ ψ ) − ( φ − tψ )( tx + y ) ≤ (1 + t ) [ ψ ( x ) − φ ( y )] − s [ J x ( ty + s J ∗ ψ ) + t ( φ + ψ )( J ∗ ψ )]using the fact that s J x ( x ) = s k x k L (Σ) ≤
0. Using the arithmetric–geometric mean inequality on all butthe last term on the right-hand side, we obtain Z Σ u ∂u ∂ν + u ∂u ∂ν ≤ C (cid:0) ǫ k P β Tr( u , u ) k H ⊞ + ǫ − k P ⊥ β Tr( u , u ) k H ⊞ (cid:1) − stψ ( J ∗ ψ )for some constant C = C ( s, t ). Finally, we note that ψ ( J ∗ ψ ) ≤ kJ ∗ kk ψ k H − / (Σ) ≤ kJ ∗ kk P ⊥ β Tr( u , u ) k H ⊞ and choose ǫ sufficiently small, and the estimate (22) follows. (cid:3) We next observe that β ( s, t ) and µ (0) are disjoint for sufficiently negative s ; this is a consequence of auniform lower bound on the Dirichlet-to-Neumann operators. Lemma 8.
There exists s ∞ < such that β ( s, t ) ∩ µ (0) = { } for any s ≤ s ∞ and t ∈ [0 , .Proof. From the definition of µ we have µ (0) = n ( f , e Λ f , f , − e Λ f ) : f , f ∈ H / (Σ) o , and so if β ( s, t ) ∩ µ (0) = ∅ there exists a function f ∈ H / (Σ) that satisfies ( e Λ + t e Λ ) f = s J f . For i ∈ { , } let u i ∈ K i denote the unique solution to Lu i = 0 with u i | Σ = f . Integrating by parts, we have s k f k L (Σ) = Z Σ (cid:18) u ∂u ∂ν + t u ∂u ∂ν (cid:19) = Z Ω (cid:2) |∇ u | + V u (cid:3) + t Z Ω (cid:2) |∇ u | + V u (cid:3) , so there exists a positive constant C , independent of s and t , such that k ( u , u ) k H ≤ C k ( u , u ) k L + s k f k L (Σ) . (23)Now suppose the conclusion of the lemma is false, so there exist sequences of real numbers s j ∈ ( −∞ , t j ∈ [0 , f j ∈ H / (Σ), such that s j → −∞ and (Λ + t j Λ ) f j = s j f j . Let u j and u j denote the unique functions in K and K that satisfy u j | Σ = u j | Σ = f j . Without loss of generality weassume that k ( u j , u j ) k L = k u j k L (Ω ) + k u j k L (Ω ) = 1 . Since s j ≤
0, (23) implies { u j } and { u j } are bounded in H , so there are subsequences with( u j , u j ) → (¯ u , ¯ u ) in L , ( u j , u j ) ⇀ (¯ u , ¯ u ) in H . ANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS 13
It follows that ¯ u ∈ K and ¯ u ∈ K , with ¯ u | Σ = ¯ u | Σ ∈ H / (Σ). Moreover, the compactness of theembedding H / (Σ) ֒ → L (Σ) implies u j | Σ → ¯ u | Σ in L (Σ). However, since s j ≤ s j → −∞ , (23)implies u j | Σ → L (Σ), hence ¯ u | Σ = 0. Since 0 / ∈ σ ( L D ), this implies ¯ u = 0. We similarly find that¯ u = 0, which contradicts the fact that k ¯ u k L (Ω ) + k ¯ u k L (Ω ) = 1and thus completes the proof. (cid:3) Remark 2.
