DISPERSION ESTIMATES FOR SPHERICAL SCHRO¨DINGER EQUATIONS: THE EFFECT OF BOUNDARY CONDITIONS 6 1 MARKUSHOLZLEITNER,ALEKSEYKOSTENKO,ANDGERALDTESCHL 0 2 Dedicated with great pleasure toPetru A. Cojuhari on the occasion of his 65th birthday t c O Abstract. We investigate the dependence of the L1 → L∞ dispersive esti- mates for one-dimensional radial Schro¨dinger operators on boundary condi- 0 tionsat0.Incontrasttothecaseofadditiveperturbations,weshowthatthe 3 changeofaboundaryconditionatzeroresultsinthechangeofthedispersive decay estimates if the angular momentum is positive, l ∈ (0,1/2). However, ] fornonpositiveangularmomenta,l∈(−1/2,0],thestandardO(|t|−1/2)decay P remainstrueforallself-adjointrealizations. S . h t a m 1. Introduction [ We are concerned with the one-dimensional Schr¨odinger equation 2 d2 l(l+1) v iψ˙(t,x)=Hαψ(t,x), Hα :=−dx2 + x2 , (t,x)∈R×R+, (1.1) 8 3 with the angular momentum l < 1 and self-adjoint boundary conditions at x=0 6 parameterized by a paramete|r|α 2[0,π) (the definition is given in Section 2, see 1 ∈ (2.1)–(2.2) — for recent discussion of this family of operators see [1, 4]). More 0 precisely,we areinterestedin the dependence ofthe L1 L∞ dispersiveestimates . 1 associated to the evolution group e−itHα on the param→eters α [0,π) and l 0 ∈ ∈ ( 1/2,1/2). 6 − 1 On the whole line such results have a long tradition and we refer to Weder [22], : Goldberg and Schlag [9], Egorova, Kopylova, Marchenko and Teschl [5], as well as v the reviews [10, 18]. On the half line, the case l =0 with a Dirichlet boundary con- i X ditionwastreatedbyWeder[23].Thecaseofgenerall andtheFriedrichsboundary r condition at 0 (α=0 in our notation) a 1 1 limxl((l+1)f(x) xf′(x))=0, l , , (1.2) x→0 − ∈ − 2 2 (cid:16) (cid:17) wasrecentlyconsideredinKovaˇr´ıkandTruc[14]andtheyproved(seeTheorem2.4 in [14]) that ke−itH0kL1(R+)→L∞(R+) =O(|t|−1/2), t→∞. (1.3) It was proved in [13] that this estimate remains true under additive perturbations. More precisely (see [13, Theorem 1.1]), let H = H +q, where the potential q is a 0 2010 Mathematics Subject Classification. Primary35Q41,34L25;Secondary81U30,81Q15. Key words and phrases. Schro¨dingerequation, dispersiveestimates,scattering. Research supported by the Austrian Science Fund (FWF) under Grants No. P26060 and W1245. OpusculaMath.36,no.6,769–786(2016). 