Another Application of Dilation Analytic Method for Complex Lieb--Thirring Type Estimates
aa r X i v : . [ m a t h . SP ] N ov ANOTHER APPLICATION OF DILATION ANALYTIC METHODFOR COMPLEX LIEB–THIRRING TYPE ESTIMATES
Norihiro Someyama
Abstract
We consider non-self-adjoint Schr¨odinger operators H c = − ∆ + V c (resp. H r = − ∆ + V r ) acting in L ( R d ), d ≥
1, with dilation analytic complex (resp. real) poten-tials. We were able to find out perhaps a new application of dilation analytic methodin [17] (N. Someyama, ”Number of Eigenvalues of Non-self-adjoint Schr¨odinger Oper-ators with Dilation Analytic Complex Potentials,” Reports on Mathematical Physics,Volume 83, Issue 2, pp.163-174 (2019).). We give a Lieb–Thirring type estimate onresonance eigenvalues of H c in the open complex sector and that on embedded eigen-values of H r in the same way as [17]. To achieve that, we derive Lieb–Thirring typeinequalities for isolated eigenvalues of H on several complex subplanes. Keywords : non-self-adjoint Schr¨odinger operator, dilation analytic complex potential, Lieb–Thirring (type) inequality, complex isolated eigenvalue, resonance eigenvalue, embedded eigen-value.
Let d ≥ L ( R d ): H := H + V, H := − ∆where the Laplacian ∆ := P dj =1 ∂ /∂x j means the distributional derivative and V is thedilation analytic complex potential (see Definition 1.1 for detailed definitions). We definethe domain D ( H ) of H as the second-order Sobolev space H ( R d ) := W , ( R d ). The L -inner product and L -norm are defined by( u, v ) := Z R d u ( x ) v ( x ) d x, k u k L ( R d ; C ) := ( u, u ) / respectively. Moreover, we consider the one-parameter unitary group { U ( θ ) : L ( R d ) → L ( R d ); θ ∈ R } defined by U ( θ ) u ( x ) := e dθ/ u ( e θ x )for u ∈ L ( R d ). We put H ( θ ) := U ( θ ) HU ( θ ) − = e − θ ( H + e θ V θ ) (1.1) V θ ( x ) := U ( θ ) V U ( θ ) − = V ( e θ x ) (1.2)1 N. Someyama and call H ( θ ) (resp. V θ ) the dilated Hamiltonian (resp. dilated potential ). We also call thetransform by U ( θ ) such as (1.1) the complex dilation . We write e H ( θ ) := H + e θ V θ . (1.3)It is of course that H (0) = e H (0) = H . Furthermore, we denote the real (resp. imaginary)part of z ∈ C by Re z (resp. Im z ). Definition 1.1 ([17]) . V is called the dilation analytic complex potential if it satisfies thefollowings: Let d, γ ≥ .i) V is the multiplication operator with the complex-valued measurable function R d ∋ x V ( x ) ∈ C obeying V ∈ L γ + d/ ( R d ; C ) .ii) V is the H -compact operator, that is, D ( V ) ⊃ D ( H ) = H ( R d ) and V ( H + 1) − is compact in L ( R d ) .iii) The function V θ with respect to θ ∈ R has an analytic continuation into the complexstrip S α := { z ∈ C : | Im z | < α } for some α > as an L γ + d/ ( R d ; C ) -valued function with respect to x .iv) The function V θ ( H + 1) − with respect to θ ∈ R can be extended to S α as a B ( L ( R d )) -valued analytic function, where B ( S ) denotes the set of bounded, ev-erywhere defined operators on the space S .We write the set of dilation analytic complex potentials by D ( S α ; C ) for convenience. Since U ( θ + φ ) and U ( θ ) are unitarily equivalent for any φ ∈ R , we can suppose that θ isthe pure-imaginary number by setting φ = − Re θ . In other words, H ( θ ) does not dependon Re θ and σ ( H ( θ )) is only dependent on Im θ . H ( θ ) is a Kato’s type-(A) function (e.g.