Marchenko V. Sturm-Liouville operators and applications, Kiev, Naukova Dumka, 1977. 7  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... (F ( x, #[ Fredholm alternative property guarantees the existence of a unique solution of (25) if and only if (24) has only the trivial solution (see [Kre78] Using the previous result we will state a theorem about the existence and uniqueness of solution of Marchenko equation (see [Mar86]) Theorem 3. Suppose that (1) For each k 0, the reflection coe#cient b(k) satisfyes b( k) b(k) b(k) 1 and b(k) O(k 1 ) a k # #. 2) The Fourier transform B(x) of b(k) defined by B(x) 1 b(k)e ikx dk exists, is real, absolutely continuous, and lies in L (R) 3) B(x) ....

....a k # #. 2) The Fourier transform B(x) of b(k) defined by B(x) 1 b(k)e ikx dk exists, is real, absolutely continuous, and lies in L (R) 3) B(x) satisfyes a B(x) dx x ) B(x) dx #. then equation (19) has an unique solution. Proof: ommiting some technical details , check [Mar86] for a complete proof) Using the previous lemma, we must show that the homogeneous Marchenko equation (24) has only the trivial solution. Fix x and consider the equation f(z) F (z y)f(y)dy = 0. Multiply this by f and integrate f(z)f(z)dz F (z y)f(z)dydz. Using (18) and ....

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V. Marchenko. Sturm-Liouville Operators and Applications. Birkhauser, 1986.

....of 2 sheets = C n [oe n and . Each sheet is a copy of the complex plane slit along oe(H) The first sheet is glued to the sheet by identifying the banks of all slits in oe(H) We define the quasimomentum k( arccos Delta( which is analytic on the Riemann surface E (H) see [M]) Recall that if a gap jfl n j = 0; then k( is analytic at = n ; if jfl n j 6= 0; then k has the branch points n and k( n i Gamma2M n ( Gamma n ) 1 o(1) as ff n ; 2 fl n (H) where SigmaM n 0 is the effective mass; here and below z 0; z 0. We ....

Marchenko V.: Sturm-Liouville operator and applications. Basel: Birkhauser 1986.

....asymptotically periodic coe#cients A. Boutet de Monvel, I.Egorova # Abstract We construct the transformation operator for the scattering problem on the periodic background under an assumption that the coe#cients of perturbation have the first finite moment. By means of the Marchenko approach ([7]) we derive an estimate on the kernel of this transformation operator that allow us to study the inverse problem solution in the prescribed class of perturbations. 1 Introduction The scattering problem for the Jacobi operator in # (Z) Ly) n = a n 1 y n 1 b n y n a n y n 1 , 1.1) with fast ....

....] # . 1.6) Here a constant C depends on inf n# a n , #L L and the sum on l. h.s. of inequality (1.2) Estimate (1. 6) looks similar to the corresponding estimate for the Schrodinger operator, but the approach to derive it, used by [11] is di#erent to that one, developed in [7]. It is based on some special properties of discrete operators. The goal of the present paper is to construct the transformation operators for the Jacobi operator with asymptotically periodic coe#cients. We study some questions of the scattering problem on the periodic background and we obtain the ....

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Marchenko V. Sturm-Liouville operators and applications, Birkhauser, 1986

....in nite number of zeros (resonances) fk n g N 1 in C , where 0 6 jk N 1 j 6 jk N 2 j 6 : The functions (x; k n ) k n 2 C are the eigenfunctions of H and we de ne the norming constants by j (x; k n )j dx; k n 2 C ; 1. 4) which are important in the inverse problem (see [M]) We have the following identities w( k) w(k) s( k) s(k) k 2 C ; 1.5) w(k)w( k) 4k s(k)s( k) k 2 C : 1.6) The scattering matrix for H; H 0 = 2 has the following form SM (k) r (k) r (k) a(k) r = s( k) b( k) a(k) k 2 R; 1.7) where 1=a is the ....

