Properties of the Scattering Transform on the Real Line

Size: px
Start display at page:

Download "Properties of the Scattering Transform on the Real Line"

Transcription

1 Journal of Mathematical Analysis and Applications 58, 3 43 (001 doi: /jmaa , available online at on Properties of the Scattering Transform on the Real Line Michael Hitrik Department of Mathematics, University of California, Berkeley, CA hitrik@math.berkeley.edu Submitted by Fritz Gesztesy Received August 8, 000 Continuity properties of the scattering transform associated to the Schrödinger operator on the real line are studied. Stability estimates of Lipschitz type are derived for the scattering and inverse scattering transforms. 001 Academic Press 1. INTRODUCTION The purpose of this paper is to analyze some of the mapping properties of the scattering transform associated to the Schrödinger operator H p u = u + pu on the real line. Here p belongs to the space L + of real-valued measurable potentials such that a 1 + p d < for any a>. We establish Lipschitz type estimates for the scattering and inverse scattering trasforms on bounded sets in L +. The scattering transform S appears naturally in connection with the Marchenko equation. Also, it is precisely this transform that linearizes the Korteweg de Vries (KdV flow. Several authors (see [4, 11, 13] have observed that p and S p have similar behavior at +. Moreover, S p p is continuous and has better bounds at + than p. When p is also well behaved at, there is a simple relation between S p and the scattering matri of p, given essentially by the Fourier transformation. In [17] the range of S on L + was characterized and it was proved that S maps L + homeomorphically onto S L +. In the contet of half-line scattering, a similar observation was made in [19]. Here we shall sharpen these X/01 $35.00 Copyright 001 by Academic Press All rights of reproduction in any form reserved.

2 4 michael hitrik results by establishing the Lipschitz continuity of the scattering and inverse scattering transformations. (See [8, 9] for stability analyses of the Cauchy problem for the KdV equation by means of scattering transforms. It should also be mentioned that stability estimates for potentials directly in terms of their scattering data have been studied in [14, 1] (see also [, 6]. The plan of the paper is as follows. In Section, after recalling the intertwining operators and the scattering transform, we state our main result in Theorem.1. The Lipschitz continuity of the forward transform is then established in Section 3, and that of the inverse transform in Section 4. Throughout the paper we make use of the operator theoretical approach to inverse scattering developed in [17]. This approach has recently been used by the author (see [6, 7] when studying error estimates in inverse scattering and the distribution of resonances in one dimension. Some of the results obtained in these papers will be useful here as well.. THE SCATTERING TRANSFORM AND STATEMENT OF THE RESULTS First, we shall briefly describe the main notation used in the paper. The space L + of real-valued measurable functions p such that p z = z 1 + p d < z R (.1 was already introduced in Section 1. This is a Fréchet space under the seminorms in (.1. We shall also use the Banach space L of real-valued measurable functions p on 0 such that p 0 <, and let L c denote its compleification. For p L, write p 0 = p. We shall now recall the intertwining operators and the scattering transform, as presented in [17]. Consider the Schrödinger operator H p u = u + pu where p L +. Associated to p is the operator A = I + R with H p A = AH 0 (. and y in the support of the distribution kernel of A. The function R y is continuous up to the boundary in the set where y>and R L 1 = R y dy < (.3 for any. Moreover, R L 1 0as +. (Much more precise estimates of R are given later. Apart from functions that are continuous on the whole of R, R y R 0 y

3 properties of the scattering transform 5 where and R 0 y = ( 1 θ + y p t dt (.4 +y / θ + t =1 when t 0 and 0 otherwise. This is the leading term in R, and R satisfies the equation R = R 0 + L p R (.5 where L p is given by L p R y = E y y p R y d dy (.6 E y = 1 when <0 y < and 0 otherwise (.7 The solution of (.5 is obtained by iteration, R = L k p R 0 (.8 k=0 It follows in view of (.4 and (.8 that the function R, for any, depends only on the restriction of p to the interval. Also note that the foregoing construction of R is valid for comple-valued potentials p satisfying (.1. To introduce the scattering transform, the kernel of the Marchenko equation Q y =Q y p is also needed. When p L +, this is a continuous function on R that satisfies Q y =Q y and solves the equation R y +Q y + R t Q t y dt = 0 y> (.9 Note that the part of Q that is linear in p is equal to R = R + 0 = 1 p y dy (.10 Let Q denote also the operator with integral kernel Q y. It is clear from the contet which interpretation is chosen at every formula. For future reference, we shall now state the operator formulation of the Marchenko equation (.9 as given in [17], A I + Q A τ = I (.11

4 6 michael hitrik Here A τ is the transpose of A A τ y =A y. [We remark here that the operator calculus developed in Section 3 of [17] makes it meaningful to study products as in (.11.] It is then well known that we have the following representation for the kernel Q, Q y = 1 +y / q s ds (.1 with q L + (see [13, 15, 11, 17]. This gives the map S p q from L + to L + obtained by solving (.11 for Q. We shall say that S is the scattering transform. According to Theorem 4.9 of [17], S is a homeomorphism between L + and its range. Note that in view of (.10, S may be thought as a nonlinear perturbation of the identity on L +. We refer to Sections and 7 of [17], where it is shown that the KdV equations of all orders are linearized when passing from p to q. We remark that when p is comple valued, the kernel Q obtained by solving (.11 is a function of + y, and the representation (.1 is true. This follows from the proof of Theorem 4.7 in [17]. In what follows, however, we are concerned mainly with real-valued potentials. It follows from (.9, (.10, and (.1 [see also (3.3] that the restriction of q to any interval z depends only on the restriction of p to this set, and, conversely, the restriction of q to z determines p on this interval. Restricting to the positive half-ais, S thus may be regarded as an injective mapping on L. (See also Remarks 5 and 7. Theorem.1. Let p 1 and p L + be two potentials such that p j M j = 1 for some M>0. Then the following holds: 1. There eists a constant C 1 depending only on M such that S p 1 S p C 1 p 1 p. The estimate p 1 p C S p 1 S p is true. Here C = C M depends only on M. Remark 1. In the proofs we shall obtain eplicit bounds for the size of the Lipschitz constants in terms of the bounded sets in question.

5 properties of the scattering transform 7 Remark. It follows easily from (.8, (.9, and (.1 that the mapping S L + L + commutes with the action of the translation group τ h p = p + h h R, so that τ h S = Sτ h. It follows therefore from Theorem.1 that when M is a bounded set in L + we have S p S p 1 z C p p 1 z p 1 p M for any z, where C depends only on z and M. Similarly, it is true that p p 1 z C z M S p S p 1 z p 1 p M for any z R. 3. ESTIMATES FOR THE FORWARD TRANSFORM In this section we shall prove the first part of Theorem.1. In our analysis we make use of the nonincreasing functions f = p t dt g = f t dt (3.1 associated with a potential p L +. Then the estimate R y 1 ( ( ( + y + y f ep g g y> (3. is true, (see [13]. When considering two potentials p 1 and p L +, set δp = p p 1 and f j = p j d g j = f j d j = 1 (3.3 Our starting point is the following result. Although giving only L bounds on the difference of the intertwining kernels, it will be sufficient for our purposes. Lemma 3.1. Suppose that p 1 p L +, and let I + R j j = 1, bethe corresponding intertwining operators. Then the estimate δr y 1 (g ep 1 +g ( + y g δp t dt y > (3.4 holds. Here δr = R R 1.