1 2 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL real integrable on R function. If in addition + 1 ∞ q(x)dx< and xmax(2,l+1) q(x)dx< , (1.4) | | ∞ | | ∞ Z0 Z1 and there is neither a resonance nor an eigenvalue at 0, then e−itHP (H) = (t−1/2), t . (1.5) c L1(R+)→L∞(R+) O | | →∞ Here P (H) is t(cid:13)he orthogona(cid:13)l projection in L2(R ) onto the continuous spectrum c (cid:13) (cid:13) + of H. The main result of the present paper shows that the decay estimates (1.3) and (1.5)arenolongertrueforα (0,π)ifl (0,1/2).Inotherwords,thismeansthat ∈ ∈ singular rank one perturbations destroy these decay estimates if l (0,1/2) (since ∈ thechangeofaboundaryconditioncanbeconsideredasarankoneperturbationin the resolvent sense). Namely, consider first the operator H , which is associated π/2 with the following boundary condition at x=0: 1 1 limx−l−1(lf(x)+xf′(x))=0, l , . (1.6) x→0 ∈ − 2 2 (cid:16) (cid:17) Theorem 1.1. Let l <1/2. Then | | ke−itHπ/2kL1(R+)→L∞(R+) =O(|t|−1/2), t→∞, (1.7) for all l ( 1/2,0], and ∈ − ke−itHπ/2kL1(R+,max(x−l,1))→L∞(R+,min(xl,1)) =O(|t|−1/2+l), t→∞, (1.8) whenever l (0,1/2). The last estimate is sharp. ∈ In the remaining case α (0,π/2) (π/2,π), the decay estimate is given by the ∈ ∪ the next theorem. Theorem 1.2. Let l <1/2 and α (0,π/2) (π/2,π). Then | | ∈ ∪ ke−itHαPc(Hα)kL1(R+)→L∞(R+) =O(|t|−1/2), t→∞, (1.9) for all l ( 1/2,0], and ∈ − ke−itHαPc(Hα)kL1(R+,max(x−l,1))→L∞(R+,min(xl,1)) =O(|t|−1/2), t→∞, (1.10) whenever l (0,1/2). ∈ Notice that in the case l (0,1/2) we need to consider weighted L1 and L∞ ∈ spaces since functions contained in the domain of H might be unbounded near 0. α Finally, let us briefly outline the content of the paper. In the next section we define the operator H and collect its basic spectral properties. Section 3 contains α the proof of Theorem 1.1. In particular, we compute explicitly the kernel of the evolution group e−itHπ/2 and this enables us to prove (1.7) and (1.8) by using the estimates for Bessel functions J (all necessary facts on Bessel functions are ν contained in Appendix A). Theorem 1.2 is proved in Section 4. Its proof is based onthe useofaversionofthe vanderCorputlemma,whichis giveninAppendix B. Also Appendix B contains necessary facts about the Wiener algebras (R) and 0 (R). In the final section we formulate some sufficient conditions forWa function W f(H) of a 1-D Schr¨odinger operator H to be an integral operator. DISPERSION ESTIMATES: THE EFFECT OF BOUNDARY CONDITIONS 3 2. Self-adjoint realizations and their spectral properties Let l ( 1/2,1/2)and denote by H the maximal operator associated with max ∈ − d2 l(l+1) τ = + −dx2 x2 in L2(R ). Note that τ is limit point at infinity and limit circle at x = 0 since + l <1/2.Therefore,self-adjointrestrictionsofH (orinotherwords,self-adjoint max |re|alizations of τ in L2(R )) form a 1-parameter family. More precisely (see, e.g., + [7] and also [1]), the following limits 1 Γ f := limW (f,xl+1), Γ f := − limW (f,x−l) (2.1) 0 x 1 x x→0 2l+1x→0 existandare finite for all f dom(H ). Self-adjointrestrictions H ofH are max α max ∈ parameterized by the following