[5, 8, 15]) which is operator-valued and analytic with respect to θ ∈ S α . Remark 1.1. (1) The dilation analytic method originally introduced in [2] and it was definedfor real potentials. We also call the dilation analytic method the complex dilation method or complex scaling method . This method and the now famous results derived by it wereorganized and customized in e.g. [5, 15]. Aguilar and Combes originally proposed dilationanalytic potentials so as to give a sufficient condition for the absence of the singularlycontinuous spectrum of the Schr¨odinger operator (then, remark that the non-negative halfline [0 , ∞ ) is the essential spectrum of it). More to say, the dilation analytic method is anatural factor that we consider and introduce complex potentials.(2) V θ has an analytic extension from S α to the closure S α of S α and H ( θ ) can be extendedfrom R to S α with respect to θ as a B ( L ( R d ))-valued analytic function, but we do not needsuch assumptions in the present paper. Throughout the present paper, we write σ ( T ), σ d ( T ), σ ess ( T ) for the spectrum, dis-crete spectrum, essential spectrum of the closed operator T respectively. Also, ‘isolatedeigenvalues’ are simply abbreviated as ‘eigenvalues’. The algebraic multiplicity m λ ( H ) of λ ∈ σ d ( H ) is defined by m λ ( H ) := sup N ∈ N (cid:0) dim ker( H − λ ) N (cid:1) . nother Application of Dilation Analytic Method for Complex LT-type Estimates V decays at infinity, it is well known that σ d ( H ) ⊂ ( −∞ , V ∈ L γ + d/ ( R d ; R ) is well known (e.g. [11, 13, 14]) asthe estimate on negative eigenvalues: X λ ∈ σ d ( H ) ⊂ ( −∞ , | λ | γ ≤ L γ,d k V − k γ + d/ L γ + d/ ( R d ; R ) , V ± := | V | ± V d obeys that γ ≥ / d = 1 ,γ > d = 2 ,γ ≥ d ≥ . (1.5)Then, L γ,d is a constant depending on d, γ and it is important for the accuracy of theestimate (see e.g. [6, 9, 10, 12]). In particular, (1.4) is well known as Cwikel–Lieb–Rozenbljum inequalities (e.g. [15, 18]) which are estimates on the number of negativeeigenvalues of H if d ≥
3. Related to this, Frank, Laptev, Lieb and Seiringer [7] gave someLieb–Thirring type inequalities for isolated eigenvalues of Schr¨odinger operators with anycomplex potentials on partial complex planes. The following inequality (1.7) is particularlythe most fundamental result for complex Lieb–Thirring inequalities.
Theorem 1.1 ([7]) . Let d, γ ≥ . Suppose V ∈ L γ + d/ ( R d ; C ) . We denote C ± ( κ ) := { z ∈ C : | Im z | < ± κ Re z } , (1.6) where these sets represent two sets, one for the upper sign and the other for the lower sign.Then, for any κ > , X λ ∈ σ d ( H ) ∩ C + ( κ ) c | λ | γ ≤ C γ,d (cid:18) κ (cid:19) γ + d/ k V k γ + d/ L γ + d/ ( R d ; C ) (1.7) and X λ ∈ σ d ( H ) ∩ C − ( κ ) | λ | γ ≤ (1 + κ ) L γ,d k (Re V ) − k γ + d/ L γ + d/ ( R d ; C ) . (1.8) Here S c is the complement set of the set S , C γ,d := 2 γ/ d/ L γ,d and L γ,d the constant of real Lieb–Thirring inequalities (1.4). We can obtain the usual Lieb–Thirring inequality (1.4) by letting κ ↓ κ → ∞ . Corollary 1.1 ([7]) . Suppose V ∈ L γ + d/ ( R d ; C ) . For d, γ ≥ , one has X λ ∈ σ d ( H ) ∩{ z ∈ C :Re z ≤ } | λ | γ ≤ C γ,d k V k γ + d/ L γ + d/ ( R d ; C ) . (1.9) N. Someyama
Remark 1.2. (1) It is now known [3] that complex Lieb–Thirring estimates on all eigenvaluesin C \ [0 , ∞ ) of H with any complex potential like (1.4) cannot hold if γ > d/ H = − ∆ + V by H ( A ) := ( − i ∇ + A ) + V with any real vector potential A and complex potential V .(So, we can read Theorem 1.2 and Theorem 2.1-2.2 described later as results for H ( A ).) Inaddition, their proofs also enable us to replace | V ( x ) | in (1.7) and (1.9) by { (Re V ( x )) − + | Im V ( x ) |} / √