6 8 michael hitrik Proof. From the proof of Lemma in [13], recall that if a function H j u v j = 1, is defined by H j u v =R j u v u + v, then H j satisfies H j u v = 1 ( v p j t dt + p j α β H j α β dβ dα u u 0 v>0 (3.5 Writing δh = H H 1,weget δh u v = 1 ( v δp t dt + p α β δh α β dβ dα u u 0 ( v + δp α β H 1 α β dβ dα v > 0 (3.6 Consider u ρ u v = Bound (3. means that and, therefore, ρ u v 1 Setting 1 0 u ( v 0 δp α β H 1 α β dβ dα H j u v 1 f j u ep g j u v g j u u u 1 u 1 ( = 1 it follows that h u v 1 ( v f 1 α δp α β ep g 1 α β g 1 α dβ dα 0 ( v f 1 α ep g 1 α v g 1 α ( f 1 α v ep g 1 α v g 1 α f 1 t ep g 1 t dt u v ( ep g 1 u v 1 u h u v = 1 u 0 δp s ds u v δp s ds u v δp t dt + ρ u v δp α β dβ dα δp s ds α v dα δp t dt + 1 ep g 1 u v 1 δp t dt u v 1 ep g 1 u v u v δp t dt

7 properties of the scattering transform 9 Solving (3.6 by iteration as in [13], Lemma 3.1.1, yields δh u v 1 ( ep g 1 u v δp t dt ep g u v g u u v This completes the proof. Remark 3. It follows from the proof of Lemma 3.1 that the following sharper bound is valid: R y R 1 y 1 ( ep δp t dt + ep g 1 1 +y / ( g g ( +y y> δp t dt We also remark that an L 1 estimate on δr can be obtained from the bound δr y 1 ( ( + y δp t dt + e g1 f 1 t δp t dt +y / ( ( + y ep g g which follows from a small modification of the proof of Lemma 3.1. Setting p 1 = 0 in these bounds, (3. is recovered. In addition to basic estimate (3. and Lemma 3.1, similar bounds are required for B = A 1, the second intertwining operator. These bounds will be needed to estimate the kernel Q in (.11 in terms of the potential. It follows from the results in Section 3 of [17] that y in the support of B, and B = I + T with T continuous in this set. Moreover, T = k=1 1 k R k. (See [17] for the definition and properties of the relevant Fréchet algebra of distribution kernels containing R and T. Proposition 3.. The estimate T y 1 ( ( ( + y + y f ep g g y y> (3.7 is true. Moreover, if p 1 and p L +, and T j correspond to p j j = 1, then T y T 1 y 1 ( δp s ds +y / ( ( ( + y + y ep g 1 + g (3.8

8 30 michael hitrik Proof. We shall make use of the intertwining relation (.. Approimating the potential p L + by test functions and making use of some arguments of Section 4 in [17], clearly B satisfies BH p = H 0 B, and the corresponding equation for the function T is then T y =T 0 y E y y p y T y d dy (3.9 where ( 1 T 0 y = θ + y p t dt +y / and E y has been defined in (.7. It follows, as in the proof of Lemma of [13], that if a function G u v is defined by G u v =T u v u + v, then G obeys ( v G u v = p t dt p α+β G α β dβ dα v>0 (3.10 u u Equation (3.10 is solved by iteration, and we have 0 T y = T n y (3.11 n=1 where T n is homogeneous of degree n in p. Working with (3.10, a simple inductive argument demonstrates that the estimate T n y 1 ( + y g + y / g y n 1 f n 1! y (3.1 holds, and this gives (3.7. Note also that T + 0 = 1 p t dt (3.13 When proving (3.8, we argue in the same way as in the proof of Lemma 3.1, using (3.10 instead of (3.5, and this completes the proof. Remark 4. Note that the bound (3.7 on the kernel T is better than the bound (3. on R, which reflects the conveity of the function g. Note also that estimate (3.8 gives both L 1 and L bounds on T T 1. Remark 5. Recall the comple Banach space L c introduced in Section. It follows from the proof of Proposition 3. (see also Theorem 4.1 in [17] that the map p T y =T y p is analytic from L c to the Banach space N. Here, following [17], we first introduce the space V of complevalued functions Q in y 0 y 0, tending to 0 as + y,

9 properties of the scattering transform 31 that are continuous up to the boundary in the sets ± y > 0 and such that if then Q z w = sup Q y (3.14 z y z +y / w Q = sup Q y + Q 0 w dw < ( y 0 0 This is a comple Banach space with the norm just defined, and we then let N be the closed subspace comprising all Q V such that y in their support. Because T y p is given as the power series (3.11, where T n is a homogeneous polynomial of degree n in p L c with values in N, the analyticity of p T follows from (3.1, which shows that the series converges in N, and the convergence is uniform on bounded sets in L c.we have the same result for the map p R y =R y p. From (.5 and (.6, recall that R y R 0 y = 1 ( y+ p R y dy y+ d (3.16 It is then easy to see that the function R R 0 is of class C 1 up to the boundary in the set where y>. Set R 1 0 y = R y and R 0 1 y = y R y when y>and = 0 when >y. The function U y =R 1 0 y R 0 1 y (3.17 restricted to the set y>etends continuously to the closure of this set. Moreover, a simple computation using (3.16 or, alternatively, (3.5 shows that U y = p R y + d (3.18 and using (3., we obtain U y 1 ( ( ( + y + y f f ep g g y> (3.19 The following related bound is needed to establish the Lipschitz continuity. Lemma 3.3. If U j correspond to p j L + j = 1, then U y U 1 y 1 ( ( + y δp t dt f 1 ep g 1 + ep g 1 +g f

10 3 michael hitrik Proof. From (3.18, U y U 1 y = p ( R y + R 1 y + d ( p p 1 R 1 y + d and the result then follows in view of (3. and Lemma 3.1. We now come to estimate the difference of the kernels Q Q 1 corresponding to p j L + j = 1. The following proposition is the starting point. We remark that a different proof of estimate (3.0 is given in Chapter 3 of [13]. However, because the argument here is short, and in what follows the identity (3.4 is needed, we present the proof. Proposition 3.4. The estimate Q j 1 f j ep g j cosh g j j = 1 (3.0 is true. Proof. Recall from (.11 that A j I + Q j A τ j = I j = 1 (3.1 Writing A 1 j = I + T j, we have Q j = A 1 ( j A 1 τ j I = I + Tj I + Tj τ I = T j + Tj τ + T j Tj τ (3. Both sides here are continuous up to the boundary in the sets ± y > 0, and because Tj τ is supported in the set where y, (3. gives Q j + 0 =Q j =T j ( T j Tj τ (3.3, because this func- Here we have written ( T j Tj τ tion is continuous in R. This gives + 0 = ( Tj T τ j Q j =T j and using (3.7 and (3.13, we obtain Q j 1 f j + 1 ( + z f 4 j ( ( ( + z ep g j 1 f j f j Tj z dz (3.4 ( + z g j z f j dz ( ( ( + z + z f j ep g j dz