boundary conditions at x=0: dom(H )= f dom(H ): sin(α)Γ f =cos(α)Γ f , α [0,π). (2.2) α max 1 0 { ∈ } ∈ Note that the case α=0 corresponds to the Friedrichs extension of H =H∗ . min max Let φ(z,x) and θ(z,x) be the fundamental system of solutions of τu=zu given by πx φ(z,x)=Cl−1 2 z−2l4+1Jl+12(√zx), r (2.3) πx z2l4+1 θ(z,x)=C J (√zx), l 2 sin((l+1)π) −l−12 r 2 where J is the Bessel function of order ν (see Appendix A) and ν √π C = . (2.4) l Γ(l+ 3)2l+1 2 The Weyl solution normalized by Γ ψ =1 is given by 0 πx ψ(z,x)=θ(z,x)+m(z)φ(z,x)=Cliz2l4+1 2 Hl(+1)1/2(√zx)∈L2(0,∞), (2.5) r where H(1) is the Hankel function of the first kind [17, Chapter X.2], and ν ( z)l+1/2 m(z)= C2 − , z C R , (2.6) − l sin((l+ 1)π) ∈ \ + 2 is the Weyl function associated with H . Here the branch cut of the root is taken 0 along the negative real axis. Notice that C2 dρ(λ)= πl 1[0,∞)(λ)λl+12dλ (2.7) is the corresponding spectral measure. It follows from (A.1) that x−l φ(z,x)=xl+1(1+o(1)), θ(z,x)= (1+o(1)), 2l+1 as x 0 and, moreover, → Γ θ =Γ φ=1, Γ θ =Γ φ=0. 0 1 1 0 4 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL Set φ (z,x):=cos(α)φ(z,x)+sin(α)θ(z,x), α (2.8) θ (z,x):=cos(α)θ(z,x) sin(α)φ(z,x), α − for all z C. Therefore, W(θ ,φ )=1 and α α ∈ m(z)cos(α)+sin(α) ψ (z,x):=θ (z,x)+m (z)φ (z,x), m (z)= , (2.9) α α α α α cos(α) m(z)sin(α) − is a Weyl solution normalized by W(ψ ,φ )=1. Hence α α φ (z,x)ψ (z,y), x y, α α G (z;x,y)= ≤ (2.10) α (φα(z,x)ψα(z,y), x y, ≥ is the Green’s function of H . The absolutely continuous spectrum remains un- α changed, σ (H )=[0, ), but there is one additional eigenvalue ac α ∞ 2 cot(α)cos(lπ) 2l+1 E = (2.11) α − C2 (cid:18) l (cid:19) if π <α<π. Finally, since 2 Imm(z) Imm (z)= , (2.12) α cos(α) m(z)sin(α)2 | − | we get the absolutely continuous partof the correspondingspectralmeasure of the operator H : α 1 ρ′ (λ)dλ= Imm (λ+i0)dλ α π α (2.13) 1 C2λl+1/2 (λ) = l 1[0,∞) dλ. π(cos(α) C2sin(α)tan(πl)λl+1/2)2+C4sin2(α)λ2l+1 − l l 3. Proof of Theorem 1.1 Similar to the case α = 0 (see [14]), the kernel of the evolution group e−itHπ/2 can be computed explicitly. Lemma 3.1. Let l < 1/2. Then the evolution group e−itHπ/2 is an integral oper- | | ator for all t=0 and its kernel is given by 6 [e−itHπ/2](x,y)= il−21t/2eix24+ty2√xyJ−l−1/2 x2yt , (3.1) (cid:16) (cid:17) for all x, y >0 and t=0. 