Electronic Journal of Differential Equations, Vol. 211 211, No. 111, pp. 1 9. ISSN: 172-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GODON TYPE THEOEM FO MEASUE PETUBATION CHISTIAN SEIFET Abstract. Generalizing the concept of Gordon potentials to measures we prove a version of Gordon s theorem for measures as potentials and show absence of eigenvalues for these one-dimensional Schrödinger operators. 1. Introduction According to [2], the one-dimensional Schrödinger operator H = + V has no eigenvalues if the potential V L 1,loc can be approximated by periodic potentials in a suitable sense. The aim of this paper is to generalize this result to measures µ instead of potential functions V ; i.e., to more singular potentials. Although all statements remain valid for complex measures we only focus on real but signed measures µ, since we are interested in self-adjoint operators. In the remaining part of this section we explain the situation and define the operator in question. We also describe the class of measures we are concerned with. Section 2 provides all the tools we need to prove the main theorem: H = +µ has no eigenvalues for suitable µ. In section 3 we show some examples for Schrödinger operators with measures as potentials. We consider a Schrödinger operator of the form H = + µ on L 2. Here, µ = µ + µ is a signed Borel measure on with locally finite total variation µ. We define H via form methods. To this end, we need to establish form boundedness of µ. Therefore, we restrict the class of measures we want to consider. Definition 1.1. A signed Borel measure µ on is called uniformly locally bounded, if µ loc := sup µ [x, x + 1] <. x We call µ a Gordon measure if µ is uniformly locally bounded and if there exists a sequence µ m m N of uniformly locally bounded periodic Borel measures with period sequence p m such that p m and for all C we have lim m ecpm µ µ m [ p m, 2p m ] = ; i.e., µ m approximates µ on increasing intervals. Here, a Borel measure is p- periodic, if µ = µ + p. 2 Mathematics Subject Classification. 34L5, 34L4, 81Q1. Key words and phrases. Schrödinger operators; eigenvalue problem; quasiperiodic measure potentials. c 211 Texas State University - San Marcos. Submitted May 31, 211. Published August 29, 211. 1
2 C. SEIFET EJDE-211/111 Clearly, every generalized Gordon potential V L 1,loc as defined in [2] induces a Gordon measure µ = V λ, where λ is the Lebegue measure on. Therefore, also every Gordon potential see the original work [4] induces a Gordon measure. Lemma 1.2. Let µ be a uniformly locally bounded measure. Then µ is -form bounded, and for all < c < 1 there is γ such that u 2 d µ c u 2 2 + γ u 2 2 u W2 1. Proof. For δ, 1 and n Z we have u 2,[nδ,n+1δ] 4δ u 2 L 2nδ,n+1δ + 4 δ u 2 L 2nδ,n+1δ by Sobolev s inequality. Now, we estimate u 2 d µ = n Z n+1δ nδ u 2 d µ n Z u 2,[nδ,n+1δ] µ loc µ loc n Z 4δ u 2 L + 4 2nδ,n+1δ δ u 2 L 2nδ,n+1δ = 4δ µ loc u 2 2 + 4 µ loc u 2 δ 2. Let µ be a Gordon measure and define Dτ := W2 1, τu, v := u v + uv dµ. Then τ is a closed symmetric semibounded form. Let H be the associated selfadjoint operator. In [1], Ben Amor and emling introduced a direct approach for defining the Schrödinger operator H = + µ. Since we will use some of their results we sum up the main ideas: For u W1,loc 1 define Au L 1,loc by where x Aux := u x ut dµt := { [,x] x ut dµt, ut dµt if x, ut dµt if x <. x, Clearly, Au is only defined as an L 1,loc -element. We define the operator T in L 2 by DT := { u L 2 ; u, Au W 1 1,loc, Au L 2 }, T u := Au. Lemma 1.3. H T.