11 properties of the scattering transform 33 = 1 f j f j ep ( g j 1 = 1 ( 1 f j ep g j + 1 = 1 f j ep g j cosh g j The proof is complete. Remark 6. Note that the map T T + T τ + TT τ is analytic from N to V. Here we have used the Banach spaces introduced in Remark 5. Because V Q y Q + 0 L 1 0 is continuous, we conclude in view of that remark and Proposition 3.4 that the mapping p Q =Q p is analytic from L c to L 1 0. Lemma 3.5. Q Q 1 1 δp t dt 1 + ep ( g 1 +g sinh g 1 +sinh g Proof. In view of (3.13 and (3.4, Q Q 1 = T y T 1 y T 1 y +T y dy 1 δp t dt A straightforward estimate using (3.7 and (3.8 then gives the result. We now come to prove the first part of Theorem.1. In doing so, we need the following representation for q = S p (see, e.g., Chapter 4 in [11]: q p =4R + 0 Q U y Q y dy (3.5 Here the function U has been introduced in (3.17. Note that (3.5 shows that q p is an absolutely continuous function vanishing at +. When proving (3.5, we observe that it follows from (.11 that y in the support of (the kernel of the operator I + R I + Q, and because this is a continuous function up to the boundary in the sets ± y > 0, we have R + 0 +Q + RQ =0 Here we may also notice that RQ y is continuous in R. Writing RQ = R y Q 0 y dy, we obtain RQ = R + 0 Q + U y Q y dy

12 34 michael hitrik and because R + 0 +Q = p q the representation (3.5 follows. If q j = S p j correspond to p j L + j = 1, then we have, in view of (3.5, h h 1 =I 1 + I + I 3 + I 4 (3.6 where we have written h j =q j p j, j = 1, and I j are given by I 1 = 4 R + 0 R Q I = 4R Q Q 1 I 3 = U y U 1 y Q y dy and I 4 = U 1 y Q y Q 1 y dy with R j Q j and U j corresponding to p j j = 1. Now it follows from Proposition 3.4 and Lemma 3.5 that and I 1 ep g cosh g f δp t dt (3.7 I 1 + ep g 1 +g sinh g 1 +sinh g f 1 δp t dt (3.8 Using Lemma 3.3 and Proposition 3.4, we further obtain ( I 3 ep g 1 +g sinh g δp t dt f 1 ( + ep g 1 +g sinh g δp t dt f (3.9 and, finally, combining (3.19 with Lemma 3.5, we arrive at I ep g 1 +g sinh g 1 +sinh g ( ep g 1 f 1 δp t dt (3.30 Because g j g j 0 p j j = 1, when 0, combining (3.6 with (3.7 (3.30 yields ( h h 1 C δp t dt f 1 +f > 0

13 properties of the scattering transform 35 It follows from (3.7 (3.30 that C = e M 3e M sinh M + cosh M can be taken when M is such that p j M j = 1. Because 1 + f j p j 0, and ( δp t dt d = t δp t dt p p 1 0 recalling the definition of h j, we finally obtain 0 q q 1 C p p 1 when p j M j = 1, and C depends only on M. This completes the proof of the first part of Theorem.1. Remark 7. We shall finish this section by making some remarks concerning analyticity properties of S. When doing this, we shall use the notation and the spaces introduced in Remark 5. Consider first the map S 1 L c L c given by ( S 1 p = p t dt Q p (3.31 When p j L c j = 0 1 p 1 1 and λ C Q p 0 + λp 1 is entirely analytic in λ with values in L 1 0. Therefore, Q p 0 + λp 1 = Q j p 0 p 1 λ j where Cauchy s inequalities together with (3.0 give that j=0 Q j L R j e p 0 +R sinh p 0 +R R > 0 It follows that with convergence in L c, we have S 1 p 0 + λp 1 = S j p 0 p 1 λ j where S j L c equals p 0 t dt Q j + p 1 t dt Q j 1, j 0, with Q 1 = 0. Because S 1 is locally bounded, analyticity of S 1 as a map from L c to L c follows (see [18]. It further follows from (3.18 that we have the estimate ( U z w p z dz R z w z and because z z p z dz p z 0, we also get that V p R, where V y = 1 + U y. Using this bound together with the fact that R y p depends analytically on p, we obtain that the map L c p j=0

14 36 michael hitrik V y =V y p N is continuously differentiable, and hence analytic. Because the analyticity of p Q y p has already been observed in Remark 6, it follows that the map p U y p Q y p dy is analytic from L c to L c. From (3.5, we conclude that q = S p L depends analytically on p L. 4. ESTIMATES FOR THE INVERSE TRANSFORM In this section we shall continue to work with potentials in the space L +. Recall the notation p = p d When p 1, p L + are such that p j M, we derive Lipschitz estimates for p p 1 in terms of S p S p 1, thereby completing the proof of Theorem.1. When doing this, we shall use the notation C for various positive constants that depend only on M. Our starting point is the following representation for the potential: ( + y p q =R R y q dy (4.1 Here the intertwining operator A = I + R corresponding to p and q = S p is related by the Marchenko equation I + Q = A 1 A τ 1 (4. and Q is defined by q as in (.1. It follows from Section 4 in [17] that each side in (4.1 is a continuous function of p L + with values in L +. When proving (4.1, it therefore may be assumed that p L + = p L+ p j L + j = 1, so that in particular p C R, and all derivatives tend to 0 at. It follows from the results of [17] that these regularity conditions imply the same regularity of q. From Section 4 of [17], we also know that the kernel of A I + Q is a smooth function up to the boundary in the sets ± y > 0, and because y in its support, we have α I + R I + Q + 0 =0 (4.3 for any multi-inde α = α α y. When α = 0 1, α RQ + 0 = 1 ( + y R y q dy 4 4

15 properties of the scattering transform 37 Now a computation using (.5 and (3.16 gives α R y = 1 ( + y 4 p + 1 p R y + d so that 1 p R + y d y> α R + 0 = 1 4 p + 1 p R + 0 d = 1 ( 4 p +1 p y dy 8 Here in the last step we have used that R + 0 = 1 p y dy In view of (4.3, the representation (4.1 now follows. Note that a representation similar to (4.1 can be found in [3], where it is related to the trace formula of inverse scattering. We now consider two potentials p 1 and p L + and let the triple R j Q j q j correspond to p j j = 1. We also write ρ j = p j q j and P j = Using (4.1, we get where I j = I j j = 1, is given by and I = p j t dt = R j + 0 j = 1 ρ ρ 1 =I 1 + I (4.4 I 1 = 1 P P 1 P +P 1 + dt ( + t R t R 1 t q ( ( ( + t + t R 1 t q q 1 Because 1 + P j p j 0 j = 1, we get that I 1 C P P 1 L 1 0 dt The required bound on I 1 in terms of q q 1 then becomes the contents of the following lemma.

