On the spectrum of non-selfadjoint Schrödinger operators with compact resolvent
aa r X i v : . [ m a t h - ph ] O c t On the spectrum of non-selfadjointSchr¨odinger operators with compact resolvent
Y. Almog and B. Helffer
Abstract
We determine the Schatten class for the compact resolvent ofDirichlet realizations, in unbounded domains, of a class of non-selfadjointdifferential operators. This class consists of operators that can be ob-tained via analytic dilation from a Schr¨odinger operator with magneticfield and a complex electric potential. As an application, we prove, ina variety of examples motivated by Physics, that the system of gener-alized eigenfunctions associated with the operator is complete, or atleast the existence of an infinite discrete spectrum.
The theory of non-selfadjoint differential operators is at a much less devel-oped state than that of selfadjoint theory. The lack of variational methodsmakes it difficult, in many interesting cases, to determine whether a non-selfadjoint operator P possesses a complete systems of generalized eigenvec-tors (by which we mean that the vector space they span is dense), or even ifthe spectrum is non-empty. In addition, the definition of a closed extensionof the differential operator, is not always a straightforward matter. (Thereare, of course, other questions of interest, such as the effectiveness of theFourier expansion [14, 7] which we do not address here.) There is, however,significant interest in these questions (cf. for instance [8, 15, 19, 1, 22] toname just a few references).In recent contributions we have considered, together with X.B. Pan, sim-ilar questions for a well defined closed extension A (or A D ) of the differentialoperator A = − (cid:16) ∇ − i x y ) + icy , (1.1)1here ˆı y is a unit vector in the y direction. We studied the spectrum of thisextension both in the entire ( x, y ) plane [3], where we show that σ ( A ) = ∅ ,and for the Dirichlet realization A D of A in the half plane R = { ( x, y ) ∈ R | y > } . In the latter case we show that the spectrum is not empty in the limit c → c → ∞ [4], where our techniques involve analytic dilation.The existence of a non-empty spectrum for general values of c remains anopen question.In another contribution [2] we show that the normal state for a super-conductor in the presence of an electric current, and the magnetic field itinduces becomes locally unstable, under some additional conditions whichare omitted here, whenever ( A − λ ) − becomes unbounded, where λ is afunction of the electric current. The analysis in [2] rests on some assump-tions on the normal electric potential, that are not the most general ones.Thus, for instance, it assumes a non vanishing electric potential gradient. Ifthe boundary conditions are such that this assumption is violated, one hasto analyze the spectrum of different linear differential operators instead of(1.1), that are still of some physical interest.Let then Ω ⊆ R d be (possibly) unbounded, ∂ Ω ∈ C ,α for some α > A = ( A , . . . , A d ) ∈ C ( ¯Ω , R d ). Let B ( x ) denote the anti-symmetric matrixassociated with curl A : B jk = ∂ k A j − ∂ j A k . We attempt to prove existence of a non-empty spectrum, and when possible,to prove completeness of the system of (generalized) eigenvectors of the op-erator − ∆ A + V in L (Ω , C ), where V may be complex valued. Since in someof the examples below we use analytic dilation, we consider a more generalclass of operators. (Note that analytical dilation can be applied, in general,only in domains that are invariant under real dilation). In particular, weconsider here the operator A α = − e − i πm ( k +1) ∂ ∂x − e i π k +1) (cid:16) ∂∂y − i x m m (cid:17) + e i π k +1) y k , (1.2)obtained via analytic dilation of the operator: A = − ∂ ∂x − (cid:16) ∂∂y − i x m m (cid:17) + iy k . (1.3)2e begin by defining the class of operators considered in this work. Letthen ( α , . . . , α d ) ∈ ( − π , π d and K def = min ≤ k ≤ d cos(2 α k ) > . (1.4)Suppose that V = V + V where V ∈ C (Ω , C ) satisfies ℜ V ≥ − λ ∗ , (1.5)for some λ ∗ ∈ R .Suppose further that there exists a constant C such that |∇ V | + max ( k,l ) ∈{ ,...,d } |∇ B kℓ | ≤ C m
B,V in Ω , (1.6)where m B,V = p | V | + | B | + 1 , (1.7)Assume in addition that V ∈ L ∞ loc (Ω , C ) is such that for every ǫ >
0, thereexists C ǫ for which | V | ≤ C ǫ + ǫ m B,V in Ω . (1.8)Finally, to assure compactness of the resolvent, we assume that m B,V −−−−−→ | x |→ + ∞ + ∞ in Ω . (1.9)Consider then the operator initially defined on C ∞ (Ω) by: P = P + V , (1.10)where P = − d X k =1 e iα k ∂ A k (1.11)and ∂ A k := ∂∂x k − iA k .
3e consider here the Dirichlet realization P D of P in Ω, i.e., some closedextension of P which should be defined properly on a subspace of functionssatisfying a Dirichlet condition on ∂ Ω, or on the entire space R d . Our def-inition, based on some generalization of the Friedrichs extension in a nonnecessarily coercive case, will coincide with the standard notion when Ω isbounded, Ω = R d or when the operator is selfadjoint and semi-bounded.Before stating our main result, we recall that, if H is a Hilbert space and p > C p ( H ) denotes the set of compact operators T suchthat kT k p = (cid:16) ∞ X n =1 µ n ( T ) p (cid:17) /p < ∞ , where µ n ( T ) are the eigenvalues of ( T ∗ T ) / repeated according to theirmultiplicity [10, 11]. For 1 ≤ p , C p is a Banach space with k · k p as its norm.For 0 < p < C p is still well defined, but is not a Banach space and k ∆ k p isnot a norm.Our main result follows Theorem 1.1.
Under the above assumptions we have, for the Dirichlet re-alizations of P and − ∆ A + | V | , that for every λ ∈ ρ ( − ∆ A + | V | ) ∩ ρ ( P )( − ∆ A + | V | − λ ) − ∈ C p ( L (Ω , C )) ⇒ ( P − λ ) − ∈ C p ( L (Ω , C )) . (1.12)The optimality of this result is also of interest. In this direction we provethe following Theorem 1.2.