2. See [7] for details. [17] shows that, if V is a dilation analytic complex potential, we can obtain the Lieb–Thirring type inequality for all eigenvalues (in C \ [0 , ∞ )) of H as follows. On and after,we write i := √− Theorem 1.2 ([17]) . Suppose that V ∈ D ( S α ; C ) with α > π/ . For d, γ ≥ , one has X λ ∈ σ d ( H ) | λ | γ ≤ C γ,d X ± k V ± iπ/ k γ + d/ L γ + d/ ( R d ; C ) . More precisely, if we write C + (resp. C − ) for the upper-half (resp. lower-half ) complexplane, we have1) the estimate on eigenvalues on C + : X λ ∈ σ d ( H ) ∩ ( C + ∪ ( −∞ , | λ | γ ≤ C γ,d k V iπ/ k γ + d/ L γ + d/ ( R d ; C ) ,
2) the estimate on eigenvalues on C − : X λ ∈ σ d ( H ) ∩ ( C − ∪ ( −∞ , | λ | γ ≤ C γ,d k V − iπ/ k γ + d/ L γ + d/ ( R d ; C ) . The above theorems and corollary indicate that improving L γ,d is an important studyto increase the accuracy of Lieb–Thirring estimates for complex potentials. Definition 1.2 (e.g. [15]) . For θ ∈ S α , elements of the complex subset σ res ( H | θ ) := σ d ( H ( θ )) \ σ d ( H ) are called resonance eigenvalues of H under complex dilation with θ ∈ S α . Remark 1.3.
Resonance eigenvalues of H are sometimes defined as isolated and non-realeigenvalues of H ( θ ). We can find that definition in [16] for instance.One of the reasons to study the eigenvalue estimates for complex potentials is toestimate the resonance eigenvalues or those number. There is for instance a precedingresult on resonance estimates in [4], Proposition 6. nother Application of Dilation Analytic Method for Complex LT-type Estimates We will prove Theorem 2.2 which is our main theorem in the same way as the proof ofTheorem 1.2. For that reason, we recall some results in [17]. Hereafter, C + (resp. C − )denotes the complex upper-half (resp. lower-half) plane. Moreover, we always write inwhat follows θ (resp. φ ) for a complex (resp. real) angle expressed in radians. Proposition 2.1 ([17]) . Suppose V ∈ D ( S α ; C ) . Then, σ ess ( e H ( iφ )) = σ ess ( H ) = [0 , ∞ ) ,σ ess ( H ( iφ )) = e − iφ [0 , ∞ ) for any iφ ∈ S α . Lemma 2.1 ([17]) . Suppose that V ∈ D ( S α ; C ) . Then, σ d ( H ) ∩ C ± = σ d ( H ( iφ )) ∩ C ± for any iφ ∈ S α ∩ C ± , where the two symbols ± correspond arbitrarily to each other. We write { λ ( iφ ) } for the eigenvalues of H ( iφ ). It is shown (e.g. [8, 15, 17]) that each λ ( iφ ) ∈ σ d ( H ( iφ )) is given by the branch of one or several analytic functions as Puiseuxseries. Then, they can be written as e λ ( iφ ) = e iφ λ ( iφ ) (2.1)by virtue of (1.1) and (1.3), if { e λ ( iφ ) } are eigenvalues of e H ( iφ ). Lemma 2.2 ([17]) . Suppose that V ∈ D ( S α ; C ) . Then, m λ ( H ) = m λ ( iφ ) ( H ( iφ )) = m e λ ( iφ ) ( e H ( iφ )) (2.2) for any iφ ∈ S α ∩ C ± . Remark 2.1. (1) It is well known [8, 15] that Proposition 2.1 and Lemma 2.1 hold for real V .Moreover Lemma 2.2 is the same.(2) As we can see from the proofs in [17], Lemma 2.1-2.2 still hold even if ‘ C ± ’ is replaced with‘any subset of C ± ’ in each statement. However, in order to replace ‘ C ± ’ by ‘(any subset of)the left-half complex plane’, we must keep in mind the range of α (see Theorem 2.1 and thatproof for details). Let us give an important theorem. The following result serves as a lemma to proveTheorem 2.2.