16 38 michael hitrik Lemma 4.1. The estimate P P 1 L 1 0 C q q 1 (4.5 is true. Proof. We shall make use of the following identity proved in Proposition 4.1 of [6]: 1 P 1 P = A 1 Q Q 1 A τ (4.6 where A j = I + R j, j = 1. We then have 1 P 1 P = Q Q 1 + R 1 y Q y Q 1 y dy + R y Q y Q 1 y dy } + R 1 y { R z Q y z Q 1 y z dz dy (4.7 Because Q y Q 1 y 1 δq t dt +y / where δq = q q 1, using (3., we get R j y Q y Q 1 y dy 1 R j L 1 1 ep g j 1 δq t dt j = 1 δq t dt Estimating the last integral on the right side of (4.7, we see that { } R 1 y R z Q y z Q 1 y z dz dy From (4.7, we obtain 1 R 1 L 1 R L 1 δq t dt 1 ep g 1 1 ep g 1 e p 1 e p 1 1 P P 1 C and this completes the proof. δq t dt δq t dt δq t dt 0 0

17 properties of the scattering transform 39 We now come to estimate I, the second term in (4.4. When doing this, when R, we consider the operators Qj f y = Q j y t f t dt y > f C 0 j = 1 (4.8 on. These are then Hilbert Schmidt operators on L, and from [17] recall that the fact that q j is in the range of S means precisely that I + Qj has a positive lower bound on L for any j = 1. The operators I + Qj are thus invertible on L, and it is also well known (see [11, 17] that I + Qj is then an isomorphism on L 1 j = 1. When deriving bounds on R y R 1 y, the norm of ( I + Q 1 on L 1 must be controlled. To this end, we introduce the operators ( R j f y = R j y t f t dt y > and ( R τ j f y = y y R j t y f t dt y > on L p p= 1. Then we have (see [13] and [14] that ( I + Q 1 j = I + R τ j I + R j j = 1 (4.9 in these spaces. A straightforward estimate using (3. shows that I + R j L 1 L1 1 + g j ep g j e g j j = 1 where L 1 = L1. The same estimate is valid for the operator I + ( R τ j, and in view of (4.9, we get ( I + Qj 1 L ep 4g 1 L1 j (4.10 Setting S j y = Q j y and R j y =R j y y> j= 1, from (.9 we obtain R R 1 = ( I +Q 1S ( I +Q 1 1S1 = ( I +Q 1 ( S S 1 + (( I +Q 1 ( I +Q 1 1 S1 = ( I +Q 1 ( S S 1 + ( I +Q 1 ( Q 1 Q ( I +Q 1 1S1 = ( I +Q 1 [ S S 1 + ( Q 1 Q R1 ] (4.11 Because Q y Q 1 y 1 +y / δq t dt

18 40 michael hitrik it follows that the absolute value of the epression in the square brackets in (4.11 does not eceed 1 δq s ds + 1 ( R 1 t δq s ds dt y+t / +y / 1 +y / where in the last step we had that δq s ds ( R 1 L 1 eg 1 R j L 1 ep g j 1 +y / δq s ds which follows from (3.. Combining this with (4.10 and (4.11, we get the L 1 bound R R 1 L 1 ep 4g ep g 1 Now we have R y R 1 y = Q 1 y Q y + + and using (4.1, we obtain Q 1 Q y y R 1 y dy t δq t dt (4.1 Q y y R 1 y R y dy y> (4.13 R y R 1 y Q y Q 1 y y / Q y y Q 1 y y R 1 y dy Q y y R y R 1 y dy δq t dt 1 + R 1 L 1 ( q t dt R R 1 L 1 +y / ( ep 4g +g 1 t δq t dt + 1 ep g 1 +y / q s ds +y / δq t dt (4.14

19 properties of the scattering transform 41 Write where I = I 1 + I I 1 = R t R 1 t q ( + t dt and ( ( ( + t + t I = R 1 t q q 1 dt Bound (4.14 then gives ( I 1 ep g 1 δq t dt q t dt ( + ep 4g +g 1 q s ds t δq t dt and we immediately get I 1 C ( q + q q q 1 (4.15 It follows now from [13] that q C ( p + p. Therefore, to complete the proof of Theorem.1, it remains to estimate I. When doing this, note that in view of (3., the estimate I ep g 1 f 1 δq t dt is true, which gives that I C q q 1. Combining this with (4.4, Lemma 4.1, and (4.15 completes the proof of Theorem.1. Remark 8. We shall finally give a representation formula for the scattering transform of a potential p L + that vanishes near +. It will then be convenient to change the notation somewhat, so that we consider p L 1 loc 0 and take the zero etension of p to the negative half-ais. Then ˇp =p L + vanishes for >0, and associated with p we introduce the scattering transform from the left, S, S p =S ˇp We also introduce the kernels R and T so that R y p = R y ˇp and I + T = I + R 1. It is then true that S p vanishes on the interval 0, and the restriction of p to any interval of the form 0 a a > 0, determines and is determined by the restriction of S p to this set. We claim that S p = 4 y T y > 0 (4.16 y=0

20 4 michael hitrik When proving (4.16, recall the Marchenko equation from the left, where I + R I + Q I + R τ =I Q y = 1 +y / S p t dt (4.17 Arguing as in the proof of Proposition 3.4, we get y Q y =T y + T y T y y dy >y (4.18 Here also note that y in the support of T y in view of the support properties of p, so that in particular T y y 0 =0, y 0. Then by (4.17 and (4.18, we get (4.16. Representation (4.16 is analogous to the epression for the A-amplitude associated with p appears in Section 9 of [5]. We recall here that the A-amplitude has been introduced in [0] as a basic ingredient of an approach to the inverse spectral theory for half-line Schrödinger operators. In [0], A was related to the Weyl m-function by means of a Fourier Laplace transformation. To recall the epression for A given in [5], set K y =R y R y 0 y and define the kernel L so that I + L = I + K 1. It is then true that A α = y L α y (4.19 y=0 (see [5]. It follows from (.5, (4.16, and (4.19 that the leading terms in S p and A agree and are both equal to p. (Alternatively, this also follows from (3.5 and the series epansion for A given in Lemma. of [0]. We also have that S p A is (absolutely continuous, and bounds on this function can now be derived using the results of [0] and this paper. ACKNOWLEDGMENTS I am grateful to Anders Melin and Barry Simon for very useful discussions. This paper was prepared when I was affiliated to the École Polytechnique. I should like to thank its Centre de Mathématiques for the generous hospitality.

21 properties of the scattering transform 43 REFERENCES 1. Z. S. Agranovich and V. A. Marchenko, The Inverse Problem of Scattering Theory, Gordon and Breach, New York, T. Aktosun, Stability of the Marchenko inversion, Inverse Probl. 3 (1987, P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 3 (1979, L. D. Faddeev, Inverse problem of quantum theory of scattering II, J. Sov. Math. 5 (1976, F. Gesztesy and B. Simon, A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure, Ann. Math. 15 (000, M. Hitrik, Stability of an inverse problem in potential scattering on the real line, Comm. P.D.E. 5 (000, M. Hitrik, Bounds on scattering poles in one dimension, Commun. Math. Phys. 08 (1999, A. Jeffrey and L. Zhang, Stability of the Cauchy problem for the Korteweg de Vries equation I, Appl. Anal. 50 (1993, A. Jeffrey and L. Zhang, Stability of the Cauchy problem for the Korteweg de Vries equation II, Appl. Anal. 5 (1994, T. Kappeler and E. Trubowitz, Properties of the scattering map, Comm. Math. Helv. 61 (1986, B. M. Levitan, Inverse Sturm Liouville Problems, VNU Science Press, Utrecht, D. Sh. Lundina and V. A. Marchenko, A refinement of the inequalities which characterize the stability of the inverse problem of scattering theory, Mat. Sb. (N.S. 78 (1969, [In Russian] 13. V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser Verlag, Basel, V. A. Marchenko, Stability of the inverse problem of scattering theory, Mat. Sb. (N.S. 77 (1968, [In Russian] 15. V. A. Marchenko, Epansion in eigenfunctions of non-self-adjoint singular differential operators of second order, AMS Transl. 5 (1963, V. A. Marchenko, Some questions in the theory of one-dimensional linear differential operators of the second order I, AMS Transl. 101 (1973, A. Melin, Operator methods for inverse scattering on the real line, Comm. P.D.E. 10 (1985, J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Boston, A. G. Ramm, Inverse scattering on half-line, J. Math. Anal. Appl. 133 (1988, B. Simon, A new approach to inverse spectral theory, I. Fundamental formalism, Ann. Math. 150 (1999,