With the notation and assumptions of the previous theorem, • When Ω is either the entire space or the half-space in R d we have ( P − λ ) − ∈ C p ( L (Ω , C )) ⇒ ( − ∆ A + | V |− λ ) − ∈ C p ( L (Ω , C )) (1.13) for every λ ∈ ρ ( − ∆ A + | V | ) ∩ ρ ( P ) . • For a general domain Ω and for any α ∈ [ − π, π ) , the Dirichlet realiza-tions of − e iα ∆ A + V and − ∆ A + | V | satisfy ( − e iα ∆ A + V − λ ) − ∈ C p ( L (Ω , C )) ⇒ ( − ∆ A + | V |− λ ) − ∈ C p ( L (Ω , C )) , (1.14) for every λ ∈ ρ ( − ∆ A + | V | ) ∩ ρ ( − e iα ∆ A + V ) .
4n this last case, we consequently have an equivalence, in the sense ofSchatten classes of the resolvents, between the Dirichlet realizations of P and − ∆ A + | V | .The complementary question is, naturally, to which Schatten class does theresolvent of − ∆ A + | V | belong? The following theorem provides a satisfactoryanswer. Theorem 1.3.
Suppose that for some p > Z Ω × R d ( | ξ | + m B,V ) − p dxdξ < ∞ . (1.15) Then, ( − ∆ A + | V | + 1) − ∈ C p ( L (Ω , C )) . Once the Schatten class for a compact operator has been obtained, onecan use the following fundamental result in operator theory to prove com-pleteness of its system of eigenvectors (cf. for instance Theorem X.3.1 in [11],or Corollary XI.9.31 in [10]).
Theorem 1.4.
Let H denote a Hilbert space, and A ∈ C p ( H ) be a compactoperator for some p > . Assume that its numerical range W A = {hA ϕ, ϕ i | ϕ ∈ H , k ϕ k = 1 } lies inside a closed angle with vertex at zero and opening π/p .Let Span ( A ) denote the closure of the vector space generated by the general-ized eigenfunctions. Then, Span ( A ) = H is complete. We apply this statement to the resolvent of P . Note that W ( P− λ ) − = {h ( P − λ ) − ϕ, ϕ i | ϕ ∈ H , k ϕ k = 1 } = {h ψ, ( P − λ ) ψ i | ψ ∈ D ( P ) , k ( P − λ ) ψ k = 1 } . Hence, if for some λ ∈ ρ ( P ), W P− λ lies in a closed angle with vertex at zeroand opening π/p , then so does W ( P− λ ) − and it would follow immediatelythat P has a complete system of eigenfunctions.The rest of the contribution is arranged as follows.In the next section, we define the Dirichlet realization of P and prove thatits resolvent is compact.In Section 3 we prove Theorems 1.1-1.3.Finally, in the last section, we use these results, together with Theorem 1.4to prove completeness of the system of (generalized)-eigenvectors, or at leastexistence of a non-empty spectrum, for a few particular cases of (1.10) mo-tivated by superconductivity problems.5 Definition of the Dirichlet realization As P is defined by (1.10) for smooth functions only, we seek a closed extension P D corresponding to its Dirichlet realization. For unbounded domains thedefinition of this extension deserves special attention. We thus consider thesesquilinear form( u, v ) a ( u, v ) := d X k =1 exp {− iα k }h ∂ A k u , ∂ A k v i L (Ω) + Z Ω V ( x ) u ( x )¯ v ( x ) dx (2.1)initially defined on C ∞ (Ω) × C ∞ (Ω).As is the common practice in such cases, it is useful to consider instead, forsome, sufficiently large, γ > u, v ) a γ ( u, v ) := P k exp {− iα k }h ∂ A k u , ∂ A k v i L (Ω) + R Ω V ( x ) u ( x )¯ v ( x ) dx + γ R u ¯ vdx , (2.2)to assure some coercivity.The Friedrichs extension of (2.2) is a continuous sesquilinear form on V × V , where V = { u ∈ H ,A (Ω) | | V | / u ∈ L (Ω , C ) } , (2.3)and H ,A (Ω) denotes the closure of C ∞ (Ω) in the magnetic Sobolev space H A (Ω) = { u ∈ L (Ω) , ∇ A u ∈ L (Ω) } . (2.4)When Ω is bounded, α k = 0 for all 1 ≤ k ≤ d , and V is real, the Dirichlet re-alization of P can be easily obtained by applying the Lax-Milgram Theorem(and the Friedrichs extension construction). When V is complex valued and ℑ V ≥
0, the same method prevails possible if we employ a minor general-ization of the Lax Milgram Theorem, where hermitianity for the sesquilinearform is no longer assumed [4]. When ℑ V has no definite sign and is notbounded by ℜ V or the magnetic field, a more elaborate generalization of theLax-Milgram Theorem is needed. In particular, it is a necessary to replacethe standard requirement for V -ellipticity (or coercivity) of (2.2) by a weakerone. This is the object of the next subsection.6 .2 A generalized Lax-Milgram Theorem Let V denote a Hilbert space. Consider a continuous sesquilinear form a defined on V × V : ( u, v ) a ( u, v ) . Recall that for a sesquilinear form continuity means that for some
C > | a ( u, v ) | ≤ C k u k V k v k V , ∀ u, v ∈ V . (2.5)We denote the associated linear map by A ∈ L ( V ), i.e., a ( u, v ) = hA u , v i V . (2.6) Theorem 2.1.
Let a be a continuous sesquilinear form on V × V . If a satisfies, for some Φ , Φ ∈ L ( V ) | a ( u, u ) | + | a ( u, Φ ( u )) | ≥ α k u k V , ∀ u ∈ V . (2.7) | a ( u, u ) | + | a (Φ ( u ) , u ) | ≥ α k u k V , ∀ u ∈ V . (2.8) then A , as defined in (2.6) , is a continous isomorphism from V onto V .Moreover A − is continuous.Proof. We split the proof into two different steps.