EJDE-211/111 GODON TYPE THEOEM 3 Proof. Let u DH. Then u W2 1 W1,loc 1 and Au L 1,loc. Let ϕ Cc Dτ. Using Fubini s Theorem, we compute x Auxϕ x dx = u x ut dµt ϕ x dx x = u xϕ x dx ut dµtϕ x dx = u xϕ x dx + [t,,t ϕ x dxut dµt = u xϕ x dx + utϕt dµt + = u ϕ + uxϕx dx = τu, ϕ = Hu ϕ = Huxϕx dx. ϕ x dxut dµt utϕt dµt Hence, Au = Hu L 2. We conclude that Au W1,loc 1 and therefore u DT, T u = Au = Hu. emark 1.4. For u DH we obtain u x = Aux + x ut dµt for a.a. x. Since Au W1,loc 1 and x x ut dµt is continuous at all x with µ{x} =, u is continuous at x for all x \ sptµ p, where µ p is the point measure part of µ. 2. Absence of eigenvalues We show that H has no eigenvalues. The proof is based on two observations. The first one is a stability result and will be achieved in Lemma 2.6, the second one is an estimate of the solution for periodic measure perturbations, see Lemma 2.8. As in [2] we start with a Gronwall Lemma, but in a more general version for locally finite measures. For the proof, see [3]. Lemma 2.1 Gronwall. Let µ be a locally finite Borel measure on [,, u L 1,loc [,, µ and α : [, [, measurable. Suppose, that ux αx + us dµs x. Then [,x] ux αx + αs exp µ[s, x] dµs x. [,x] For x we abbreviate := [x, x ], t := [t, x] [x, t] t. Let µ be uniformly locally bounded. Then µ x + 1 µ loc x.
4 C. SEIFET EJDE-211/111 Furthermore, if µ is periodic and locally bounded, µ is uniformly locally bounded. Let H := + µ and E. Then u W1,loc 1 = DA is a solution of Hu = Eu, if Au = Eu in the sense of distributions i.e., u satisfies the eigenvalue equation but without being an L 2 -function. Lemma 2.2. Let µ 1, µ 2 be two uniformly locally bounded measures, E and u 1 and u 2 solutions of H 1 u 1 = Eu 1, H 2 u 2 = Eu 2 subject to u 1 = u 2, u 1+ = u 2+, u 1 2 + u 1+ 2 = 1. Then there are C, C such that for all x u1 x u2 x 1x u 2x C + u 2 t d µ 1 µ 2 t I x + C C + u 2 d µ 1 µ 2 e Cλ+ µ1 Eλ Ixt dλ + µ 1 Eλ t. I t Proof. Write and u 1 x u 2 x = x u 1t u 2t dt u 1x u 2x = u 1+ u 2+ u 1 µ 1 {} u 2 µ 2 {} + x u 1 t dµ 1 t x = u 2 µ 2 {} µ 1 {} + x u 2 t dµ 1 µ 2 t + u 2 t dµ 2 t x x Eu 1 t u 2 t dt u 1 t u 2 t dµ 1 Eλt. Hence, x u1 x u 2 x u 1x u = + dµ 2x u 2 µ 2 {} µ 1 {} u 2 t 1 µ 2 t x 1 u1 t u + 2 t λ 1 u 1t u d t. 2t µ 1 Eλ We conclude, that u1 x u2 x 1x u 2x C + u 2 t d µ 1 µ 2 t I x u1 t u2 t + C dλ + 1t u µ1 Eλ t. 2t An application of Lemma 2.1 with αx = C + u 2 t d µ 1 µ 2 t and µ = Cλ + µ 1 Eλ yields the assertion. emark 2.3. egarding the proof of Lemma 2.2 we can further estimate C u 2 µ 1 µ 2 x.
EJDE-211/111 GODON TYPE THEOEM 5 Lemma 2.4. Let E and u be a solution of u = Eu. Then there is C such that u x Ce C x for all x. In the following lemmas and proofs the constant C may change from line to line, but we will always state the dependence on the important quantities. Lemma 2.5. Let µ 1 be a locally bounded p-periodic measure, E, u 1 a solution of H 1 u 1 = Eu 1. Then there is C such that u 1 x Ce C x x. Proof. Let u be a solution of u = Eu subject to the same boundary conditions at as u 1. By Lemma 2.2 we have u 1 x u x C + u t d µ 1 t I x + C C + u s d µ 1 s e Cλ+ µ1 Eλ Ixt dλ + µ 1 Eλ t I t C + µ 1 Ce C x + C + C µ 1 I t e C t e λ+ µ1 Eλ Ixt dλ + µ 1 Eλ t C + C µ 1 e C x 1 + e λ+ µ1 Eλ Ix λ + µ 1 Eλ. Since µ 1 is periodic and locally bounded it is uniformly locally bounded and we have µ 1 x + 1 µ 1 loc. Furthermore, µ 1 Eλ is periodic and uniformly locally bounded, so µ 1 Eλ x + 1 µ 1 Eλ loc. We conclude that C + C x + 1 µ 1 loc e C x u 1 x u x 1 + e x +11+ µ1 Eλ loc x + 11 + µ 1 Eλ loc Ce C x, where C is depending on E, µ 1 loc and µ 1 Eλ loc. Hence, u 1 x u 1 x u x + u x Ce C x. Lemma 2.6. Let µ be a Gordon measure and µ m the p m -periodic approximants, E. Let u be a solution of Hu = Eu, u m a solution of H m u m = Eu m for m N obeying the same boundary conditions at. Then there is C such that ux u x um x u mx Ce C x µ µ m x. Proof. By Lemma 2.2 and emark 2.3 we know that ux um x x u um µ µ m + u m t d µ µ m t mx I x + C u m µ µ m I t + u m d µ µ m I t e Cλ+ µ Eλ Ixt dλ + µ Eλ t.