An Inverse Problem for the Matrix Schrödinger Equation

An Inverse Problem for the Matrix Schrödinger Equation Journal of Mathematical Analysis and Applications 267, 564 575 (22) doi:1.16/jmaa.21.7792, available online at http://www.idealibrary.com on An Inverse Problem for the Matrix Schrödinger Equation Robert

More information

CONSTRUCTION OF THE HALF-LINE POTENTIAL FROM THE JOST FUNCTION

CONSTRUCTION OF THE HALF-LINE POTENTIAL FROM THE JOST FUNCTION CONSTRUCTION OF THE HALF-LINE POTENTIAL FROM THE JOST FUNCTION Tuncay Aktosun Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762 Abstract: For the one-dimensional

More information

PLEASE SCROLL DOWN FOR ARTICLE

PLEASE SCROLL DOWN FOR ARTICLE This article was downloaded by: [Ingrid, Beltita] On: 2 March 29 Access details: Access Details: [subscription number 9976743] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered

More information

Piecewise Smooth Solutions to the Burgers-Hilbert Equation

Piecewise Smooth Solutions to the Burgers-Hilbert Equation Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang

More information

ZEROS OF THE JOST FUNCTION FOR A CLASS OF EXPONENTIALLY DECAYING POTENTIALS. 1. Introduction

ZEROS OF THE JOST FUNCTION FOR A CLASS OF EXPONENTIALLY DECAYING POTENTIALS. 1. Introduction Electronic Journal of Differential Equations, Vol. 20052005), No. 45, pp. 9. ISSN: 072-669. URL: http://ejde.math.tstate.edu or http://ejde.math.unt.edu ftp ejde.math.tstate.edu login: ftp) ZEROS OF THE

More information

On the least values of L p -norms for the Kontorovich Lebedev transform and its convolution

On the least values of L p -norms for the Kontorovich Lebedev transform and its convolution Journal of Approimation Theory 131 4 31 4 www.elsevier.com/locate/jat On the least values of L p -norms for the Kontorovich Lebedev transform and its convolution Semyon B. Yakubovich Department of Pure

More information

Rolle s Theorem for Polynomials of Degree Four in a Hilbert Space 1

Rolle s Theorem for Polynomials of Degree Four in a Hilbert Space 1 Journal of Mathematical Analysis and Applications 265, 322 33 (2002) doi:0.006/jmaa.200.7708, available online at http://www.idealibrary.com on Rolle s Theorem for Polynomials of Degree Four in a Hilbert

More information

Commutator estimates in the operator L p -spaces.

Commutator estimates in the operator L p -spaces. Commutator estimates in the operator L p -spaces. Denis Potapov and Fyodor Sukochev Abstract We consider commutator estimates in non-commutative (operator) L p -spaces associated with general semi-finite

More information

Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005

Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER

More information

Chapter One. The Calderón-Zygmund Theory I: Ellipticity

Chapter One. The Calderón-Zygmund Theory I: Ellipticity Chapter One The Calderón-Zygmund Theory I: Ellipticity Our story begins with a classical situation: convolution with homogeneous, Calderón- Zygmund ( kernels on R n. Let S n 1 R n denote the unit sphere

More information

COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED TIME SCALES. Hüseyin Tuna

COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED TIME SCALES. Hüseyin Tuna Indian J. Pure Appl. Math., 47(3): 535-544, September 2016 c Indian National Science Academy DOI: 10.1007/s13226-016-0196-1 COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED

More information

ON POISSON BRACKETS COMPATIBLE WITH ALGEBRAIC GEOMETRY AND KORTEWEG DE VRIES DYNAMICS ON THE SET OF FINITE-ZONE POTENTIALS

ON POISSON BRACKETS COMPATIBLE WITH ALGEBRAIC GEOMETRY AND KORTEWEG DE VRIES DYNAMICS ON THE SET OF FINITE-ZONE POTENTIALS ON POISSON BRACKETS COMPATIBLE WITH ALGEBRAIC GEOMETRY AND KORTEWEG DE VRIES DYNAMICS ON THE SET OF FINITE-ZONE POTENTIALS A. P. VESELOV AND S. P. NOVIKOV I. Some information regarding finite-zone potentials.

More information

On compact operators

On compact operators On compact operators Alen Aleanderian * Abstract In this basic note, we consider some basic properties of compact operators. We also consider the spectrum of compact operators on Hilbert spaces. A basic

More information

IMRN Inverse Spectral Analysis with Partial Information on the Potential III. Updating Boundary Conditions Rafael del Rio Fritz Gesztesy

IMRN Inverse Spectral Analysis with Partial Information on the Potential III. Updating Boundary Conditions Rafael del Rio Fritz Gesztesy IMRN International Mathematics Research Notices 1997, No. 15 Inverse Spectral Analysis with Partial Information on the Potential, III. Updating Boundary Conditions Rafael del Rio, Fritz Gesztesy, and Barry

More information

SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES

SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES MARIA J. ESTEBAN 1 AND MICHAEL LOSS Abstract. Distinguished selfadjoint extension of Dirac operators are constructed for a class of potentials

More information

Recall that any inner product space V has an associated norm defined by

Recall that any inner product space V has an associated norm defined by Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner

More information

A uniqueness result for one-dimensional inverse scattering

A uniqueness result for one-dimensional inverse scattering Math. Nachr. 285, No. 8 9, 941 948 (2012) / DOI 10.1002/mana.201100101 A uniqueness result for one-dimensional inverse scattering C. Bennewitz 1,B.M.Brown 2, and R. Weikard 3 1 Department of Mathematics,

More information

DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT

DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT Electronic Journal of Differential Equations, Vol. 217 (217, No. 227, pp. 1 42. ISSN: 172-6691. URL: http://ejde.math.tstate.edu or http://ejde.math.unt.edu DIFFERENTIAL EQUATIONS WITH A DIFFERENCE QUOTIENT

More information

ON THE BOUNDARY CONTROL APPROACH TO INVESE SPECTRAL AND SCATTERING THEORY FOR SCHRODINGER OPERATORS

ON THE BOUNDARY CONTROL APPROACH TO INVESE SPECTRAL AND SCATTERING THEORY FOR SCHRODINGER OPERATORS ON THE BOUNDARY CONTROL APPROACH TO INVESE SPECTRAL AND SCATTERING THEORY FOR SCHRODINGER OPERATORS ALEXEI RYBKIN Abstract. We link boundary control theory and inverse spectral theory for the Schrödinger

More information

An inverse problem for the non-selfadjoint matrix Sturm Liouville equation on the half-line

An inverse problem for the non-selfadjoint matrix Sturm Liouville equation on the half-line c de Gruyter 2007 J. Inv. Ill-Posed Problems 15 (2007), 1 14 DOI 10.1515 / JIIP.2007.002 An inverse problem for the non-selfadjoint matrix Sturm Liouville equation on the half-line G. Freiling and V. Yurko

More information

1.4 The Jacobian of a map

1.4 The Jacobian of a map 1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p

More information

The following definition is fundamental.