Theorem 2.1.
Let d, γ ≥ . Suppose that V ∈ D ( S α ; C ) with α > π − Arctan κ for any κ > . Then, one has X λ ∈ σ d ( H ) ∩ U ± κ | λ | γ ≤ (1 + κ ) L γ,d (cid:13)(cid:13)(cid:13)(cid:13)h Re (cid:16) e ± i ( π − Arctan κ ) V ± i ( π − Arctan κ ) (cid:17)i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) (2.3) N. Someyama where these represent two inequalities, one for the upper sign and the other for the lowersign, and U κ := { z ∈ C : π/ < arg z < π/ κ } , U − κ := { z ∈ C : − π/ − κ < arg z < − π/ } . Proof.
Fix κ > U κ . The same can be saidfor them in U − κ . We write λ for an eigenvalue of H and denote the complex left-halfplane by C < . We can first show that λ ( iφ ) = λ for any iφ ∈ S α ∩ C < as well as Lemma2.1. We can next show, from (2.1), that e λ ( iφ ) = e iφ λ for any iφ ∈ S α ∩ C < . We can alsosee (2.2) for any iφ ∈ S α ∩ C < as well as Lemma 2.2. Thus, we should estimate { e λ ( iφ ) } instead of { λ } , because of these facts and (2.1). Let us set φ = π − Arctan κ . It follows,from the above, that e i (2 φ ) ( σ d ( H ) ∩ U κ ) = σ d ( e H ( iφ )) ∩ C − ( κ )by recalling (1.6) for C − ( κ ). So, we have X λ ∈ σ d ( H ) ∩ U κ | λ | γ = X λ ∈ σ d ( H ) ∩ U κ | e iφ λ | γ = X e λ ( iφ ) ∈ σ d ( e H ( iφ )) ∩ C − ( κ ) | e λ ( iφ ) | γ ≤ (1 + κ ) L γ,d k [Re( e i (2 φ ) V iφ )] − k γ + d/ L γ + d/ ( R d ; C ) . Hence, this completes the proof.We write C II (resp. C III ) for the second (resp. third) quadrant of C . Because ofTheorem 2.1, we can easily know Lieb–Thirring type inequalities for eigenvalues on C II or C III as follows.
Corollary 2.1.
Let d, γ ≥ . Suppose that V ∈ D ( S α ; C ) with α > π/ . Then,1) Eigenvalue estimate on C II : X λ ∈ σ d ( H ) ∩ C II | λ | γ ≤ L γ,d (cid:13)(cid:13)(cid:13)(cid:13)h Re( e iπ/ V iπ/ ) i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) ,
2) Eigenvalue estimate on C III : X λ ∈ σ d ( H ) ∩ C III | λ | γ ≤ L γ,d (cid:13)(cid:13)(cid:13)(cid:13)h Re( e − iπ/ V − iπ/ ) i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) . Proof.
It is obvious from (2.3), since we have κ = 1 by setting Arctan κ = π/ nother Application of Dilation Analytic Method for Complex LT-type Estimates We now would like to estimate the complex eigenvalues which appear newly by complexdilation. We focus on eigenvalues of H ( iφ ) appear in open complex sector { z ∈ C : − φ < arg z < } . For convenience, let us call them complex resonance eigenvalues of Hhereinafter. The following result is our main theorem. The idea of that proof is the waywhich can be called ‘double complex dilation.’ We denote V θ ,...,θ n ( x ) := [ U ( θ n ) · · · U ( θ ) U ( θ ) V U ( θ ) − U ( θ ) − · · · U ( θ n ) − ]( x )= V θ + ··· + θ n ( x ) ,H ( θ , . . . , θ n ) := U ( θ n ) · · · U ( θ ) U ( θ ) HU ( θ ) − U ( θ ) − · · · U ( θ n ) − = H ( θ + · · · + θ n )for any n ∈ N . The same applies to e H ( θ , . . . , θ n ). Theorem 2.2.