6 Proof. First, notice that φ (z,x)=θ(z,x), m (z)= 1/m(z), π/2 π/2 − and then define the spectral transformation U: L2(R ) L2(R ;ρ ) by + + π/2 → U: f fˆ, fˆ(λ):= θ(λ,x)f(x)dx, 7→ R Z + for every f L2(R ). Notice that U extends to an isometry on L2(R ) and its inverse U−1:∈L2(cR +;ρ ) L2(R ) is given by + + π/2 + → U−1: g gˇ, gˇ(x):= θ(λ,x)g(λ)dρ (λ), π/2 7→ R Z + DISPERSION ESTIMATES: THE EFFECT OF BOUNDARY CONDITIONS 5 for all g L2(R ;ρ ). Therefore, we get by using (2.3) and (2.13) ∈ c + π/2 (e−(it+ε)Hπ/2f)(x)=(U−1e−(it+ε)λUf)(x)=(U−1e−(it+ε)λfˇ)(x) = θ(λ,x)e−(it+ε)λ θ(λ,y)f(y)dydρ (λ) π/2 R R Z + Z + √xy = e−(it+ε)λ J (√λx)J (√λy)f(y)dydλ. R R 2 −l−21 −l−12 Z +Z + Since l <1/2, (A.1) implies that | | 2l+1/2 J (k) (1+ (k)) (3.2) | −l−1/2 |≤ Γ(1/2 l)kl+1/2 O − as k 0. Noting that f L2(R ) and using (3.2), Fubini’s theorem implies → ∈ c + √xy (e−(it+ε)Hπ/2f)(x)= R f(y) R e−(it+ε)λ 2 J−l−12(√λx)J−l−21(√λy)dλdy. Z + Z + (3.3) The integral √xy ∞ [e−(it+ε)Hπ/2](x,y):= 2 Z0 e−itλJ−l−21(√λx)J−l−21(√λy)dλ (3.4) is knownas Weber’s second exponential integral[21, 13.31](cf. also [6, (4.14.39)]) § and hence (e−(it+ε)Hπ/2f)(x)= ε+1itZ0∞e−4x(2ε++yit2)√2xyI−l−21(cid:16)2(εx+yit)(cid:17)f(y)dy, where I is the modified Bessel function (see [17, Chapter X] and in particular ν formula (10.27.6) there) ∞ (z/2)ν+2n I (z)= =e∓iνπ/2J ( iz), π arg(z) π/2. (3.5) ν ν n!Γ(ν+m+1) ± − ≤ ≤ n=0 X The estimate (A.2) implies J (k) k−1/2(1+ (k−1)) (3.6) −l−1/2 | |≤ O as k . Therefore, there is C >0 which depends only on l and such that →∞ 1+k l √kJ (k) C , k >0. (3.7) | −l−1/2 |≤ k (cid:18) (cid:19) By (3.7) we deduce 2|√ε+xyit|(cid:12)(cid:12)e−4x(2ε++yit2)I−l−12(cid:16)2(εx+yit)(cid:17)(cid:12)(cid:12)≤Cs|ε+1it|(cid:12)(cid:12)1+ 2(εx+yit)(cid:12)(cid:12)l, whichisuniforml(cid:12)(cid:12)y(wrt.ε)boundedoncom(cid:12)(cid:12)pactsetsK (cid:12)(cid:12) R+ R+.Th(cid:12)(cid:12)uswecan apply dominated convergence and hence the claim follow⊂s⊂. × (cid:3) In particular, we immediately arrive at the following estimate. 6 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL Corollary 3.2. Let l <1/2. Then there is a constant C >0 which depends only | | on l and such that the inequality C 2t+xy l [e−itHπ/2](x,y) (3.8) ≤ √2t xy (cid:18) (cid:19) (cid:12) (cid:12) holds for all x, y >0 an(cid:12)d t>0. (cid:12) Proof. Applying (3.7) to (3.1), we arrive at (3.8). (cid:3) Remark 3.3. For any fixed x and y R , we get from (A.1) + ∈ √xy xy −l−1/2 1 xy −l e−itHπ/2(x,y) = (3.9) ∼ 2t 4t t1/2−l 2 (cid:12) (cid:12) (cid:16) (cid:17) (cid:16) (cid:17) Moreover, in vi(cid:12)ew of (A.1) o(cid:12)ne can see that (cid:12) (cid:12) xy −l e−itHπ/2(x,y) cltl−1/2 , (3.10) ≥ 2 (cid:12) (cid:12) (cid:16) (cid:17) whenever xy <t with so(cid:12)me constant c(cid:12)l >0, which depends only on l. (cid:12) (cid:12) Now we are ready to prove our first main result. Proof of Theorem 1.1. If l ( 1/2,0], then ∈ − 2t+xy l 1 xy ≤ (cid:18) (cid:19) for all x,y >0 and t 0. This immediately implies (1.7). ≥ Assume now that l (0,1/2). Clearly, ∈ 2t+xy t =1+2 3tmax(x−1,1)max(y−1,1) xy xy ≤ for all t 1 and x, y >0. Indeed, the latter