The following definition is fundamental. 1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic

More information

FREMLIN TENSOR PRODUCTS OF CONCAVIFICATIONS OF BANACH LATTICES

FREMLIN TENSOR PRODUCTS OF CONCAVIFICATIONS OF BANACH LATTICES FREMLIN TENSOR PRODUCTS OF CONCAVIFICATIONS OF BANACH LATTICES VLADIMIR G. TROITSKY AND OMID ZABETI Abstract. Suppose that E is a uniformly complete vector lattice and p 1,..., p n are positive reals.

More information

Inverse spectral problems for first order integro-differential operators

Inverse spectral problems for first order integro-differential operators Yurko Boundary Value Problems 217 217:98 DOI 1.1186/s13661-17-831-8 R E S E A R C H Open Access Inverse spectral problems for first order integro-differential operators Vjacheslav Yurko * * Correspondence:

More information

MINIMAL NORMAL AND COMMUTING COMPLETIONS

MINIMAL NORMAL AND COMMUTING COMPLETIONS INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 4, Number 1, Pages 5 59 c 8 Institute for Scientific Computing and Information MINIMAL NORMAL AND COMMUTING COMPLETIONS DAVID P KIMSEY AND

More information

Global well-posedness for KdV in Sobolev spaces of negative index

Global well-posedness for KdV in Sobolev spaces of negative index Electronic Journal of Differential Equations, Vol. (), No. 6, pp. 7. ISSN: 7-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Global well-posedness for

More information

Integrable operator representations of R 2 q and SL q (2, R)

Integrable operator representations of R 2 q and SL q (2, R) Seminar Sophus Lie 2 (1992) 17 114 Integrable operator representations of R 2 q and SL q (2, R) Konrad Schmüdgen 1. Introduction Unbounded self-adjoint or normal operators in Hilbert space which satisfy

More information

This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author s institution, sharing

More information

Semigroups and Linear Partial Differential Equations with Delay

Semigroups and Linear Partial Differential Equations with Delay Journal of Mathematical Analysis and Applications 264, 1 2 (21 doi:1.16/jmaa.21.675, available online at http://www.idealibrary.com on Semigroups and Linear Partial Differential Equations with Delay András

More information

Scattering Theory. Erik Koelink

Scattering Theory. Erik Koelink Scattering Theory Erik Koelink e.koelink@math.ru.nl Spring 28, minor corrections 28 ii Contents Introduction. Scattering theory...................................2 Inverse scattering method.............................

More information

ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION. In Ho Jeon and B. P. Duggal. 1. Introduction

ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION. In Ho Jeon and B. P. Duggal. 1. Introduction J. Korean Math. Soc. 41 (2004), No. 4, pp. 617 627 ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION In Ho Jeon and B. P. Duggal Abstract. Let A denote the class of bounded linear Hilbert space operators with

More information

Research Article Some Estimates of Certain Subnormal and Hyponormal Derivations

Research Article Some Estimates of Certain Subnormal and Hyponormal Derivations Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 362409, 6 pages doi:10.1155/2008/362409 Research Article Some Estimates of Certain

More information

Presenter: Noriyoshi Fukaya

Presenter: Noriyoshi Fukaya Y. Martel, F. Merle, and T.-P. Tsai, Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gkdv Equations, Comm. Math. Phys. 31 (00), 347-373. Presenter: Noriyoshi

More information

Chapter 3: Baire category and open mapping theorems

Chapter 3: Baire category and open mapping theorems MA3421 2016 17 Chapter 3: Baire category and open mapping theorems A number of the major results rely on completeness via the Baire category theorem. 3.1 The Baire category theorem 3.1.1 Definition. A

More information

Ann. Polon. Math., 95, N1,(2009),

Ann. Polon. Math., 95, N1,(2009), Ann. Polon. Math., 95, N1,(29), 77-93. Email: nguyenhs@math.ksu.edu Corresponding author. Email: ramm@math.ksu.edu 1 Dynamical systems method for solving linear finite-rank operator equations N. S. Hoang

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

Euler Equations: local existence

Euler Equations: local existence Euler Equations: local existence Mat 529, Lesson 2. 1 Active scalars formulation We start with a lemma. Lemma 1. Assume that w is a magnetization variable, i.e. t w + u w + ( u) w = 0. If u = Pw then u

More information

Journal of Mathematical Analysis and Applications

Journal of Mathematical Analysis and Applications J. Math. Anal. Appl. 381 (2011) 506 512 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Inverse indefinite Sturm Liouville problems

More information

Where is matrix multiplication locally open?

Where is matrix multiplication locally open? Linear Algebra and its Applications 517 (2017) 167 176 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Where is matrix multiplication locally open?

More information

Complex symmetric operators

Complex symmetric operators Complex symmetric operators Stephan Ramon Garcia 1 Complex symmetric operators This section is a brief introduction to complex symmetric operators, a certain class of Hilbert space operators which arise

More information

arxiv:math/ v1 [math.sp] 18 Jan 2003

arxiv:math/ v1 [math.sp] 18 Jan 2003 INVERSE SPECTRAL PROBLEMS FOR STURM-LIOUVILLE OPERATORS WITH SINGULAR POTENTIALS, II. RECONSTRUCTION BY TWO SPECTRA arxiv:math/31193v1 [math.sp] 18 Jan 23 ROSTYSLAV O. HRYNIV AND YAROSLAV V. MYKYTYUK Abstract.

More information

ON THE CHERN CHARACTER OF A THETA-SUMMABLE FREDHOLM MODULE.

ON THE CHERN CHARACTER OF A THETA-SUMMABLE FREDHOLM MODULE. ON THE CHERN CHARACTER OF A THETA-SUMMABLE FREDHOLM MODULE. Ezra Getzler and András Szenes Department of Mathematics, Harvard University, Cambridge, Mass. 02138 USA In [3], Connes defines the notion of

More information

An Asymptotic Property of Schachermayer s Space under Renorming

An Asymptotic Property of Schachermayer s Space under Renorming Journal of Mathematical Analysis and Applications 50, 670 680 000) doi:10.1006/jmaa.000.7104, available online at http://www.idealibrary.com on An Asymptotic Property of Schachermayer s Space under Renorming

More information

A note on the σ-algebra of cylinder sets and all that

A note on the σ-algebra of cylinder sets and all that A note on the σ-algebra of cylinder sets and all that José Luis Silva CCM, Univ. da Madeira, P-9000 Funchal Madeira BiBoS, Univ. of Bielefeld, Germany (luis@dragoeiro.uma.pt) September 1999 Abstract In

More information

08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms

08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms (February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops

More information

ON THE DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS

ON THE DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume, Number, Pages S -9939(XX- ON THE DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS N. S. HOANG AND A. G. RAMM (Communicated

More information

A PERIODICITY PROBLEM FOR THE KORTEWEG DE VRIES AND STURM LIOUVILLE EQUATIONS. THEIR CONNECTION WITH ALGEBRAIC GEOMETRY

A PERIODICITY PROBLEM FOR THE KORTEWEG DE VRIES AND STURM LIOUVILLE EQUATIONS. THEIR CONNECTION WITH ALGEBRAIC GEOMETRY A PERIODICITY PROBLEM FOR THE KORTEWEG DE VRIES AND STURM LIOUVILLE EQUATIONS. THEIR CONNECTION WITH ALGEBRAIC GEOMETRY B. A. DUBROVIN AND S. P. NOVIKOV 1. As was shown in the remarkable communication

More information

Journal of Algebra 226, (2000) doi: /jabr , available online at on. Artin Level Modules.