Step 1 : A is injective, and has a closed range.Choose u ∈ V , such that A u = 0. This implies hA u, u i = 0 and hA u, Φ ( u ) i = 0 . (2.9)It, however, follows from (2.7) that |hA u , u i V | + |hA u , Φ ( u ) i V | ≥ α k u k V , ∀ u ∈ V . Hence, (1 + k Φ k ) kA u k V · k u k V ≥ α k u k V , ∀ u ∈ V , and consequently, for some ˜ α > kA u k V ≥ ˜ α k u k V , ∀ u ∈ V , (2.10)7rom which injectivity readily follows. Closedness of the range easily followsfrom (2.10) and the continuity of A . Step 2 : A ( V ) is dense in V , and A − is continuous.Consider u ∈ V such that hA v , u i V = 0 , ∀ v ∈ V . In particular, we canchoose v = u and v = Φ ( u ) to obtain a ( u, u ) = 0 and a (Φ ( u ) , u ) = 0.Hence, by (2.8) we must have u = 0. Thus, A is a bijection, A − : V → V exists and is continous by (2.10).We now consider two Hilbert spaces V and H such that V ⊂ H , and thatfor some
C > u ∈ V , we have k u k H ≤ C k u k V . (2.11)Suppose further that V is dense in H . (2.12)Let D ( S ) = { u ∈ V | v a ( u, v ) is continuous on V in the norm of H} . (2.13)We can now define the operator S : D ( S ) → H by a ( u, v ) = h Su , v i H , ∀ u ∈ D ( S ) and ∀ v ∈ V . (2.14)We can now prove
Theorem 2.2.
Let a be a continuous sesquilinear form satisfying (2.7) and (2.8) . Suppose, in addition, that V ⊂ H and that (2.11) and (2.12) hold.Assume further that Φ extends into a continuous linear map in L ( H ) . Let S be defined by (2.13) - (2.14) . Then1. S is bijective from D ( S ) onto H and S − ∈ L ( H ) .2. D ( S ) is dense in both V and H S is closed.4. Let b denote the conjugate sesquilinear form of a , i.e. ( u, v ) b ( u, v ) := a ( v, u ) . et S denote the closed linear operator associated with b by the sameconstruction . Then S ∗ = S and S ∗ = S . (2.15)
Proof.
We show here only that S is injective. This is a consequence, for all u ∈ D ( S ) , of α k u k H ≤ C α k u k V ≤ C ( | a ( u, u ) | + | a ( u, Φ ( u ))) | = C |h Su , u i H | + |h Su , Φ ( u ) i H |≤ ˆ C k Su k H · k u k H , which leads to α k u k H ≤ C k Su k H , ∀ u ∈ D ( S ) , (2.16)and α k u k V ≤ C k Su k H k u k H , ∀ u ∈ D ( S ) . (2.17)Injectivity easily follows.We omit the rest of the proof, as it does not deviate from the proof of thestandard Lax-Milgram Theorem. Interested readers may find a presentationof the missing details of the standard case in [13] P D We return to the operator P introduced in (1.10) on C ∞ (Ω) and describehow the previous abstract theory applies to the construction P D . V = 0Let H = L (Ω , C ) and V as introduced in (2.3). Initially, we equip V withthe norm: u
7→ k u k V := s k u k H A + Z p | V | + 1 | u ( x ) | dx .
9e later prove (see (2.23)) that u
7→ k u k V ,B := s k u k H A + Z m B,V | u ( x ) | dx , is an equivalent norm on V .To apply the previous results to the sesquilinear form introduced in (2.2) weneed to establish first that a satisfies (2.7) and (2.8). To this end we setΦ ( u ) = Φ ( u ) = φ u , where φ = ℑ V m B,V . Clearly Φ belongs to L ( V ) and L ( H ), since it is a multiplication operatorby a function in W , ∞ (Ω) . Note that by (1.8) ∇ φ belongs to L ∞ (Ω) .It can be easily verified that a is continuous on V ×V . To use Theorem 2.2,we thus need to establish (2.7) and (2.8). We first observe that for any u ∈ V ,we have: ℜ a γ ( u, u ) ≥ K Z Ω |∇ A u ( x ) | dx + Z ( ℜ V ( x ) + γ ) | u ( x ) | dx , (2.18)where K is defined in (1.4). Furthermore, ℑ a ( u, φ u ) = ℑ d X k =1 Z Ω e iα k ∂ A k u ( x ) ∂ A k ( φ u ) dx + Z |ℑ V | ( m B,V ) − | u | dx . After some simple manipulation we arrive at ℑ a ( u, φ u ) = ℑ d X k =1 Z Ω e iα k ∂ A k u ( x ) ∂ k φ ¯ u dx + Z |ℑ V | ( m B,V ) − | u | dx + d X k =1 Z Ω φ sin(2 α k ) | ∂ A k u | dx . Clearly, as k φ k ∞ ≤ (cid:12)(cid:12)(cid:12) ℑ d X k =1 Z Ω e iα k ∂ A k u ( x ) ∂ k φ ¯ u dx (cid:12)(cid:12)(cid:12) ≤ ǫ k∇ A u k + C ǫ k u k . γ and C such that for γ > γ C ( ℑ a γ ( u, φ u ) + ℜ a γ )( u, u )) ≥ k∇ A u ( x ) k dx + Z ℜ V ( x ) | u ( x ) | dx + Z |ℑ V | ( m B,V ) − | u | dx + Z | u ( x ) | dx . (2.19)To complete the proof of (2.7) we need an estimate for k B m − / B,V u k . Tothis end we introduce the operator identity ([ · , · ] being the Poisson bracket) B kℓ = i [ ∂ A k , ∂ A l ] . (2.20)We then use (2.20) to obtain Z Ω B kℓ ( x ) ( m B ,v ) − | u | dx = i Z Ω { [ ∂ A k , ∂ A l ] u } · B kℓ ( x )( m B,V ) − u dx ≤ C n k ∂ A l u k k ∂ A k u k + ( k ∂ A k u k + k ∂ A k u k ) k u k sup x ∈ Ω |∇ ( B kℓ ( m B,V ) − ) | o . As before this leads to Z Ω B kℓ ( x ) ( m B,V ) − | u | dx ≤ C ( k∇ A u k + k u k ) ≤ ˜ C ℜ a ( u, u ) . (2.21)We can now deduce from (2.21) and (2.19) that for some C > k u k V ,B ≤ C ( |ℑ a ( u, φ u ) | + |ℜ a ( u, u ) | ) , (2.22)establishing both (2.7) and (2.8) and also k u k V ,B ≤ e C k u k V . (2.23)Hence, the linear operator S γ associated with a γ can be defined over the set(2.13), and is an isomorphism from D ( S γ ) into L (Ω , C ). It can be easilyverified that a γ ( u, v ) = hP γ u, v i for all u ∈ C ∞ (Ω) (with P γ = P + γ ) andhence S /C ∞ (Ω) = P γ . We can then define the extension of P on D ( S γ ) by P D = S γ − γI .