6 C. SEIFET EJDE-211/111 We have M := sup µ m loc <, m N since µ m approximates µ. Hence, also and Lemma 2.5 yields sup µ m Eλ loc < m N u m x Ce C x, where C can be chosen independently of m. Therefore ux um x x u Ce C x µ µ m + Ce C x µ µ m mx 1 + e Cλ+ µ Eλ Ix λ + µ Eλ. Since µ Eλ x + 1 µ Eλ loc, we further estimate ux x um x Ce u C x µ µ m mx where C is depending on µ Eλ loc and of course on M, µ loc and E. Lemma 2.6 can be regarded as a stability or continuity result: if the measures converge in total variation, the corresponding solutions converge as well. Now, we focus on periodic measures and estimate the solutions. This will then be applied to the periodic approximations of our Gordon measure µ. emark 2.7. a Let f, g be two solutions of the equation Hu = Eu. Define their Wronskian by W f, gx := fxg x+ f x+gx. By [1], Proposition 2.5, W f, g is constant. b Let u be a solution of the equation Hu = Eu. Define the transfer matrix T E x mapping u, u + to ux, u x+. Consider now the two solutions u N, u D subject to un 1 ud u N + =, u D + =. 1 Then un x u T E x = D x u N x+ u D x+. We obtain det T E x = W u N, u D x and det T E is constant, hence equals 1 for all x. Lemma 2.8. Let µ be p-periodic and E. Let u be a solution of Hu = Eu subject to u 2 + u + 2 = 1. Then {,, } u p up u2p max p+ u p+ u 1 2p+ 2. The proof of this lemma is completely analoguous to the proof of [2, Lemma 2.2].
EJDE-211/111 GODON TYPE THEOEM 7 Lemma 2.9. Let v L 2 BV loc and assume that for all r > we have Then vx as x. vx vx + r x. Proof. Without restriction, we can assume that v. We prove this lemma by contradiction. Assume that vx does not hold for x. Then we can find δ > and q k in with q k such that vq k δ for all k N. By square integrability of v we have v1 [qk,q k +1] 2. Therefore, we can find a subsequence r n of q k satisfying v1 [rn,r n+1] 2 2 3n/2 Now, Chebyshev s inequality implies n N. λ { x [r n, r n + 1]; vx 2 n} 2 2n v1 [rn,r n+1] 2 2 2 n n N. Denote A n := { x [r n, r n + 1]; vx 2 n} r n [, 1]. Then λa n 2 n and λ n 3 A n λa n 2 2 < 1. n 3 Hence, G := [, 1] \ n 3 A n has positive measure. For r G, r > it follows vr n + r 2 n n 3. Therefore, a contradiction. lim inf n vr n vr n + r δ >, Lemma 2.1. Let µ be a Gordon measure, E, u DH a solution of Hu = Eu. Then ux as x and u x as x. Proof. Since u DH Dτ W2 1 we have ux as x. Lemma 1.3 yields u DT and Au = Hu = Eu. Let r >. Then, for almost all x, u x + r u x = Aux + r Aux + ut dµt Hence, = x+r x x,x+r] Au y dy + ut dµt. x,x+r] x+r u x + r u x E uy dy + ut d µ t x x,x+r] E r u,[x,x+r] + u,[x,x+r] µ [x, x + r] u,[x,x+r] E r + r + 1 µ loc. By Sobolev s inequality, there is C depending on r, but r is fixed anyway such that u,[x,x+r] C u W 1 2 x,x+r x. Thus, u x + r u x x. An application of Lemma 2.9 with v := u yields u x as x. Now, we can state the main result of this paper.