follows from the weaker estimate ≥ t tmax(x−1,1)max(y−1,1), t 1, x,y >0, xy ≤ ≥ which is equivalent to 1 max(x,1)max(y,1) for all x, y >0. Therefore, ≤ l 2t+xy 3tlmax(x−l,1)max(y−l,1), t 1, x,y >0, xy ≤ ≥ (cid:18) (cid:19) which proves (1.8). Remark 3.3 shows that (1.8) is sharp. (cid:3) 4. Proof of Theorem 1.2 Let us consider the following improper integrals: I (t;x,y):=√xy e−itk2J (kx)J (ky)Imm (k2)k−2ldk, (4.1) 1 l+1 l+1 α R 2 2 Z + I (t;x,y):=√xy e−itk2J (kx)J (ky)Imm (k2)kdk, (4.2) 2 l+1 −l−1 α R 2 2 Z + I (t;x,y):=√xy e−itk2J (kx)J (ky)Imm (k2)k2l+2dk, (4.3) 3 −l−1 −l−1 α R 2 2 Z + where x, y > 0 and t = 0. Moreover, here and below we shall use the convention Imm (k2) := Imm (k6 2 +i0) = lim Imm (k2 +iε) for all k R. Denote the α α ε↓0 α ∈ corresponding integrand by A , that is, I (t)= e−itk2A (k;x,y)dk. Our aim is j j R j + R DISPERSION ESTIMATES: THE EFFECT OF BOUNDARY CONDITIONS 7 to use Lemma B.2 (plus the remarks after this lemma) and hence we need to show thateachA belongstothe Wieneralgebra (R),thatis,coincidewithafunction j W which is the Fourier transform of a finite measure. We also need the following estimates, which follow from (2.13) C2 k 2l+1, α=0, Imm (k2)= l| | k , (4.4) α  cos2(πl) k −2l−1+ (k −4l−2), α=0, →∞ Cl2sin2(α)| | O | | 6 and  Cl2 k 2l+1+ (k 4l+2), α=π/2, Imm (k2)= cos(α)2| | O | | 6 k 0. (4.5) α C−2cos2(πl)k −2l−1, α=π/2, →  l | | 4.1. The integral I1. Consider the function rl+1 ∞ ( r2/4)n J(r):=√rJ (r)= − , r 0. l+12 2l+1/2 n!Γ(ν+n+1) ≥ n=0 X Note that J(r) rl+1 as r 0 and J(r) = 2 sin(r lπ)+O(r−1) as r + ∼ → π − 2 → ∞ (see (A.2)). Moreover, J′(r) rl as r 0 aqnd J′(r) = 2 cos(r lπ)+O(r−1) ∼ → π − 2 as r + (see (A.4)). In particular, J˜(r):=J(r) 2qsin(r lπ) is in H1(R ). → ∞ − π − 2 + Moreover, we can define J(r) for r < 0 such that itqis locally in H1 and J(r) = 2 sin(r lπ) for r < 1. By construction we then have J˜ H1(R) and thus π − 2 − ∈ qJ˜ is the Fourier transform of an integrable function (see Lemma B.3). Moreover, sin(r lπ) is the Fourier transform of the sum of two Dirac delta measures and so − 2 J is the Fourier transformofafinite measure.By scaling,the totalvariationofthe measures corresponding to J(kx) is independent of x. Next consider the function Imm (k2) C2 F(k):= α = l . k 2l+1 (cos(α) C2sin(α)tan(πl)k 2l+1)2+C4sin2(α)k 4l+2 | | − l | | l | | By Corollary B.6, F is in the Wiener algebra (R). 0 W Now it remains to note that I (t)= e−itk2A (k2;x,y)dk = e−itk2J(kx)J(ky)F(k)dk, (4.6) 1 1 R R Z + Z + and applying Lemma B.2 we end up with the estimate I (t;x,y) Ct−1/2, t>0, (4.7) 1 | |≤ with a positive constant C >0 independent of x, y >0. 