11e have, hence, defined a closed extension of P , on a set of functions satis-fying, as D ( S ) ⊂ V , a Dirichlet boundary condition.As a matter of fact, it can be easily verified that D ( P D ) = D ( S γ ) = { u ∈ V | P u ∈ L (Ω) } (2.24)where P u is defined as a distribution on Ω.Hence, P D defines the Dirichlet realization of P in Ω.Additionally, as D ( S γ ) ⊂ V which is compactly embedded in L (Ω , C ) inview of (1.9), it follows that P D has a compact resolvent. We conclude this section by establishing the same results for P when V isnot necessarily 0 but satisfies (1.8). We define to this end the sesquilinearform b γ : V × V → C b γ ( u, v ) := P k e iα k R Ω ∂ A k u ( x ) ∂ A k v ( x ) dx + R Ω ( V ( x ) + γ ) u ( x ) ¯ v ( x ) dx = a γ ( u, v ) + R Ω V ( x ) u ( x ) ¯ v ( x ) dx . In view of (1.8), b γ is continuous. Furthermore, for any u ∈ V we have by(2.22) that k u k H A + R m B,V | u ( x ) | dx ≤ C ( ℑ a γ ( u, φ u ) + ℜ a γ ( u, u )) ≤ C ( ℑ b γ ( u, φ u ) + ℜ b γ ( u, u )) + C h| V | u, u i . We thus obtain, by (1.8), the existence of γ and C such that: k u k V ,B ≤ C ( ℑ b γ ( u, φ u ) | + ℜ b γ + γ ( u, u )) . It therefore follows, that we can apply to b γ + γ the same construction whichwas applied to a γ , to obtain the same domain in the general case (and thesame form domain) and that its resolvent is compact. In this section we attempt to obtain the optimal value of p for the Schattenclass of the resolvent of the Dirichlet realization P D . We begin by showingthat if ( − ∆ A + | V | + 1) − ∈ C p then ( P D − λ ) − ∈ C p , thereby allowing us12o use techniques from selfadjoint theory. Then, we provide a criterion on V and B which can be used to determine whether the resolvent of P D is in agiven Schatten class. For convenience of notation we omit from now on thesuperscript D and write P instead of P D . We begin with the following comparison result
Proposition 3.1.
Let { µ n } ∞ n =1 denote the n ′ th eigenvalue of ( P ∗ P ) / , where P ∗ P is the linear operator associated with the sesquilinear form q : D ( P ) × D ( P ) → C given by q ( u, v ) = hP u, P v i . The domain of P ∗ P is given by D ( P ∗ P ) = { u ∈ D ( P ) | P u ∈ D ( P ∗ ) } . Consider then the Dirichlet realization in Ω of − ∆ A + m B,V , and let σ ( − ∆ A + m B,V ) = { ν j } ∞ j =1 . Then, there exists
C > such that ν n ≤ C (1 + µ n ) . (3.1) Proof.
For any u ∈ V we have h ( − ∆ A + m B,V ) u, u i ≤ C (cid:0) | b ( u, φ u ) | + | b ( u, u ) | + k u k (cid:1) . (3.2)Hence, for any u ∈ D ( P ) , we can write h ( − ∆ A + m B,V ) u, u i ≤ C (cid:0) |hP u, φ u i| + |hP u, u i| + k u k (cid:1) . (3.3)Consequently, for all u ∈ D ( P ), h ( − ∆ A + m B,V ) u, u i ≤ C (cid:0) kP u k k u k + k u k (cid:1) . (3.4)It thus follows that for each j ≥ E j ofdimension j in D ( P ∗ P ) (hence also in D ( P )) such that, for all u ∈ E j , wehave h ( − ∆ A + m B,V ) u, u i ≤ C ( µ j + 1) k u k . (3.5)By Proposition 11.9 in [13] applied to the Dirichlet realization of( − ∆ A + m B,V ) (observing that the domain of the operator can be replacedby the form domain V of this Dirichlet realization and that D ( P ) ⊂ V ), wethen obtain (3.1). 13 .2 Proof of Theorem 1.3 By (2.23) we have that h u, ( − ∆ A + m B,V ) u i ≤ ˜ C h u, ( − ∆ A + | V | + 1) u i . Thus, by the min-max principle, the resolvents of the operators − ∆ A + m B,V and − ∆ A + | V | + 1 always belong to the same Schatten class. We thereforebegin by restating the theorem in the following equivalent form Theorem 3.2.
Suppose that Z Ω × R d ( | ξ | + m B,V ) − p dxdξ < ∞ . (3.6) Then, ( − ∆ A + m B,V ) − ∈ C p ( L (Ω , C )) . We note that by (3.1) it follows that whenever (3.6) is satisfied, then(
P − λ ) − ∈ C p . Proof.