8 C. SEIFET EJDE-211/111 Theorem 2.11. Let µ be a Gordon measure. Then H has no eigenvalues. Proof. Let µ m be the periodic approximants of µ. Let E and u be a solution of Hu = Eu. Let u m be the solutions for the measures µ m. By Lemma 2.6 we find m N such that ux x um x 1 u mx 4 for m m and almost all x [ p m, 2p m ]. By Lemma 2.8 we have lim sup ux 2 + u x 2 1 x 4 >. Hence, u cannot be in DH by Lemma 2.1. 3. Examples emark 3.1 periodic measures. Every locally bounded periodic measure on is a Gordon measure. Thus, for µ := n Z δ n+ 1 the operator H := + µ has 2 no eigenvalues. Some examples of quasi-periodic L 1,loc -potentials can be found in [2]. For a measure µ and x let T x µ := µ x. If µ is periodic with period p, then T p µ = µ. Example 3.2. Let α, 1 \ Q. There is a unique continued fraction expansion α = 1 a 1 + 1 a 2+ 1 a 3 + 1... with a n N. For m N we set = pm q m, where p =, p 1 = 1, p m = a m p m 1 + p m 2, q = 1, q 1 = a 1, q m = a m p m 1 + q m 2. The number α is called Liouville number, if there is B such that α Bm qm. The set of Liouville numbers is a dense G δ. Let ν, ν be 1-periodic measures and assume that there is γ > such that ν x ν [, 1] x γ x. Define µ := ν + ν α and µ m := ν + ν for m N. Then µ m is q m -periodic and µ µ m [ q m, 2q m ] = ν α ν [ q m, 2q m ] = ν α ν [ pm, 2p m ] 2p m 1 n= p m ν α ν [n, n + 1].
EJDE-211/111 GODON TYPE THEOEM 9 Now, we have ν α ν [n, n + 1] = T n ν α ν [, 1] = T n ν α T n ν [, 1] = T n ν α ν [, 1]. With g m,n y := y + α 1y + n and using periodicity of ν we obtain T n ν α = ν g m,n. Hence, ν α ν [n, n + 1] = ν g m,n ν [, 1]. For y [, 1] and n { p m,..., 2p m 1} we have α 1 y + n α 1 2q m Bm qm. Thus, ν g m,n ν [, 1] 2q m Bm qm γ = 2qm B γ m qmγ. We conclude that and therefore for arbitrary C µ µ m [ q m, 2q m ] 3p m 2q m B γ m qmγ e Cqm µ µ m [ q m, 2q m ] m. Hence µ is a Gordon potential and H := + µ does not have any eigenvalues. Acknowledgements. We want to thank Peter Stollmann for advices and hints and especially for providing a proof of Lemma 2.9. Many thanks to the anonymous referee for the useful comments. eferences [1] A. Ben Amor and C. emling; Direct and Inverse Spectral Theory of One-dimensional Schrödinger Operators with Measures. Integr. Equ. Oper. Theory 52 25 395 417. [2] D. Damanik and G. Stolz; A Generalization of Gordon s Theorem and Applications to quasiperiodic Schrödinger Operators. El. J. Diff. Equ. 55 2 1 8. [3] S. N. Ethier and T. G. Kurtz; Markov processes, characterization and convergence. Wiley Series in Probability and Statistics, 25. [4] A. Gordon; On the point spectrum of the one-dimensional Schrödinger operator. Usp. Math. Nauk 31 1976 257 258. [5] S. Klassert, D. Lenz and P. Stollmann; Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals. DCDS-A 294 211 1553 1571. [6] B. Simon; Schrödinger Semigroups. Bull. Amer. Math. Soc. 73 1982 447 526. [7] P. Stollmann and J. Voigt; Perturbation of Dirichlet forms by measures. Pot. An. 5 1996 19 138. Christian Seifert Fakultät Mathematik, Technische Universität Chemnitz, 917 Chemnitz, Germany E-mail address: christian.seifert@mathematik.tu-chemnitz.de