Let d, γ ≥ . Suppose that V ∈ D ( S α ; C ) with α > | φ − π | . Then,complex resonance eigenvalues of H are estimated as X µ ∈ σ res ( H | iφ ) \ [0 , ∞ ) | µ | γ ≤ (1 + tan φ ) L γ,d (cid:13)(cid:13)(cid:13)(cid:13)h Re (cid:16) e i ( φ − π ) V i ( φ − π ) (cid:17)i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) (2.4) for any iφ ∈ S α .Proof. This proof is similar to the proofs of Theorem 1.2 and Theorem 2.1. The key toproof is to apply Theorem 2.1 as κ = tan φ . Then, Lemma 2.1-2.2 and (2.1) imply that e − iπ/ σ res ( H | iφ ) = e − iπ/ h σ d ( H ( iφ )) ∩ { z ∈ C : − φ < arg z < } i = σ d (cid:0) e H ( iφ, − iπ/ (cid:1) ∩ U − tan φ = σ d (cid:0) e H (cid:0) i ( φ − π ) (cid:1)(cid:1) ∩ U − tan φ . Hence, it follows that X µ ∈ σ res ( H | iφ ) | µ | γ ≤ X e µ ( i ( φ − π ∈ σ d ( e H ( i ( φ − π ∩ U − tan φ | e µ ( i ( φ − π )) | γ ≤ (1 + tan φ ) L γ,d (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) Re (cid:18) e − i ( π − φ ) V i ( φ − π , − i ( π − φ (cid:19)(cid:21) − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) = (1 + tan φ ) L γ,d (cid:13)(cid:13)(cid:13)(cid:13)h Re (cid:16) e i ( φ − π ) V i ( φ − π ) (cid:17)i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) by applying (2.3) for eigenvalues in U − κ . Remark 2.2.
We write σ p ( T ) for the point spectrum of the closed operator T . If V is a dilationanalytic real potential, the spectral decomposition theorem implies that σ p ( H ) ∩ (0 , ∞ ) = σ d ( H ( iφ )) ∩ (0 , ∞ ) (2.5)for φ ∈ (0 , min { α, π/ } ) (e.g. [15]). In this sense, embedded eigenvalues (in the essential orabsolutely continuous spectrum [0 , ∞ )) of H are invariant under complex dilation. (In the case of N. Someyama dilation analytic complex potentials, we cannot however use the spectral decomposition theoremand we have no idea if the same is true.) Thus, all eigenvalues which appear newly by complexdilation belong to { z ∈ C : − φ < arg z < } if embedded eigenvalues of H exist. Moreover, then,Lemma 2.2 also holds for embedded eigenvalues and the proof is similar. We derived Corollary 2.1 by complex dilation, but we can produce the following resultsby double complex dilation and Corollary 2.1. Here, C I (resp. C IV ) denotes the first (resp.fourth) quadrant of C . Proposition 2.2.
Let d, γ ≥ . Suppose that V ∈ D ( S α ; C ) with α > π/ . Then,1) Eigenvalue estimate on C I : X λ ∈ σ d ( H ) ∩ C I | λ | γ ≤ L γ,d (cid:13)(cid:13)(cid:13)(cid:13)h Re (cid:16) e πi/ V πi/ (cid:17)i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) ,
2) Eigenvalue estimate on C IV : X λ ∈ σ d ( H ) ∩ C IV | λ | γ ≤ L γ,d (cid:13)(cid:13)(cid:13)(cid:13)h Re (cid:16) e − πi/ V − πi/ (cid:17)i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) . Proof.
We should apply Corollary 2.1 to e λ ( iπ/
4) = iλ ∈ C II if λ ∈ C I . In fact, we have 1)as follows: X λ ∈ σ d ( H ) ∩ C I | λ | γ ≤ L γ,d (cid:13)(cid:13)(cid:13)(cid:13)h Re (cid:16) e iπ/ e iπ/ V iπ/ ,iπ/ (cid:17)i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) = 2 L γ,d (cid:13)(cid:13)(cid:13)(cid:13)h Re (cid:16) e πi/ V πi/ (cid:17)i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) .