For Ω = R d , (3.6) has been established in [6]. To extend it to theDirichlet realization of − ∆ A + m B,V for general domains we extend U = m B,V to R d in the following manner: U ρ,M = ( U in Ω ρ (1 + x ) M in Ω c , where M is chosen so that Z R d × R d ( | ξ | + (1 + x ) M ) − p dxdξ < ∞ . and ρ ≥ σ ( − ∆ A + U ρ,M ) = { µ j } ∞ j =1 , σ ( − ∆ A + U ) = { λ j } ∞ j =1 , where − ∆ A + U ρ,M is the unique self-adjoint extension on R d (by Kato’stheorem) and − ∆ A + U is the Dirichlet realization in Ω.It can be easily verified by comparison of the form domains that µ j ≤ λ j , ∀ j ≥ . ρ ≥ − ∆ A + U + 1) − p ≤ Trace( − ∆ A + U ρ,M + 1) − p ≤ (2 π ) − n Z R d × R d ( ξ + U ρ,M + 1) − p dxdξ . (3.7)Taking the limit ρ → ∞ yieldsTrace( − ∆ A + U + 1) − p ≤ (2 π ) − n Z Ω × R d ( ξ + U + 1) − p dxdξ . It follows that if (3.6) holds true, then ( − ∆ A + U ) − ∈ C p ( L (Ω , C )). (3.6) Let D = { u ∈ C ∞ ( ¯Ω , C ) ∩ H (Ω , C ) | Supp u is a compact subset of Ω } . It is clear that D is in D ( P D ). We first show Lemma 3.3.
Let D ( P D ) be given by (2.24) . Then, D is dense in D ( P D ) under the norm kP · k + k · k V .Proof. Let u ∈ D ( P D ). By (2.17), (2.8) and (2.23) we have k u k V ,B ≤ C ( k u k + kP u k ) . (3.8)Let η ∈ C ∞ ( R + , [0 , η ( t ) = ( t < t > . Denote then by η k : Ω → R + the restriction to Ω of the cutoff functiondefined by η k ( x ) = η ( | x | /k ) for all x ∈ Ω. By our assumption on ∂ Ω we havethat η k ∈ C (Ω) for all k ≥ { v k } ∞ k =1 through v k = η k u . Using the localregularity of the Dirichlet problem we easily conclude that v k ∈ H (Ω , C )15as compact support, and it can be readily verified that k v k − u k V → k → + ∞ .We now prove that v k → u in the graph norm. To prove that kP ( v k − u ) k → P ( v k − u ) = ( η k − P u + u d X m =1 e iα m ∂ η k ∂x m + 2 d X m =1 e iα m ∂η k ∂x m ∂ A m u . The first two terms on the right-hand-side tend to 0 (in L sense) since both u and P u are in L (Ω , C ). For the last term we have (cid:13)(cid:13)(cid:13) d X m =1 e iα m ∂η k ∂x m ∂ A m u (cid:13)(cid:13)(cid:13) ≤ Ck k∇ A u k . Hence, by (3.8), we obtain (3.9).The result of the foregoing discussion is that˜ D = { u ∈ H ( ¯Ω , C ) ∩ H (Ω , C ) | Supp u is a compact subset of Ω } , is dense in D ( P D ) with respect to the norm introduced in the lemma. Onecan now complete the proof invoking standard arguments that show that D is dense in ˜ D under the same norm.We continue this subsection by the following lemma. Lemma 3.4.
Under Assumptions (1.5)-(1.11), there exists C (Ω , P ) suchthat for all u ∈ D we have k Bu k ≤ C ( k ∆ A u k + k V u k + k u k ) . (3.10) Proof.
Let u ∈ D . We use the identity (2.20) to obtain, after integration byparts k B km u k = ℑh [ ∂ A k , ∂ A m ] u, B km u i ≤ |h ∂ A m u, ∂ A k ( B km u ) i| + |h ∂ A k u, ∂ A m ( B km u ) i| . (3.11)As h ∂ A m u , ∂ A k ( B km u ) i = h B km ∂ A m u , ∂ A k u i + h ∂ A m u , u∂ k B km i ,
16t follows from (1.6) and (3.11) that k B km u k ≤ k| B km | / ∂ A m u k k| B km | / ∂ A k u k + C k m B,V u k ( k ∂ A m u k + k ∂ A k u k ) . Consequently, for any ǫ > k B km u k ≤ k| B km | / ∂ A m u k k| B km | / ∂ A k u k + C h ǫ k m B,V u k + 14 ǫ ( k ∂ A m u k + k ∂ A k u k + k u k ) i . Hence, for all ǫ > C ǫ > k B km u k − ǫ k Bu k ≤ C ǫ (cid:0) k| B km | / ∂ A m u k + k| B km | / ∂ A k u k + k V u k + k ∂ A m u k + k ∂ A k u k + k u k (cid:1) . Summing over k and m we then obtain, using the standard inequality k∇ A u k ≤
12 ( k ∆ A u k + k u k ) , (3.12)that for sufficiently small ǫ , k Bu k ≤ C ( k| B | / ∇ A u k + k ∆ A u k + k V u k + k u k ) , and hence, k m B,V u k ≤ C ( k| B | / ∇ A u k + k ∆ A u k + k V u k + k u k ) . (3.13)Next, we use integration by parts to show that −h m B,V u , ∆ A u i = h m B,V ∇ A u , ∇ A u i + h u ∇ m B,V , ∇ A u i . By (1.6) we have that |∇ m B,V | ≤ C m
B,V . Consequently, −h m B,V u , ∆ A u i ≥ k m / B,V ∇ A u k − C k m B,V u k k∇ A u k . With the aid of (3.12), we then obtain that k m B,V u k k ∆ A u k ≥ k m / B,V ∇ A u k − C k m B,V u k k ( k ∆ A u k + k u k ) . k m / B,V ∇ A u k ≤ C k m B,V u k ( k ∆ A u k + k u k ) , from which we conclude that k m / B,V ∇ A u k ≤ ǫ k m B,V u k + C ǫ ( k ∆ A u k + k u k ) , (3.14)which, combined with (3.13), easily yields (3.10).We continue with the following comparison result Lemma 3.5.