If we assume that the metric tensor g is Lipschitz, instead of just L ∞ , the above compactnessargument is not needed. With this additional regularity hypothesis, Theorem 4.25 of [19] gives k ( u , u ) k L ≤ C k f k L (Σ) which, together with (23) , immediately establishes Lemma 8. We have thus shown that β ( s, t ) is a smooth curve in the Fredholm–Lagrangian Grassmannian F Λ µ (0) ( H ⊞ ),so it has a well-defined Maslov index with respect to either s or t . Proposition 2. If t ∈ [0 , is fixed, then Mas( β ( s, t ); µ (0)) = Mor (Λ + t Λ ) .Proof. As in the proof of Lemma 8, we have that β ( s, t ) ∩ µ (0) = ∅ if and only if there is a function f ∈ H / (Σ) that satisfies ( e Λ + t e Λ ) f = s J f . This implies dim[ β ( s, t ) ∩ µ (0)] = dim ker( e Λ + t e Λ − s J ) =dim ker(Λ + t Λ − s ), where the last equality is a consequence of Lemma 5.We claim that the path s β ( s, t ) is positive definite. Assuming the claim, it follows from Lemma 8that Mas( β ( s, t ); µ (0)) = X s ≤ dim[ β ( s, t ) ∩ µ (0)] = Mor (Λ + t Λ ) . To prove the claim, suppose that s ∗ is a crossing time, so there is a path z ( s ) = ( x, t φ + sx, t x, φ ) in β ( s ) with z ( s ∗ ) ∈ β ( s ∗ , t ) ∩ µ (0). We compute ω ⊞ (cid:18) z, dzds (cid:19) = ω ⊞ (( x, t φ + sx, t x, φ ) , (0 , x, , ω (( x, t φ ) , (0 , x )) − ω (( t x, φ ) , (0 , k x k L (Σ) , which is positive unless x = 0. But if x = 0, then t φ + s ∗ x = e Λ x = 0, which is not possible because z ( s ∗ ) = 0, so the claim is proved. (cid:3) Proof of Theorem 2.
Since the boundary of β : [ s ∞ , × [0 , → F µ (0) Λ( H ⊞ ) is null-homotopic, its Maslovindex vanishes, hence Mas( β ( s, µ (0)) + Mas( β (0 , t ); µ (0)) = Mas( β ( s, µ (0)) . Since β (0 , t ) = β ( t ), Proposition 2 impliesMas( β ( t ); µ (0)) = Mor (Λ + Λ ) − Mor (Λ ) . and the proof is complete. (cid:3) Lemma 9.
Suppose / ∈ σ ( L D ) ∪ σ ( L D ) , so Λ and Λ are well defined. Then Mor(Λ + Λ ) ≤ Mor(Λ ) + Mor(Λ ) . and Mor (Λ + Λ ) ≤ Mor (Λ ) + Mor (Λ ) . In fact the proof yields the stronger result that Mor(Λ + Λ ) is bounded above by the dimension of thesum of the negative subspaces for Λ and Λ , which is bounded above by the sum of the dimensions. Proof.
Since Λ is selfadjoint with compact resolvent, there is a spectral decomposition L (Σ) = E − ⊕ E ⊕ E +1 , and similarly for Λ . Let p = dim( E − ) and q = dim( E − ). Then V := E − + E − has dimension r ≤ p + q ,and Λ + Λ is nonnegative on V ⊥ . Thereforesup dim( U )= r inf (cid:26) h (Λ + Λ ) f, f ik f k L (Σ) : f ∈ U ⊥ (cid:27) ≥ inf (cid:26) h (Λ + Λ ) f, f ik f k L (Σ) : f ∈ V ⊥ (cid:27) ≥ λ r +1 (Λ + Λ ) ≥
0, so Mor(Λ + Λ ) ≤ r and the proof is complete.Replacing E − i by E − i ⊕ E i for i ∈ { , } yields λ r +1 (Λ + Λ ) >
0, and the second inequality follows. (cid:3) Applications
We now present several applications of the results proved above. The results in Sections 5.3 and 5.4 arenot immediate consequences of Theorems 1 and 2 but follow from similar constructions, so we only sketchthe proofs.5.1.
The nodal deficiency.
Our first application is Corollary 2, which gives an explicit formula for thenodal deficiency of an eigenfunction φ k corresponding to the simple eigenvalue λ k . Proof of Corollary 2.
Define Ω + = { φ k > } , Ω − = { φ k < } , and let n ± ( φ k ) denote the number of connected components of Ω ± , so the total number of nodal domains is n ( φ k ) = n + ( φ k ) + n − ( φ k ).For ǫ > L ( ǫ ) = − ∆ − ( λ k + ǫ ). On each nodal domain the first Dirichlet eigenvalue of L is λ k , sofor small enough ǫ > L D ± ( ǫ ) are invertible, withMor( L D ± ( ǫ )) = n ± ( φ k ) . Moreover, the global realization L G ( ǫ ) is invertible as long as λ k + ǫ < λ k +1 , and soMor( L G ( ǫ )) = k. Now let Λ ± ( ǫ ) denote the Dirichlet-to-Neumann maps for L ( ǫ ) on Ω ± . It follows immediately fromCorollary 1 that the nodal deficiency δ ( φ k ) = k − n ( φ k ) is given by δ ( φ k ) = Mor (Λ + ( ǫ ) + Λ − ( ǫ )) , and so the proof is complete. (cid:3) Almost doubled manifolds.