2) can be shown in the same way.We write C > for the right-half complex plane. Proposition 2.2 immediately derives thefollowing estimate. Corollary 2.2.
Let d, γ ≥ . Suppose that V ∈ D ( S α ; C ) with α > π/ . Then, theeigenvalues of H on C > \ [0 , ∞ ) are estimated as follows: X λ ∈ σ d ( H ) ∩ C > | λ | γ ≤ L γ,d X ± (cid:13)(cid:13)(cid:13)(cid:13)h Re (cid:16) e ± πi/ V ± πi/ (cid:17)i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) . (2.6)Thus, we can obtain an estimate on all eigenvalues different form Theorem 1.2 asfollows. Corollary 2.3 (cf. [17]) . Let d, γ ≥ . Suppose that V ∈ D ( S α ; C ) with α > π/ . Then,one has X λ ∈ σ d ( H ) | λ | γ ≤ C γ,d k V k γ + d/ L γ + d/ ( R d ; C ) + 2 L γ,d X ± (cid:13)(cid:13)(cid:13)(cid:13)h Re (cid:16) e ± πi/ V ± πi/ (cid:17)i − (cid:13)(cid:13)(cid:13)(cid:13) γ + d/ L γ + d/ ( R d ; C ) . (2.7) Proof.
The desired estimate follows by combining (1.9) and (2.6). nother Application of Dilation Analytic Method for Complex LT-type Estimates We would like to estimate embedded eigenvalues of real Schr¨odinger operators as Lieb–Thirring type. We assume in this subsection that H = H + V with real potentials V haveembedded eigenvalues { λ e } in σ ac ( H ) = [0 , ∞ ). Moreover, we write in this subsection H c (resp. H r ) for the Schr¨odinger operator with the complex (resp. real) potential to preventcunfusion.We first derive the following estimate on isolated eigenvalues on the upper imaginaryaxis i R + := { z ∈ C : Re z = 0 , Im z > } of complex Schr¨odinger operators. Theorem 2.3.
Let d, γ ≥ . Suppose that V ∈ D ( S α ; C ) with α > π/ . Then, theeigenvalues of H c on i R + are estimated as follows: X λ ∈ σ d ( H c ) ∩ i R + | λ | γ ≤ L γ,d k (Im V iπ/ ) + k γ + d/ L γ + d/ ( R d ; C ) . (2.8) Proof.
We take eigenvalues { λ } of H c on i R + . We obtain { λ } = { λ ( iφ ) } by virtue ofLemma 2.1, and { e λ ( iφ ) } = { e i (2 φ ) λ ( iφ ) } by virtue of (2.1). So, setting φ = π/ { e λ ( iπ/ } = { e iπ/ λ ( iπ/ } = { iλ } and all of iλ lie on R − . Also, these iλ correspondto negative real eigenvalues under the potential iV iπ/ . Thus, the standard Lieb–Thirringinequality implies that X λ ∈ σ d ( H c ) ∩ i R + | λ | γ = X e λ ( iπ/ ∈ σ d ( H r ) | e λ ( iπ/ | γ ≤ L γ,d k Re( iV iπ/ ) − k γ + d/ L γ + d/ ( R d ; R ) = L γ,d k (Im V iπ/ ) + k γ + d/ L γ + d/ ( R d ; C ) . Here the above estimate is discussed with the property of multiplicities of eigenvalues:Lemma 2.2. This completes the proof.We next prove the following estimate by double complex dilation in the same way asTheorem 2.2.
Theorem 2.4.
Let d, γ ≥ . Suppose that V ∈ D ( S α ; R ) with α > π/ . Then, theembedded eigenvalues of H r in [0 , ∞ ) are estimated as follows: X λ e ∈ σ pp ( H r ) ∩ [0 , ∞ ) λ γ e ≤ L γ,d k (Re V iπ/ ) + k γ + d/ L γ + d/ ( R d ; R ) . (2.9) Proof.