Let A = − e iα ∆ A + V , and let { µ n } ∞ n =1 denote the n ′ th eigen-value of ( A ∗ A ) / . Let further σ (( − ∆ A + | V | )) = { ν j } ∞ j =1 . Then, there exists
C > and n > , such that for all n > n we have µ n ≤ C (1 + ν n ) . (3.15) Proof.
Clearly, for every u ∈ D , k ( − ∆ A + | V | ) u k = k ∆ A u k + k V u k − ℜh| V | u, ∆ A u i . (3.16)As 2 ℜh ( | V + V | − | V | ) u, ∆ A u i ≤ k| V | u k k ∆ A u k , it follows, after integration by parts, that2 ℜh| V | u, ∆ A u i ≤ − h| V |∇ A u, ∇ A u i +2 |ℜh u ∇| V | , ∇ A u i| +2 k| V | u k k ∆ A u k . (3.17)For every ǫ > |ℜh u ∇| V | , ∇ A u i ≤ ǫ k u ∇ V k + 1 ǫ k∇ A u k ≤ Cǫ ( k V u k + k| B | u k ) − ǫ ℜh u, ∆ A u i . (3.18)By (3.10), for each positive ǫ , there exists C ǫ such that2 |ℜh u ∇| V | , ∇ A u i| ≤ ǫ ( k V u k + k ∆ A u k ) + C ǫ k u k . (3.19)18or the last term on the right-hand-side we have, by (1.8), that for any ǫ > C ǫ > k V u k k ∆ A u k ≤ ǫ ( k V u k + k| B | u k + k ∆ A u k ) + C ǫ k u k . With the aid of (3.10), we then conclude that for every ǫ > C ǫ > k V u k k ∆ A u k ≤ ǫ ( k ∆ A u k + k V u k ) + C ǫ k u k . Substituting the above together with (3.19) into (3.17) yields that for every ǫ > C ǫ > ℜh| V | u, ∆ A u i ≤ ǫ (cid:0) k| V | u k + k ∆ A u k (cid:1) + C ǫ k u k . Substituting the above into (3.16) we obtain, if we choose ǫ sufficiently small, k ∆ A u k + k V u k ≤ C k ( − ∆ A + | V | ) u k . (3.20)From here we easily obtain that there exists C > u ∈ Dk ( − e iα ∆ A + V ) u k ≤ C ( k ( − ∆ A + | V | ) u k + k u k ) . (3.21)We now use Lemma 3.3 to conclude that D is dense in D ( − ∆ A + | V | ). Hencefor all n ≥ { u k } nk =1 ⊂ D such that h ( − ∆ A + | V | ) u, u i ≤ ν n k u k , ∀ u ∈ span { u k } nk =1 . We can now obtain (3.15) by using once again Proposition 11.9 in [13] (as inthe proof of (3.1)) together with (3.21) and the fact that
D ⊂ D ( − e iα ∆ A + V ). Remark 3.6.
The above lemma, together with (3.1) proves that for any p > and λ ∈ ρ ( − e iα ∆ A + V ) ∩ ρ ( − ∆ A + | V | ) , ( − ∆ A + | V | − λ ) − ∈ C p ⇔ ( − e iα ∆ A + V − λ ) − ∈ C p , (3.22) whereas Theorem 3.2 provides us only with an upper bound for the optimalvalue of p . While the Dirichlet realization of − ∆ A + | V | in Ω is self-adjoint,the authors are unaware of an asymptotic expansion for the counting functionassociated with it in the necessary generality. There is, however, reason tobelieve that (3.6) is optimal. An example where such an asymptotic expansionhas been derived is given in [16, 17] for the case Ω = R d , and where A and V are polynomials.
19e now prove optimality in the general case for the particular case whenΩ is the half-space in R d . Lemma 3.7.
Let
Ω = R d + where R d + = { x ∈ R d | x d > } . Let further D = { u ∈ D , | P u ∈ D} . Then, there exists C (Ω , P ) > , such that for all u ∈ D we have kP u k ≤ C [ k (∆ A + | V | ) u k + k u k ] . (3.23) Proof.