We now consider the setting of Theorem 4, in which (cid:13)(cid:13)(cid:13)e Λ − e Λ − cI (cid:13)(cid:13)(cid:13) B ( H / (Σ)) < c (24)for some constant c . Proof of Theorem 4.
As in the proof of Proposition 2, we have that t ∗ is a crossing time, i.e. β ( t ∗ ) ∩ µ (0) = { } , if and only if ker( I + t ∗ e Λ − e Λ ) is nontrivial. We compute I + t e Λ − e Λ = I + ct I + t e Λ − e Λ − ct I = (1 + ct ) (cid:20) I + t ct (cid:16)e Λ − e Λ − cI (cid:17)(cid:21) , hence I + t ∗ e Λ − e Λ is invertible if t ∗ ct ∗ (cid:13)(cid:13)(cid:13)e Λ − e Λ − cI (cid:13)(cid:13)(cid:13) H / (Σ) < t t / (1 + ct ) is increasing, it suffices to verify the above condition at t = 1. This isjust the inequality (24), so the result follows. (cid:3) ANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS 15
A simple case is when τ : M → M is an involution such that τ | Σ = id, τ (Ω ) = Ω , and L ( u ◦ τ ) = ( Lu ) ◦ τ for all u ∈ H ( M ). If ∂M is nonempty, it is necessary to assume that τ (Σ D ) = Σ D and τ (Σ N ) = Σ N . Forinstance, if the cylinder shown in Figure 2 is given by [0 , π ] × S , with the involution τ ( x, θ ) = (2 π − x, θ ), werequire either ∂M = Σ D or ∂M = Σ N , so both { } × S and { π } × S have the same boundary conditions.It follows immediately that e Λ = e Λ , so Theorem 4 implies Mas( β ( t ); µ (0)) = 0.If the involution τ does not preserve the boundary conditions, it may not be the case that e Λ = e Λ ,and Mas( β ( t ); µ (0)) may be nonzero. This can be seen from an elementary computation for the operator L = − ( d/dx ) − C on [0 , ℓ ], with C a positive constant. We let Ω = (0 , ℓ/
2) and Ω = ( ℓ/ , ℓ ), and computeMas( β ( t ); µ (0)) assuming Dirichlet boundary conditions at x = 0 and Neumann conditions at x = ℓ .A basis for the weak solution space of the equation Lu = 0 issin( √ Cx ) , cos( √ C ( x − ℓ ))and so the space of two-sided Cauchy data at ℓ/ µ (0) = n(cid:16) a sin( √ Cℓ/ , a √ C cos( √ Cℓ/ , b cos( √ Cℓ/ , b √ C sin( √ Cℓ/ (cid:17) : a, b ∈ R o . A crossing occurs at time t ∗ precisely when a = b and t ∗ = cot( √ Cℓ/ , π nπ ≤ √ Cℓ ≤ π nπ (25)for some integer n ≥
0. The crossing form is given by2 a √ C sin ( √ Cℓ/ > , and so the Maslov index is either 1 or 0, depending on whether or not (25) is satisfied.5.3. The Dirichlet-to-Neumann map.