Journal of Algebra 226, (2000) doi: /jabr , available online at   on. Artin Level Modules. Journal of Algebra 226, 361 374 (2000) doi:10.1006/jabr.1999.8185, available online at http://www.idealibrary.com on Artin Level Modules Mats Boij Department of Mathematics, KTH, S 100 44 Stockholm, Sweden

More information

Sobolev spaces. May 18

Sobolev spaces. May 18 Sobolev spaces May 18 2015 1 Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references

More information

TADAHIRO OH 0, 3 8 (T R), (1.5) The result in [2] is in fact stated for time-periodic functions: 0, 1 3 (T 2 ). (1.4)

TADAHIRO OH 0, 3 8 (T R), (1.5) The result in [2] is in fact stated for time-periodic functions: 0, 1 3 (T 2 ). (1.4) PERIODIC L 4 -STRICHARTZ ESTIMATE FOR KDV TADAHIRO OH 1. Introduction In [], Bourgain proved global well-posedness of the periodic KdV in L T): u t + u xxx + uu x 0, x, t) T R. 1.1) The key ingredient

More information

SPECTRAL PROPERTIES OF JACOBI MATRICES OF CERTAIN BIRTH AND DEATH PROCESSES

SPECTRAL PROPERTIES OF JACOBI MATRICES OF CERTAIN BIRTH AND DEATH PROCESSES J. OPERATOR THEORY 56:2(2006), 377 390 Copyright by THETA, 2006 SPECTRAL PROPERTIES OF JACOBI MATRICES OF CERTAIN BIRTH AND DEATH PROCESSES JAOUAD SAHBANI Communicated by Şerban Strătilă ABSTRACT. We show

More information

APPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES

APPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES APPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES S. H. KULKARNI AND G. RAMESH Abstract. Let T be a densely defined closed linear operator between

More information

Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on

Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department

More information

A G Ramm, Implicit Function Theorem via the DSM, Nonlinear Analysis: Theory, Methods and Appl., 72, N3-4, (2010),

A G Ramm, Implicit Function Theorem via the DSM, Nonlinear Analysis: Theory, Methods and Appl., 72, N3-4, (2010), A G Ramm, Implicit Function Theorem via the DSM, Nonlinear Analysis: Theory, Methods and Appl., 72, N3-4, (21), 1916-1921. 1 Implicit Function Theorem via the DSM A G Ramm Department of Mathematics Kansas

More information

S chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1.

S chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1. Sep. 1 9 Intuitively, the solution u to the Poisson equation S chauder Theory u = f 1 should have better regularity than the right hand side f. In particular one expects u to be twice more differentiable

More information

How large is the class of operator equations solvable by a DSM Newton-type method?

How large is the class of operator equations solvable by a DSM Newton-type method? This is the author s final, peer-reviewed manuscript as accepted for publication. The publisher-formatted version may be available through the publisher s web site or your institution s library. How large

More information

EXPLICIT UPPER BOUNDS FOR THE SPECTRAL DISTANCE OF TWO TRACE CLASS OPERATORS

EXPLICIT UPPER BOUNDS FOR THE SPECTRAL DISTANCE OF TWO TRACE CLASS OPERATORS EXPLICIT UPPER BOUNDS FOR THE SPECTRAL DISTANCE OF TWO TRACE CLASS OPERATORS OSCAR F. BANDTLOW AND AYŞE GÜVEN Abstract. Given two trace class operators A and B on a separable Hilbert space we provide an

More information

The Gram matrix in inner product modules over C -algebras

The Gram matrix in inner product modules over C -algebras The Gram matrix in inner product modules over C -algebras Ljiljana Arambašić (joint work with D. Bakić and M.S. Moslehian) Department of Mathematics University of Zagreb Applied Linear Algebra May 24 28,

More information

Division of Physics, Astronomy and Mathematics California Institute of Technology, Pasadena, CA 91125

Division of Physics, Astronomy and Mathematics California Institute of Technology, Pasadena, CA 91125 OPERATORS WITH SINGULAR CONTINUOUS SPECTRUM: II. RANK ONE OPERATORS R. del Rio 1,, N. Makarov 2 and B. Simon Division of Physics, Astronomy and Mathematics California Institute of Technology, 25-7 Pasadena,

More information

FRAMES AND TIME-FREQUENCY ANALYSIS

FRAMES AND TIME-FREQUENCY ANALYSIS FRAMES AND TIME-FREQUENCY ANALYSIS LECTURE 5: MODULATION SPACES AND APPLICATIONS Christopher Heil Georgia Tech heil@math.gatech.edu http://www.math.gatech.edu/ heil READING For background on Banach spaces,

More information

Computations of Critical Groups at a Degenerate Critical Point for Strongly Indefinite Functionals

Computations of Critical Groups at a Degenerate Critical Point for Strongly Indefinite Functionals Journal of Mathematical Analysis and Applications 256, 462 477 (2001) doi:10.1006/jmaa.2000.7292, available online at http://www.idealibrary.com on Computations of Critical Groups at a Degenerate Critical

More information

4 Hilbert spaces. The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan

4 Hilbert spaces. The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan Wir müssen wissen, wir werden wissen. David Hilbert We now continue to study a special class of Banach spaces,

More information

AALBORG UNIVERSITY. Schrödinger operators on the half line: Resolvent expansions and the Fermi Golden Rule at thresholds

AALBORG UNIVERSITY. Schrödinger operators on the half line: Resolvent expansions and the Fermi Golden Rule at thresholds AALBORG UNIVERSITY Schrödinger operators on the half line: Resolvent expansions and the Fermi Golden Rule at thresholds by Arne Jensen and Gheorghe Nenciu R-2005-34 November 2005 Department of Mathematical

More information

Meromorphic continuation of zeta functions associated to elliptic operators

Meromorphic continuation of zeta functions associated to elliptic operators Meromorphic continuation of zeta functions associated to elliptic operators Nigel Higson November 9, 006 Abstract We give a new proof of the meromorphic continuation property of zeta functions associated

More information

INVERSE SCATTERING TRANSFORM, KdV, AND SOLITONS

INVERSE SCATTERING TRANSFORM, KdV, AND SOLITONS INVERSE SCATTERING TRANSFORM, KdV, AND SOLITONS Tuncay Aktosun Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762, USA Abstract: In this review paper, the

More information

Derivatives of Harmonic Bergman and Bloch Functions on the Ball

Derivatives of Harmonic Bergman and Bloch Functions on the Ball Journal of Mathematical Analysis and Applications 26, 1 123 (21) doi:1.16/jmaa.2.7438, available online at http://www.idealibrary.com on Derivatives of Harmonic ergman and loch Functions on the all oo

More information

The elliptic sinh-gordon equation in the half plane

The elliptic sinh-gordon equation in the half plane Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 8 25), 63 73 Research Article The elliptic sinh-gordon equation in the half plane Guenbo Hwang Department of Mathematics, Daegu University, Gyeongsan

More information

1 Math 241A-B Homework Problem List for F2015 and W2016

1 Math 241A-B Homework Problem List for F2015 and W2016 1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let

More information

BLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE

BLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE BLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE OSCAR BLASCO, PABLO GALINDO, AND ALEJANDRO MIRALLES Abstract. The Bloch space has been studied on the open unit disk of C and some

More information

arxiv:math/ v1 [math.ap] 28 Oct 2005

arxiv:math/ v1 [math.ap] 28 Oct 2005 arxiv:math/050643v [math.ap] 28 Oct 2005 A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation Hans Lindblad and Avy Soffer University of California at San Diego and Rutgers

More information

Math 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.