Clearly, kP u k = d X k,m =1 e i ( α k − α m ) h ∂ A m u, ∂ A k u i . We now write h ∂ A m u, ∂ A k u i = −h ∂ A k ∂ A m u, ∂ A k u i + Z ∂ Ω ∂ A m u · ∂ A k u ν k ds . Since u ∈ D , we have that P u = 0 on ∂ Ω, and hence d X m =1 e i ( α k − α m ) Z ∂ Ω ∂ A m u · ∂ A k u ν k ds = 0 , ∀ ≤ k ≤ d . Consequently, kP u k = − d X k,m =1 e i ( α k − α m ) h ∂ A k ∂ A m u, ∂ A k u i . (3.24)Next we write h ∂ A k ∂ A m u, ∂ A k u i = h ∂ A m ∂ A k ∂ A m u, ∂ A k u i + i h B km ∂ A m u, ∂ A k u i = h ∂ A m ∂ A k u, ∂ A k u i + i ( h B km ∂ A m u, ∂ A k u i + h ∂ A m B km u, ∂ A k u i ) . (3.25)Integration by parts yields − h ∂ A m ∂ A k u, ∂ A k u i = k ∂ A m ∂ A k u k + Z ∂ Ω ν m ∂ A m ∂ A k u ∂ A k u ds . (3.26)20s u = 0 on the boundary, the surface integral on the right-hand-side vanisheswhenever ( m, k ) = ( d, d ). If m = k = d , we use the fact that P u = 0 on theboundary to obtain Z ∂ Ω ∂ A d ∂ A d u ∂ A d u ds = d − X n =1 e i ( α d − α n ) Z ∂ Ω ∂ A n ∂ A n u ∂ A d u ds = 0 . (3.27)Combining the above with (3.25), (3.26), and (3.24), yields kP u k = ℜ d X k,m =1 e i ( α k − α m ) (cid:2) k ∂ A m ∂ A k u k − i ( h B km ∂ A m u, ∂ A k u i + h ∂ A m B km u, ∂ A k u i ) (cid:3) . (3.28)We proceed by estimating the two rightmost terms in (3.28). To this endwe use (3.14) to obtain that |h B km ∂ A m u, ∂ A k u i| ≤ k| B | / ∇ A u k ≤ C ( k ∆ A u k + k V u k + k u k ) . (3.29)For the last term on the right-hand-side we have h ∂ A m B km u, ∂ A k u i = h B km ∂ A m u, ∂ A k u i + h u∂ m B km , ∂ A k u i . The first term on the right-hand-side of the above equation has already beenestimated by (3.29). For the second term we use (1.6) and (3.10) to obtain |h u∂ m B km , ∂ A k u i| ≤ C k m B,V u k k∇ A u k ≤ C ( k ∆ A u k + k V u k + k u k ) . (3.30)Substituting the above together with (3.29) into (3.28) yields kP u k ≤ ℜ d X k,m =1 e i ( α k − α m ) k ∂ A m ∂ A k u k + C ( k ∆ A u k + k V u k + k u k ) . Hence, kP u k ≤ ℜ d X k,m =1 k ∂ A m ∂ A k u k + C ( k ∆ A u k + k V u k + k u k ) . (3.31)21t is easy to show that if u ∈ D then ∆ A u = 0 on ∂ Ω: As ∂ A n u = 0 on ∂ Ω for all 1 ≤ n ≤ d , both ∆ A u | ∂ Ω = 0 and P u | ∂ Ω = 0 are equivalent to ∂ d u | ∂ Ω = 0. Hence, we may conclude from (3.28) that k ∆ A u k = ℜ d X k,m =1 (cid:2) k ∂ A m ∂ A k u k − i ( h B km ∂ A m u, ∂ A k u i + h ∂ A m B km u, ∂ A k u i ) (cid:3) . Hence, by (3.31) kP u k ≤ k ∆ A u k − ℑ d X k,m =1 (cid:2) ( h B km ∂ A m u, ∂ A k u i + h ∂ A m B km u, ∂ A k u i ) (cid:3) + C ( k ∆ A u k + k V u k + k u k ) . Using (3.29) and (3.30) once again yields kP u k ≤ C ( k ∆ A u k + k V u k + k u k ) , and hence kP u k ≤ C ( k ∆ A u k + k V u k + k u k ) . The proof of (3.30) can now be easily completed with the aid of (3.20)(whichis valid for all u ∈ D ).An immediate conclusion is Corollary 3.8.
Let { µ n } ∞ n =1 and { ν n } ∞ n =1 be defined as in Lemma 3.1, andlet Ω = R d + or Ω = R d . Then, there exists C > and n ∈ N such that µ n ≤ C (1 + ν n ) . (3.32) Thus, if the resolvent of P is in C p ( L (Ω , C )) for some p > , the the sameconclusion follows for the resolvent of − ∆ A + | V | .Proof. Let u ∈ D . We need to show first that ( − ∆ A + | V | ) u ∈ D , orequivalently that ∆ A u | ∂ Ω = 0 in the case Ω = R d + (where ∂ Ω is the hyperplane x d = 0). Since u ∈ D , it follows that ∂ A k u | ∂ Ω = 0 for all 1 ≤ k ≤ d −
1. Itis easy to show (see (3.27)) that ∂ A d u | ∂ Ω = 0 using the fact that P u | ∂ Ω = 0for all u ∈ D . 22ince by Lemma 3.3 D is dense in D ( − ∆ A + | V | ), it follows that D isdense in D (cid:0) ( − ∆ A + | V | ) (cid:1) under the same graph norm. We can now usestatement 2 of Theorem 2.2 with S = ( − ∆ A + | V | ) and V = D ( − ∆ A + | V | )to prove that D is dense in D ( − ∆ A + | V | ). The proof of (3.32) can nowbe completed by using Proposition 11.9 in [13] as in the proof of Lemma 3.5. Consider the case Ω = R and L = − d dx + e iθ v ( x ) + v ( x ) , (4.1)where θ ∈ ( − π, π ), defined initially on C ∞ ( R ).In the above, v ( x ) ∈ C ( R , R ) has the asymptotic behaviour | v | ∼ | x | α as | x | → ∞ , (4.2)for some α > | v ′ | ≤ C (1 + | x | α − ) , (4.3)and v ∈ L ∞ loc ( R , C ) satisfies v = o ( | x | α ) as | x | → ∞ . (4.4)Set P = i sign θ e − iθ L . Then P meets (1.5)-(1.10), and we may ap-ply Theorems 1.1 and 1.3. Henry [14] considers this example for the case v ( x ) = x k , v ≡ L − is in C p for all p > / /α . Asthe numerical range of L + µ , for sufficiently large µ ∈ R + , lies in the sectorarg λ ∈ [ θ − π, θ ], Theorem 1.4 can be applied to obtain, for any θ ∈ ( − π, π ),that Span ( L ) = L ( R , C ) whenever α > v → + ∞ , as | x | → ∞ , the numericalrange lies inside the sector arg λ ∈ [0 , θ ] and hence for any θ ∈ ( − π, π ) and α > | θ | π −| θ | , Span ( L ) = L ( R , C ). 23he complex cubic oscillator is an example included in the class we in-troduce in (4.1) (for the general case where v can change its sign between −∞ and ∞ ) which has been frequently addressed in the literature (cf. [20],[12], [9], [15] to name just a few references). In this case, θ = π/ v ( x ) = x , v ( x ) = β x + β x , where β , β ∈ C . In [21], the existence of an infinite sequence of eigenvalueshas been established (cf. [20] for more details). Completeness of the systemof eigenfunctions in L ( R , C ) has been established in [19] for the case β = β = 0. In [15] completeness is extended to the case iβ ∈ R , β = 0. Herewe show it in greater generality, without the need to rely on the symmetriesof the particular cases addressed in [15, 19].Another case that has been addressed in [8, 7] is L = − d dx + ce iθ | x | α , (4.5)where c > θ ∈ ( − π/ , π/ L is dense in L ( R , C ) when either α ≥