We next use the Maslov index to prove Theorem 3. A geometricproof of this result was given by Mazzeo in [18]; here we observe that Mazzeo’s proof can be formulated interms of the Maslov index.The na¨ıve idea is to define a one-parameter family of Lagrangian subspaces that moves between { } ⊕ H − / (Σ) and H / (Σ) ⊕{ } , which correspond to Dirichlet and Neumann boundary conditions, respectively.By a homotopy argument the Maslov index of this path equals the difference of the Dirichlet and NeumannMorse indices of L . On the other hand, a direct computation shows that the Maslov index equals the Morseindex of the Dirichlet-to-Neumann map Λ. This depends on a monotonicity property for the eigenvalues ofΛ, which has a natural interpretation via the Maslov index.However, this approach suffers from the fact that the path of Lagrangian subspaces that interpolatesbetween Dirichlet and Neumann boundary conditions fails to be continuous at the Dirichlet endpoint; seeRemark 3. To overcome this obstacle, we interpolate between Neumann and “almost Dirichlet” boundaryconditions, then use asymptotic results for the Robin boundary value problem to relate the Dirichlet andalmost Dirichlet spectra.Since i ∈ { , } is fixed, we restrict our attention to a single domain, which we call Ω, with Lipschitzboundary Σ. We first define the subspace β ( t ) = { ( x, φ ) ∈ H : (cos t ) J x + (sin t ) φ = 0 } (26)in H = H / (Σ) ⊕ H − / (Σ), where J : H / (Σ) → H − / (Σ) denotes the compact inclusion. We let K λ ⊂ H (Ω) denote the space of weak solutions to Lu = λu and define the space of Cauchy data µ ( λ ) = (cid:26) (cid:18) u, ∂u∂ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Σ : u ∈ K λ (cid:27) . Note that µ (0) is the graph of e Λ : H / (Σ) → H − / (Σ). Lemma 10.
There exists λ ∞ < such that β ( t ) ∩ µ ( λ ) = { } for every t ∈ (0 , π/ and λ ≤ λ ∞ .Proof. Suppose β ( t ) ∩ µ ( λ ) = { } . By definition there exists a function u ∈ H (Ω) such that Lu = λu weakly and (cos t ) u | Σ + (sin t ) ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Σ = 0 , t = ǫ λt = π/ tλ = λ ∞ Figure 4.
An illustration of the homotopy in the proof of Theorem 3. A crossing at t = 0or t = π/ λ is an eigenvalue for the Dirichlet or Neumann problem, respectively.A crossing at λ = 0 occurs when − cot t is an eigenvalue of the Dirichlet-to-Neumann map.The assumption 0 / ∈ σ ( L D ) precludes the existence of a crossing at the origin.hence Z Σ u ∂u∂ν = − (cot t ) Z Σ u ≤ t ≥ t ∈ (0 , π/ λ Z Ω u = − Z Σ u ∂u∂ν + Z Ω (cid:2) |∇ u | + V ( x ) u (cid:3) ≥ Z Ω (cid:2) |∇ u | + V ( x ) u (cid:3) and the result follows with any λ ∞ < inf V ( x ). (cid:3) Proposition 3.
There exists ǫ > such that the subspaces µ ( λ ) and β ( t ) are smooth, Lagrangian, and forma Fredholm pair for ( λ, t ) ∈ [ λ ∞ , × [ ǫ, π/ . Moreover, Mas( µ ( λ ); β ( ǫ )) = − Mor( L D ) (27)Mas( µ ( λ ); β ( π/ − Mor( L N ) (28)Mas( β ( t ); µ ( λ ∞ )) = 0 (29)Mas( β ( t ); µ (0)) = Mor (Λ) . (30)Referring to the square in Figure 4, the paths in (27)–(30) correspond to the bottom, top, left and rightsides, respectively. Proof.