....the zeros of w and s. Recall that rougthly speaking the zeros of s determine a unique potential q 2 Q, but the zeros of w determine a unique potential q 2 Q if we add the sequence (see Theorem 1. 2) A great number of papers are devoted to the inverse problem for the Schr odinger operator, see [M] for ref. We use the Marchenko result from the book published in 1977 (in Russian) Later this book was translated into English, see [M] In fact he really reproved correctly the well known Faddeev s result [F] where there were some mistakes (see [DT] Remark that in the papers [H] S] Z] ....

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Marchenko V.: Sturm-Liouville operator and applications. Basel: Birkhauser 1986

....: sup #u#W1,p 1 with norm #u#M #u#W 1,p 1 . 1.15) Under condition (1. 5) H has no positive or zero eigenvalues, it has a finite number of negative eigenvalues, it has no singular continuos spectrum, and the absolutely continuous spectrum consists of [0, #) For these results see [5] and [7] In what follows we designate by u(t) the function u( t) where u(x, t) is defined in R R. THEOREM 1.1. Suppose that Assumptions A and B are satisfied and that H has no negative eigenvalues. Then, there is a # 0 such that for all f N and all # with ## #W2,2 # ....

....1.2 the scattering operator, S, determines uniquely the potential V 0 . Proof: It follows from Theorem 1.2 that S determines uniquely S L . From S L we uniquely reconstruct the scattering phase shift, see (2.10) in Section 2. Finally, we uniquely reconstruct V 0 from the scattering phase shift [5]. 5 As we show below, in the case where F (x, t, u) # j=1 V j (x, t) u we also uniquely reconstruct the . THEOREM 1.4. Suppose that the conditions of Theorem 1.2 are satisfied, and furthermore, that F (x, t, u) # j=1 V j (x, t) u u, for #, for some # 0, with j 0 an integer ....

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V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

....solubility of Simon s fundamental equation. 1. Introduction Consider a one dimensional Schr odinger equation, x) V (x)y(x) zy(x) 1.1) In inverse spectral theory, we study the problem of recovering the potential V from spectral data. Much has been done on this; we refer the reader to [4, 5] for expositions of classical work, especially the approach of Gelfand and Levitan and to [6, 9] for rather di erent views of the subject. In this paper, we are concerned with two recently developed methods: the approach of Simon [10] and the treatment given by myself in [8] More speci cally, we ....

V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

....ruled out. To the best of our knowledge, up until now there were no constructive examples of Schr odinger operators with imbedded singular continuous spectrum in any setting. We notice that the fact that potentials leading to such operators exist follows from the inverse spectral theory (see, e.g. [13, 14]) This classical result has been recently improved by Denisov [8] to give existence of potentials leading to imbedded singular continuous spectrum in the L class. Killip and Simon [9] have subsequently found necessary and sucient conditions for a measure to be a spectral measure of a free ....

V. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel 1986.

....problem (as defined above or in the narrower sense of Weyl theory) then # for all N 0. In particular, Theorem 1.1 really addresses the question raised at the beginning of this paper. The measures from are sometimes also called spectral measures of the problem on [0, N ] for example in [5, 7]) but here it is better to avoid this usage of the term, in order to avoid confusion with the spectral measures # # , # [0, #) introduced above, which form a (small) subset of . This (very classical) material on the spectral representation of H # also admits a more function theoretic ....

V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

.... the classical work of Gelfand Levitan mentioned above [14] Important improvements are due to Levitan and Gasymov [22] and further developments of this line of attack may be found in [28, 29] For modern expositions of the Gelfand Levitan theory, we refer the reader to chapter 2 of either [21] or [23]. A di erent approach which so far has been used to attack uniqueness questions, but in principle also gives a procedure for reconstructing the potential from the spectral data was recently developed by Simon, partly in collaboration with Gesztesy [15, 27] This approach emphasizes the role of ....

....theory of such operators. 19. Dirichlet boundary conditions We now consider the Schr odinger equation (3.1) with Dirichlet boundary condition at the origin: y(0) 0. In inverse spectral theory, the case of Dirichlet boundary conditions often poses additional technical problems (for instance, in [14, 22, 23], Dirichlet boundary conditions are not discussed) This seems to hold to a lesser extent for the approach developed in this paper. All results presented so far have direct analogs, and in most cases, no new ideas are needed. We will now give a very sketchy exposition of these results. In fact, I ....