Math 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces. Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,

More information

WAVELET EXPANSIONS OF DISTRIBUTIONS

WAVELET EXPANSIONS OF DISTRIBUTIONS WAVELET EXPANSIONS OF DISTRIBUTIONS JASSON VINDAS Abstract. These are lecture notes of a talk at the School of Mathematics of the National University of Costa Rica. The aim is to present a wavelet expansion

More information

On an inverse problem for Sturm-Liouville Equation

On an inverse problem for Sturm-Liouville Equation EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 1, No. 3, 17, 535-543 ISSN 137-5543 www.ejpam.com Published by New York Business Global On an inverse problem for Sturm-Liouville Equation Döne Karahan

More information

The Calderon-Vaillancourt Theorem

The Calderon-Vaillancourt Theorem The Calderon-Vaillancourt Theorem What follows is a completely self contained proof of the Calderon-Vaillancourt Theorem on the L 2 boundedness of pseudo-differential operators. 1 The result Definition

More information

Polarization constant K(n, X) = 1 for entire functions of exponential type

Polarization constant K(n, X) = 1 for entire functions of exponential type Int. J. Nonlinear Anal. Appl. 6 (2015) No. 2, 35-45 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.252 Polarization constant K(n, X) = 1 for entire functions of exponential type A.

More information

STURM-LIOUVILLE OPERATORS AND APPLICATIONS (AMS CHELSEA PUBLISHING) BY VLADIMIR A. MARCHENKO

STURM-LIOUVILLE OPERATORS AND APPLICATIONS (AMS CHELSEA PUBLISHING) BY VLADIMIR A. MARCHENKO STURM-LIOUVILLE OPERATORS AND APPLICATIONS (AMS CHELSEA PUBLISHING) BY VLADIMIR A. MARCHENKO DOWNLOAD EBOOK : STURM-LIOUVILLE OPERATORS AND APPLICATIONS (AMS Click link bellow and free register to download

More information

CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION

CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION HANS CHRISTIANSON Abstract. This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation.

More information

Determinant of the Schrödinger Operator on a Metric Graph

Determinant of the Schrödinger Operator on a Metric Graph Contemporary Mathematics Volume 00, XXXX Determinant of the Schrödinger Operator on a Metric Graph Leonid Friedlander Abstract. In the paper, we derive a formula for computing the determinant of a Schrödinger

More information

ON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS. Srdjan Petrović

ON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS. Srdjan Petrović ON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS Srdjan Petrović Abstract. In this paper we show that every power bounded operator weighted shift with commuting normal weights is similar to a contraction.

More information

SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT

SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC

More information

Hilbert space methods for quantum mechanics. S. Richard

Hilbert space methods for quantum mechanics. S. Richard Hilbert space methods for quantum mechanics S. Richard Spring Semester 2016 2 Contents 1 Hilbert space and bounded linear operators 5 1.1 Hilbert space................................ 5 1.2 Vector-valued

More information

Means of unitaries, conjugations, and the Friedrichs operator

Means of unitaries, conjugations, and the Friedrichs operator J. Math. Anal. Appl. 335 (2007) 941 947 www.elsevier.com/locate/jmaa Means of unitaries, conjugations, and the Friedrichs operator Stephan Ramon Garcia Department of Mathematics, Pomona College, Claremont,

More information

Energy transfer model and large periodic boundary value problem for the quintic NLS

Energy transfer model and large periodic boundary value problem for the quintic NLS Energy transfer model and large periodic boundary value problem for the quintic NS Hideo Takaoka Department of Mathematics, Kobe University 1 ntroduction This note is based on a talk given at the conference

More information

Geometric Multigrid Methods

Geometric Multigrid Methods Geometric Multigrid Methods Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University IMA Tutorial: Fast Solution Techniques November 28, 2010 Ideas

More information

On Generalized Set-Valued Variational Inclusions

On Generalized Set-Valued Variational Inclusions Journal of Mathematical Analysis and Applications 26, 23 240 (200) doi:0.006/jmaa.200.7493, available online at http://www.idealibrary.com on On Generalized Set-Valued Variational Inclusions Li-Wei Liu

More information

HOMOLOGICAL LOCAL LINKING

HOMOLOGICAL LOCAL LINKING HOMOLOGICAL LOCAL LINKING KANISHKA PERERA Abstract. We generalize the notion of local linking to include certain cases where the functional does not have a local splitting near the origin. Applications

More information

SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY

SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS

More information

Math Subject GRE Questions

Math Subject GRE Questions Math Subject GRE Questions Calculus and Differential Equations 1. If f() = e e, then [f ()] 2 [f()] 2 equals (a) 4 (b) 4e 2 (c) 2e (d) 2 (e) 2e 2. An integrating factor for the ordinary differential equation

More information

The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge

The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge Vladimir Kozlov (Linköping University, Sweden) 2010 joint work with A.Nazarov Lu t u a ij

More information

L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS IS INDEPENDENT OF p

L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS IS INDEPENDENT OF p L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS IS INDEPENDENT OF p RALPH CHILL AND SACHI SRIVASTAVA ABSTRACT. If the second order problem ü + B u + Au = f has L p maximal regularity for some p

More information

Weak Formulation of Elliptic BVP s

Weak Formulation of Elliptic BVP s Weak Formulation of Elliptic BVP s There are a large number of problems of physical interest that can be formulated in the abstract setting in which the Lax-Milgram lemma is applied to an equation expressed

More information

arxiv:math/ v1 [math.fa] 26 Oct 1993

arxiv:math/ v1 [math.fa] 26 Oct 1993 arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological

More information

DETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION

DETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION DETERMINATION OF THE LOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION y FRANK MERLE and HATEM ZAAG Abstract. In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity.

More information

EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS

EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Cezar Avramescu Abstract The problem of existence of the solutions for ordinary differential equations vanishing at ± is considered. AMS

More information

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,

More information

A proof of the abstract limiting absorption principle by energy estimates

A proof of the abstract limiting absorption principle by energy estimates A proof of the abstract limiting absorption principle by energy estimates Christian Gérard Laboratoire de Mathématiques, Université de Paris-Sud 91405 Orsay Cedex France Abstract We give a proof in an

More information

On a Boundary-Value Problem for Third Order Operator-Differential Equations on a Finite Interval

On a Boundary-Value Problem for Third Order Operator-Differential Equations on a Finite Interval Applied Mathematical Sciences, Vol. 1, 216, no. 11, 543-548 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.216.512743 On a Boundary-Value Problem for Third Order Operator-Differential Equations

More information

at time t, in dimension d. The index i varies in a countable set I. We call configuration the family, denoted generically by Φ: U (x i (t) x j (t))

at time t, in dimension d. The index i varies in a countable set I. We call configuration the family, denoted generically by Φ: U (x i (t) x j (t)) Notations In this chapter we investigate infinite systems of interacting particles subject to Newtonian dynamics Each particle is characterized by its position an velocity x i t, v i t R d R d at time

More information

Regularity of the density for the stochastic heat equation

Regularity of the density for the stochastic heat equation Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department

More information

From the N-body problem to the cubic NLS equation

From the N-body problem to the cubic NLS equation From the N-body problem to the cubic NLS equation François Golse Université Paris 7 & Laboratoire J.-L. Lions golse@math.jussieu.fr Los Alamos CNLS, January 26th, 2005 Formal derivation by N.N. Bogolyubov

More information