That µ ( λ ) is a smooth curve of Lagrangian subspaces was established in Proposition 3.5 of [10]. For t = 0 we can express β ( t ) as the graph of the family of selfadjoint operators A ( t ) = (cot t ) R − J on theLagrangian subspace ρ = H / (Σ), where R : H / (Σ) → H − / (Σ) is the Riesz duality operator. That is, β ( t ) = G ρ ( A ( t )) := { x + JA ( t ) x : x ∈ ρ } , with J ( x, φ ) := ( R − φ, − Rx ). It follows (cf. the proof of Lemma 6) that β ( t ) is a smooth family ofLagrangian subspaces for t = 0.We next show that µ ( λ ) and β ( t ) comprise a Fredholm pair, first observing that the result is alreadyknown for t = π/ P denote the orthogonal projection onto β ( π/
2) = H / (Σ) and define A ( t ) = I − (cot t ) J P : H → H . Since A ( t ) β ( π/
2) = β ( t ) and A ( t ) is a compactperturbation of the identity, the result follows from Lemma 3.2 in [11].The monotonicity of µ with respect to λ , as was shown in Lemma 4.2 of [10], immediately yields (28) andthe equality Mas( µ ( λ ); β ( ǫ )) = − Mor( L Rǫ ), where L Rǫ is the realization of L with β ( ǫ ) boundary conditions, ∂u∂ν = − (cot ǫ ) u. ANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS 17
We now apply Proposition 3 of [1], which gives that the ordered eigenvalues of L Rǫ converge to the orderedeigenvalues of L D as ǫ →
0. Since 0 / ∈ σ ( L D ) this implies Mor( L Rǫ ) = Mor( L D ) for sufficiently small ǫ , and(27) follows.Equality (29) is an immediate consequence of Lemma 10, so only (30) remains.Suppose t ∗ ∈ [ ǫ, π/
2] is a crossing time, so β ( t ∗ ) ∩ µ (0) = { } . Then there exists a path z ( t ) = ( x ( t ) , φ ( t ))in H such that z ( t ) ∈ β ( t ) for | t − t ∗ | ≪ z ( t ∗ ) ∈ µ (0). Since t ∗ = 0 we have φ ( t ) = − (cot t ) J x ( t ). Itfollows that φ ′ ( t ) = (csc t ) J x ( t ) − (cot t ) J x ′ ( t )and so the crossing form Q ( z ( t ∗ )) = ω ( z, z ′ ) | t = t ∗ = (csc t ∗ ) k x ( t ∗ ) k L (Σ) is strictly positive. Since β ( t ) ∩ µ (0) = { } if and only if − (cot t ) is an eigenvalue of Λ, we conclude thatMas( β ( t ) , µ (0)) = X t ∈ ( ǫ,π/ dim [ β ( t ) ∩ µ (0)] = σ (Λ) ∩ ( − cot ǫ,
0] = Mor (Λ)for ǫ > sufficiently small. (cid:3) Having established (27)–(30), Theorem 3 follows from a homotopy argument as in the proof of Theorem1. We conclude the section by justifying the decision to restrict the path β ( t ) to the interval [ ǫ, π/ Remark 3.
The path β ( t ) defined in (26) is discontinuous at t = 0 . By definition this means the correspond-ing family of orthogonal projections P ( t ) is discontinuous. It is easy to see that S ( t )( x, φ ) := x − (cot t ) J x defines a projection of H onto β ( t ) for t = 0 . Using Lemma 12.8 of [5] we compute the orthogonal projection P ( t ) = SS ∗ [ SS ∗ + ( I − S ∗ )( I − S )] − = (cid:18) I H / (Σ) − (cot t ) J ∗ − (cot t ) J (cot t ) J J ∗ (cid:19) (cid:2) I H / (Σ) + (cot t ) J ∗ J (cid:3) − (cid:2) I H − / (Σ) + (cot t ) J J ∗ (cid:3) − ! . Now consider the component in the upper left-hand corner, P ( t ) := (cid:2) I H / (Σ) + (cot t ) J ∗ J (cid:3) − : H / (Σ) −→ H / (Σ) for t = 0 . Since P (0) = 0 , it suffices to prove that k P ( t ) k . The operator J ∗ J : H / (Σ) → H / (Σ) is compact and injective (because J is), and hence has a sequence of eigenvalues tending to zero. Thus thereexists { f n } in H / (Σ) such that kJ ∗ J f n k H / (Σ) ≤ n − k f n k H / (Σ) . Letting g n = f n + (cot t ) J ∗ J f n , wehave P ( t ) g n = f n and k g n k H / (Σ) ≤ (cid:2) t ) /n (cid:3) k f n k H / (Σ) , therefore k P ( t ) g n k H / (Σ) = k f n k H / (Σ) ≥ k g n k H / (Σ) t ) /n . Letting n → ∞ implies k P ( t ) k ≥ for each t = 0 , hence P ( t ) . Periodic boundary conditions.
We finally treat the case of a single domain M , with boundary ∂M = Γ ∪ Γ glued to itself via a map τ : Γ → Γ , as described in Theorem 5 and shown in Figure 3. Themotivating example is the n -torus T n , which can be viewed as a cube in R n with opposite faces identified.We can similarly consider any compact, orientable surface of genus g , which is just a 2 g -gon with oppositefaces identified.We first establish some analytic properties of the periodic Dirichlet-to-Neumann map Λ τ defined in (16).Throughout this section we assume the hypotheses of Theorem 5. Proposition 4.