4.2. The integral I . Assume first that l (0,1/2) and write 2 ∈ χ (k) Imm (k2) A (k2;x,y)=J(kx)Y(ky) l α , 2 χ (ky) χ (k) l l where rl J(r)=√rJ (r), Y(r)=χ (r)√rJ (r), χ (r)= | | . l+12 l −l−12 l 1+ rl | | 8 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL The asymptotic behavior (4.4) and (4.5) of Imm shows that α Imm (k2) k 1+l, k 0, α M(k)= = | | → χl(k) (k −2l−1, k , | | | |→∞ and hence M H1(R), which implies that M is in the Wiener algebra (R). 0 ∈ W We continue J(r), Y(r) to the region r < 0 such that they are continuously differentiable and satisfy 2 πl 2 πl J(r)= sin r , Y(r)= cos r+ , π − 2 π 2 r (cid:18) (cid:19) r (cid:18) (cid:19) forr < 1.ThenJ˜(r):=J(r) 2 sin(r πl)andY˜(r):=Y(r) 2 cos r+ πl − − π − 2 − π 2 are in H1(R). In fact, they are qcontinuously differentiable and henqce it suffices to (cid:0) (cid:1) look at their asymptotic behavior.For r < 1 they are zero and for r >1 they are − O(r−1) andtheir derivativeis O(r−1)as canbe seenfromthe asymptoticbehavior of Bessel functions (see Appendix A). Hence both J and Y are Fourier transforms of finite measures. By scaling the total variation of the measures corresponding to J(kx) and Y(ky) are independent of x and y, respectively. It remains to consider the function χ (k)/χ (ky). Observe that l l χ (k) 1+ ky l 1 y−l h (k):=1 l =1 | | = − =(1 y−l)(1 χ (k)). y,l − χ (ky) − yl+ ky l 1+ k l − − l l | | | | By CorollaryB.6, 1 χ (R). Therefore, applying Lemma B.2, we obtain the l 0 − ∈W following estimate I (t;x,y) Ct−1/2max(1,y−l), t>0, (4.8) 2 | |≤ whenever l (0,1/2). ∈ Consider now the remaining case l ( 1/2,0]. Write ∈ − A (k2;x,y)=J(kx)Y(ky)Imm (k2), 2 α where J(r)=√rJ (r), Y(r)=√rJ (r). l+1 −l−1 2 2 Noting that Y(r) r−l as r 0 and using Lemma B.3, we can continue J and ∼ → Y to the region r < 0 such that both J and Y are Fourier transforms of finite measures. It remains to consider Imm (k2) given by (2.13). However, by Corollary B.6, α this function is in the Wiener algebra (R) and hence applying Lemma B.2, we 0 W end up with the estimate I (t;x,y) Ct−1/2, t>0, (4.9) 2 | |≤ whenever l ( 1/2,0]. ∈ − 4.3. The integral I . Again let us consider two cases. Assume first that l 3 ∈ ( 1/2,0]and then write − A (k2;x,y)=Y(kx)Y(ky)Imm (k2)k2l+1, 3 α where Y(r)=√rJ (r), r>0. −l−1 2 DISPERSION ESTIMATES: THE EFFECT OF BOUNDARY CONDITIONS 9 Notice that C2k4l+2 k 2l+1Imm (k2)= l , | | α (cos(α) C2sin(α)tan(πl)k2l+1)2+C4sin2(α)k4l+2 − l l which is the sum of a constant and a function of the form (B.5), and hence it belongs to the Wiener algebra (R) by Corollary B.6. Arguing as in the previous W subsection and applying Lemma B.2, we arrive at the following estimate I (t;x,y) Ct−1/2, t>0, (4.10) 3 | |≤ whenever l ( 1/2,0]. ∈ − If l (0,1/2), write ∈ χ (k) χ (k) Imm (k2) A (k2;x,y)=Y(kx)Y(ky) l l α , 3 χ (kx)χ (ky) χ2(k) l