It is sufficient to consider positive embedded eigenvalues { λ e } of H r , because P λ e ∈ [0 , ∞ ) | λ e | γ = P λ e ∈ (0 , ∞ ) | λ e | γ . We obtain { λ e } = { λ ( iφ ) } by virtue of (2.5), and { e λ ( iφ ) } = { e i (2 φ ) λ ( iφ ) } by virtue of (2.1). So, setting φ = π/ { e λ ( iπ/ } = { iλ e } and all of iλ e lie on i R + . Also, these λ e correspond to purely imaginary eigenvaluesunder the potential iV iπ/ . Thus, (2.8) implies that X λ ∈ σ pp ( H r ) ∩ [0 , ∞ ) λ γ e = X e λ ( iπ/ ∈ σ d ( H r ) | e λ ( iπ/ | γ ≤ L γ,d k Im( iV iπ/ ,iπ/ ) + k γ + d/ L γ + d/ ( R d ; R ) = L γ,d k (Re V iπ/ ) + k γ + d/ L γ + d/ ( R d ; C ) . N. Someyama
Here the above estimate is discussed with the property of multiplicities of (positive) em-bedded eigenvalues: Lemma 2.2 and Remark 2.2. Hence we have gained the proof.
We are interested in how the complex dilation affects the accuracy of eigenvalue estimates.In this appendix, we investigate the L p -norms of dilated potentials via examples. k V k L γ + d/ ( R d ; C ) v.s. k V iφ k L γ + d/ ( R d ; C ) We first argue the comparison of values of k V k L ( R d ; C ) and k V iφ k L ( R d ; C ) . Recall that thereal potential V which belongs to L p ( R d ) for p > max { − ε, d/ } with any ε > H -compact. That is, we only need to show that V ∈ L p ( R d ) if p ≥ p > d/ V is H -compact, as is well known.We feel that complex dilation may increase the norm of the potential in general. (One ofsuch examples can be actually seen in [17]. See also Proposition 3.2 that will be mentionedlater.) However, the following example gives us that our feeling is not always true. Proposition 3.1.
Let d ≥ and γ ≥ max { − d/ , } . Suppose that the potential V isdefined as a multiplication operator with a Gauss-type function V ( x ) = e − cx , c ∈ { z ∈ C : Re z > } (3.1) on R d . Then, V ∈ D ( S α ; C ) for any iφ ∈ S α obeying (Re c ) cos 2 φ > (Im c ) sin 2 φ, (3.2) and the followings hold:1) If Re c ≥ (Re c ) cos 2 φ − (Im c ) sin 2 φ , then one has k V k L γ +1 / ( R d ; C ) ≤ k V iφ k L γ +1 / ( R d ; C ) .
2) If Re c ≤ (Re c ) cos 2 φ − (Im c ) sin 2 φ , then one has k V iφ k L γ +1 / ( R d ; C ) ≤ k V k L γ +1 / ( R d ; C ) . Proof.
It is not difficult to see that V is dilation analytic on S α for all γ ≥ max { − d/ , } with any d ≥
1. It is however sufficient to prove this proposition for d = 1 by virtue ofthe exponential law. We assume (3.2). Then, V iφ ∈ L γ +1 / ( R ; C ) and we have k V k γ +1 / L γ +1 / ( R ; C ) = Z ∞−∞ | e − cx | γ +1 / d x = (cid:18) π (Re c )( γ + 1 / (cid:19) / , k V iφ k γ +1 / L γ +1 / ( R ; C ) = Z ∞−∞ | e − c ( e iφ x ) | γ +1 / d x = (cid:18) π { (Re c ) cos 2 φ − (Im c ) sin 2 φ } ( γ + 1 / (cid:19) / . (3.3)Hence, the proof of this theorem completes. nother Application of Dilation Analytic Method for Complex LT-type Estimates k V iφ k L γ + d/ ( R d ; C ) We finally investigate whether k V iφ k L γ + d/ ( R d ; C ) is monotonic with respect to the dilationangle φ . We feel that the more complex dilation we give, the bigger the values of normsof dilation analytic potentials may be. In fact, we can see an example that affirms ourfeeling as follows. Proposition 3.2 (cf. [17]) . Let d = 1 and γ ≥ / . We define the potential V as amultiplication operator by V ( x ) = c (1 + x ) s , s > γ + 1 , c ∈ C . Then, V ∈ D ( S α ; C ) and {k V iφ k L γ +1 / ( R ; C ) } φ ∈ [0 ,π/ , iφ ∈ S α , is always monotone in-creasing.Proof. It is easy to see that V ∈ L γ +1 / ( R ; C ) and V is H -compact, if s > / (2 γ + 1) and γ ≥ /
2. Since | V iφ ( x ) | γ +1 / = | c | γ +1 / ( x + 2(cos 2 φ ) x + 1) s (2 γ +1) ≤ C γ x s (2 γ +1) (3.4)for a suitable constant C γ > γ , we also have V iφ ∈ L γ +1 / ( R ; C ) becauseof s > / (2 γ + 1). It is not difficult to see that V is dilation analytic from the above. Wenow consider the function F ( φ ) := 2 x cos 2 φ + ( x + 1) with respect to φ by fixing x ∈ R .Since F is monotone decreasing on [0 , π/ not always true as follows. Proposition 3.3.