The periodic Dirichlet-to-Neumann map Λ τ on L (Γ ) is bounded below and selfadjointwith compact resolvent.Proof. Let f ∈ H / (Γ ). By Theorems 3.23 and 3.40 of [19], f | Γ i ∈ H / (Γ i ) and f | Γ i ◦ τ − ∈ H / (Γ i )for each i . Therefore the function given by f on Γ and f ◦ τ − on Γ is contained in H / ( ∂M ), so theboundary value problem Lu = 0 , u | Γ = f, u | Γ = f ◦ τ −
18 GRAHAM COX, CHRISTOPER K.R.T. JONES, AND JEREMY L. MARZUOLA has a unique solution u ∈ H ( M ). Moreover, k u k H ( M ) ≤ C k f k H / ( ∂M ) for some C > f , and so Q ( f ) = Z M (cid:2) |∇ u | + V u (cid:3) (31)defines a bounded quadratic form on H / ( ∂M ). Since the trace map H ( M ) → H / ( ∂M ) is bounded and V ∈ L ∞ ( M ), there are positive constants c and c such that Q ( f ) ≥ c k f k H / (Γ ) − c k u k L ( M ) . By a standard compactness argument, there exists for any ǫ > c > k u k L ( M ) ≤ ǫ k u k H ( M ) + c k u | ∂M k L (Γ ) for any u ∈ H ( M ). It follows that Q ( f ) ≥ c ′ k f k H / (Γ ) − c ′ k f k L (Γ ) for every f ∈ H / (Γ ), so Q is bounded over H / (Γ ) and coercive over L (Γ ).With B denoting the bilinear form corresponding to Q , there exists a selfadjoint operator T , with domain D ( T ) ⊂ H / (Γ ), such that B ( f, g ) = h T f, g i L (Γ ) for all f ∈ D ( T ) and g ∈ H / (Γ ). Integrating byparts and using the hypothesis τ ∗ dµ = dµ , we find that B ( f, g ) = Z Γ g ∂u∂ν + Z Γ ( g ◦ τ − ) ∂u∂ν = Z Γ g (cid:18) ∂u∂ν + ∂u∂ν ◦ τ (cid:19) for any g ∈ H / (Γ ), hence T f = ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Γ + ∂u∂ν (cid:12)(cid:12)(cid:12)(cid:12) Γ ◦ τ. (32)Therefore T = Λ τ , as defined in (16), and the proof is complete. (cid:3) We now sketch the proof of Theorem 5, which closely follows the proofs of Theorems 1, 2 and 3.
Proof of Theorem 5.
We first define the symplectic Hilbert spaces H = H / (Γ ) ⊕ H − / (Γ ), with theusual symplectic form ω , then let H p = H ⊕ H , with the form ω p = ω ⊕ ( − ω ).By K λ ⊂ H ( M ) we denote the space of weak solutions to Lu = λu (with no boundary conditionsimposed). We define the space of Cauchy data µ ( λ ) = ( (cid:18) u, ∂u∂ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Γ , (cid:18) u, − ∂u∂ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Γ ◦ τ − ! : u ∈ K λ ) and the path of boundary conditions β ( t ) = { ( x, tφ, tx, φ ) : ( x, φ ) ∈ H } as in (10) and (11).By a (now familiar) homotopy argument we find thatMor( L P ) = Mor( L DN ) + Mas( β ( t ); µ (0)) , (33)where L DN denotes the “mixed realization” of L with Neumann conditions on Γ and Dirichlet on Γ . Next,following the proof of Theorem 2, we define the two-parameter family of Lagrangian subspaces β ( s, t ) = { ( x, tφ + s J x, tx, φ ) : ( x, φ ) ∈ H } . and consequently obtain Mas( β ( t ); µ (0)) = Mor (Λ τ ) − Mor (Λ ) , (34)where Λ is the “partial Dirichlet-to-Neumann map” on Γ , obtain by mapping a function f on Γ to ∂u∂ν (cid:12)(cid:12) Γ ,where u uniquely solves the boundary value problem Lu = 0 , u | Γ = f, u | Γ = 0 . ANIFOLD DECOMPOSITIONS AND INDICES OF SCHR ¨ODINGER OPERATORS 19
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