l l where rl Y(r)=χ (r)√rJ (r), χ (r)= | | . l −l−21 l 1+ rl | | Notice that Imm (k2)k 2l+1 α M(k):= | | χ2(k) l C2 k 2l+2(1+kl)2 = l| | (cos(α) C2sin(α)tan(πl)k 2l+1)2+C4sin2(α)k 4l+2 − l | | l | | Clearly,byCorollaryB.6,M (R).Therefore,similartotheprevioussubsection, ∈W we end up with the estimate I (t;x,y) Ct−1/2max(1,x−l)max(1,y−l), t>0, (4.11) 3 | |≤ whenever l (0,1/2). ∈ 4.4. Proof of Theorem 1.2. We begin with the representation of the integral kernel of the evolution group. Lemma 4.1. Let l <1/2 and α [0,π). Then the evolution group e−itHαPc(Hα) | | ∈ is an integral operator and its kernel is given by 2 [e−itHαP (H )](x,y)= e−itk2φ (k2,x)φ (k2,y)Imm (k2)kdk, (4.12) c α α α α π R Z + where the integral is to be understood as an improper integral. Proof. By (2.3) and (2.8), φ (k2,x)=cos(α)φ(k2,x)+sin(α)θ(k2,x) α πx sin(α) = C−1cos(α)k−l−1/2J (kx)+C kl+1/2 J (kx) , 2 l l+21 l cos(πl) −l−21 r (cid:18) (cid:19) 10 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL and hence π cos2(α) φ (k2,x)φ (k2,y)= √xy k−2l−1J (kx)J (ky) (4.13) α α 2 C2 l+21 l+21 (cid:18) l sin(2α) + (J (kx)J (ky)+J (kx)J (ky)) (4.14) 2cos(πl) l+12 −l−21 −l−12 l+21 sin2(α) +C2k2l+1 J (kx)J (ky) . (4.15) l cos2(πl) −l−12 −l−21 (cid:19) By our considerations in the previous subsections, we have φ (k2,x)φ (k2,y)Imm (k2)k (R) α α α ∈W with norm uniformly bounded for x,y restricted to any compact subset of (0, ). ∞ Moreover, we have e−i(t−iε)HαPc(Hα) e−itHαPc(Hα) as ε 0 in the strong op- → ↓ erator topology. By Lemma C.1, e−i(t−iε)HαPc(Hα) is an integral operator for all ε > 0 and, moreover, the kernel converges uniformly on compact sets by Lemma C.2.Hencee−itHαPc(Hα)isanintegraloperatorwhosekernelisgivenbythelimits of the kernels of the approximating operators,that is, by (4.12). (cid:3) Proof of Theorem 1.2. Combining(4.7),(4.8),(4.9),(4.10)and(4.11),wearriveat the following decay estimate for the kernel of the evolution group 1, l ( 1/2,0], [e−itHαP (H )](x,y) Ct−1/2 ∈ − c α ≤ ×(max(1,x−l)max(1,y−l), l (0,1/2). (cid:12) (cid:12) ∈ (cid:12) (cid:12) (4.16) Thi(cid:12)s completes the proo(cid:12)f of Theorem 1.2. (cid:3) Appendix A. Bessel functions Here we collect basic formulas and information on Bessel functions (see, e.g., [17, 21]). We start with the definition: z ν ∞ ( z2/4)n J (z)= − . (A.1) ν 2 n!Γ(ν+n+1) (cid:16) (cid:17) nX=0 The asymptotic behavior as z is given by | |→∞ 2 J (z)= cos(z νπ/2 π/4)+e|Imz| (z −1) , argz <π. (A.2) ν πz − − O | | | | r (cid:16) (cid:17) Noting that ν ν J′(z)= J (z)+ J (z)=J (z) J (z), (A.3) ν − ν+1 z ν ν−1 − z ν one can show that the derivative of the reminder satisfies ′ πz 1 1 J (z) cos(z νπ π) =e|Imz| (z −1), z . (A.4) ν 2 − − 2 − 4 O | | | |→∞ (cid:18)r (cid:19)