Let d = 1 and γ ≥ / . For V ∈ D ( S α ; C ) defined as a multiplicationoperator with a Gauss-type function (3.1) for any iφ ∈ S α obeying (3.2), the followingshold:1) If Im c > , then {k V iφ k L γ +1 / ( R ; C ) } φ is monotone increasing.2) If Im c < and φ ∈ [0 , p ) (resp. [ p, π/ ), then {k V iφ k L γ +1 / ( R ; C ) } φ is monotonedecreasing (resp. monotone increasing). Here p := 12 Arctan (cid:18) − Im c Re c (cid:19) . (3.5) Proof.
We consider the function F ( φ ) := (Re c ) cos 2 φ − (Im c ) sin 2 φ with respect to φ ∈ [0 , π/ c >
0. Since we have F ′ ( φ ) = − { (Re c ) sin 2 φ + (Im c ) cos 2 φ } , weobtain the critical point p defined by (3.5) by solving F ′ ( φ ) = 0.1) We assume Im c >
0. Then, p < F (0) = Re c > φ ↑ π/ F ( φ ) = − a < . (3.7)2 N. Someyama
Thus, F is monotone decreasing on [0 , π/ {k V iφ k L γ +1 / ( R ; C ) } φ is monotone increasing on [0 , π/ c <
0. Then, p >
0, (3.6), (3.7) and F ( p ) = (Re c ) cos (cid:18) Arctan (cid:18) − Im c Re c (cid:19)(cid:19) − (Im c ) sin (cid:18) Arctan (cid:18) − Im c Re c (cid:19)(cid:19) > c >
0. Thus, F is monotone decreasing on [0 , p ) and is monotone increasingon [ p, π/ {k V iφ k L γ +1 / ( R ; C ) } φ is monotone increasing on [0 , p )and is monotone decreasing on [ p, π/ Acknowledgement
The author would like to thank referees for giving him suitable and valuable advice. He alsoappreciates the researchers who listened to and commented my presentation of the present paperat various conferences.
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Norihiro Someyama
Shin-yo-ji Buddhist Temple, 5-44-4 Minamisenju, Arakawa-ku, Tokyo 116-0003 JapanE-mail: [email protected]
ORCID iD: https://orcid.org/0000-0001-7579-5352He received a M.Sc. degree from Gakushuin University in 2014 and completed the Ph.D pro-gram without a Ph.D. degree the same university in 2017. He is a head priest of Shin-yo-ji BuddhistTemple in Japan. His research interests are the spectral theory of Schr¨odinger operators and thetheory of Schr¨odinger equations on the fuzzy spacetime. He received the Member EncouragementAward of Biomedical Fuzzy System Association for his lecture entitled ‘
Characteristic Analysisof Fuzzy Graph and its Application IV ’ in November 2018 and the Excellent Presentation Awardof National Congress of Theoretical and Applied Mechanics / JSCE Applied Mechanics Sympo-sium for his lecture entitled ‘