1, or α ∈ (0 ,
1) and θ ≤ πα . Since (4.5) is a particular case of (4.1), it follows that whenever θ ∈ ( − π, π ), L − must be in C p for all p > / /α . The numerical range of L however,is confined in arg z ∈ (0 , θ ). Hence, the eigenfunctions of L form a completesystem whenever | θ | < παα + 2 . In particular, if θ ∈ ( − π/ , π/
2) completeness of the eigensystem of L isguaranteed for all α > /
3. Note that for 0 < α < πα < παα + 2 . Hence, our method provides greater domains for θ and α where the eigenspaceof L is dense in L ( R , C ). 24 .1.2 The positive real line case: Here we consider the same differential operator as in (4.1), but this timedefined on C ∞ ( R + ) and consider the Dirichlet realization. Assuming that v and v satisfy (4.2), (4.3), and (4.4), we have by Theorem 3.2 that L − ∈ C p whenever p > / /α as before. However, as in the case where L is givenbt (4.5), since every direction outside [0 , θ ] is a direction of minimal growthfor ( L + − λ ) − . Thus, we have that Span L + = L ( R , C ) whenever θ < απα + 2 . In particular, for the case v = x and v = 0, which is known as the complexAiry’s equation, we obtain that Span L + = L ( R , C ) whenever θ < π/ Let A and V be such that V = iφ and curl A + iφ is a holomorphic functionof z = x + iy , x and y being the planar coordinates. Such a choice is in linewith the steady state Faraday’s law, which in two dimensions read (assumingall constants are equal to 1) ∇ ⊥ curl A + ∇ φ = 0 . We further narrow our choice by settingcurl A + iφ = z n . (4.6)Consider the differential operator − ∆ A + iφ with Ω = R (which is clearlya particular case of (1.10)). By Theorem 3.2, we have that P − ∈ C p forall p > /n . It can be readily verified that, independently of β , everydirection outside [ − π/ , π/
2] is a direction of minimal growth for (
P − λ ) − .Unfortunately, the condition of validity of our theorems will lead to thecondition 1 < n , which is never satisfied. However, if we consider insteadof R , a smooth domain Ω which is contained in a sector whose opening issmaller than π/ ( n +2), and such that the positive real axis is contained in thissector, then the numerical range of P is contained in a sector whose openingis less than π/ (1 + 2 /n ) and we can use Theorem 1.4 to prove completenessof the eigenspace. 25 .2.2 Analytically dilated operators Next, for m ≥ k ≥
1, let A = − ∂ ∂x − (cid:16) ∂∂y − i x m m (cid:17) + iy k , (4.7)Note that the numerical range lies in the sector arg λ ∈ [0 , π/ u → ( U u )( x, y ) = e − i m − α u ( e iα x, e − imα y )where α = − π m ( k + 1) , (4.8)to obtain that A α = U − AU = − e iα ∂ ∂x − e − imα (cid:16) ∂∂y − i x m m (cid:17) + e ikmα + iπ/ y k . (4.9)We next verify A α meets the conditions Theorem 1.1. Since V = e ikmα + iπ/ y k and 2 kmα + π − kπ k + 1) + π π k + 1) , we obtain that ℜ V = cos π k + 1) y k . Hence the conditions of Theorem 1.1 can be easily verified. We thus obtainthat ( A α ) − is in C p for all p > k + 1 m − k + 1) m − k ( m − . (4.10)Next we observe that h u, ( e − iα A α − λ ) u i = k u x k + e − i ( m +1) α (cid:16)(cid:13)(cid:13)(cid:13)(cid:16) ∂ y − i x m m (cid:17) u (cid:13)(cid:13)(cid:13) + k y k u k (cid:17) − λ k u k . It can be easily verified from the above and (4.8) that the numerical rangeof A α lies in the sector arg λ ∈ h , ( m + 1) π m ( k + 1) i . , π/
8] for the assumed range of m and k values. Consequently, the effect of this analytic dilation has permitted us toreduce the angle of the sector in which the numerical range lies. Since( m + 1)2 m ( k + 1) < k ( m − k + 1) m − , (4.11)we obtain that Span A α = L ( R , C ). By the same arguments of [4] we thenobtain that A has an infinite sequence of eigenvalues for all k ≥ m ≥ Remark 4.1. • It is not clear whether analytic dilation preserves the completeness ofthe system of generalized eigenfunctions. Hence, the best we can obtainis the existence of an infinite discrete spectrum for A . • Had we abandoned analytic dilation, we would have obtained, insteadof (4.11) , the condition < k ( m − k + 1) m − which implies m > and k > m − m −
2) (4.12) to achieve completeness. • Note that the results presented in [16] can be applied to (4.9) to con-clude the optimality of (4.10) , as they can be used to obtain the preciseSchatten class of the resolvent of − ∆ A + y k in R . Let R = { ( x, y ) ∈ R | y > } . Consider the case where V = e iθ y and A = x ˆ i y /
2. We define P + = − ∆ A + V with Ω = R (which is once again a particular case of (1.10)). Once again wehave that P − ∈ C p for all p > /n . For n = 1 and β = 0 it can be easilyshown that the numerical range of P + is confined within the sector [0 , θ ] in C . Hence, every direction outside [0 , θ ] is a direction of minimal growth for( P + − λ ) − , and consequently, Span P + = L ( R , C ) for all θ < π/ cknowledgements Y. Almog was supported by NSF grant DMS-1109030and by US-Israel BSF grant no. 2010194.
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