V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

....orthogonal polynomials are not completely disjoint. The Ricatti solution gives the an s and b n s as continued fractions. The connection between continued fractions and orthogonal polynomials goes back a hundred years to Stieltjes work on the moment problem [18] The Gel fand Levitan Marchenko [7,11,12,13] approach to the continuum case is a direct analog of this orthogonal polynomial case. One looks at solutions U(x, k)of U ## q(x)U = k 2 U(x) 1.19) obeying U(0) 1, U # (0) ik, and proves that they obey a representation U(x, k) e ikx # x x K(x, y)e iky dy, 1.20) the analog of ....

....) 5.1) where (a) If h = #,thenA j =2and B j = 2j # b 0 q(y) dy. b) If h #,thenA j =2( 1) j and B j =2( 1) j 1 j[2h # b 0 q(y) dy] Remarks. 1. The combination 2h # b 0 q(y) dy is natural when h #. Italsoenters into the formula for eigenvalue asymptotics [11,13]. 16 B. SIMON 2. One can think of (5.1) as saying that m( # 2 ) # # a 0 A(#)e 2## d# O(e 2a# ) for any a where now A is only a distribution of the form A(#) A(#) 1 2 ## j=1 A j # # (# jb) # # j=1 B j #(# jb)where# # is the derivative of a delta function. 3. As a ....

V. Marchenko, Sturm-Liouville Operators and Applications,Birkhauser, Basel, 1986.

....problem for step like Jacobi operator I.Egorova # Abstract The direct inverse problem is solved for the step like Jacobi operator in the prescribed class of convergence of the operator coe#cients to their limits. The characterization of scattering data is given by means of Marchenko approach ([7]) 1 Introduction Under the step like Jacobi operator (Jacobi matrix) we means the operator in l 2 (Z) Ly) n = a n 1 y n 1 b n y n a n y n 1 , 1.1) where inf n# a n 0 and a n 1 2 # 0, b n sign(n)b # 0 (n # #) Here b # R 0 is some constant. The discrete operator in ....

....of the Jost solutions imply relevant properties of their Wronskian W (#) Since the discrete spectrum of operator coincides with the set of zeros of this Wronskian, it can have the only accumulation points at the edges of continuous spectrum. But from lemma 2. 1, using the same arguments as in [7], pp.266 269 we obtain, that the discrete spectrum is finite. Property D follows from lemma 2.2 and formula (1.19) Consider now the behavior of the function W (#) at the edges of continuous spectrum. Note, that W (#) #= 0 as # # ( 1 b, 1 b) It follows from the independence of solutions ....

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Marchenko V. Sturm-Liouville operators and applications, Kiev, Naukova Dumka, 1977.

....we obtain h V [N Gamma1] x) 2 N Gamma [N ] x) 2 Upsilon [N ]0 (x) i [f [N Gamma1] r (i N ; x) f [N Gamma1] l (i N ; x) 0; x 2 I; 3.7) where [F ; G] FG 0 Gamma F 0 G denotes the Wronskian. Note that the Wronskian in (3. 7) is equal (see e.g. 2] [4], 5] to Gamma2 N =T [N Gamma1] i N ) which is a negative constant. Thus, from (3.7) we get [N ] x) 2 Gamma V [N Gamma1] x) Gamma 2 N = Upsilon [N ]0 (x) x 2 I: 3.8) Using (3.3) we can write (3.8) as 1 2 [V [N ] x) Gamma V [N Gamma1] x) Upsilon [N ]0 (x) x ....

V. A. Marchenko, Sturm-Liouville operators and applications, Birkhauser, Basel, 1986.

....potentials in V 2 L 1 1 (R) By this we mean the problem of finding necessary and sufficient conditions on the scattering data which guarantee that there is exactly one real valued potential V 2 L 1 1 (R) corresponding to that data. Such characterizations were given by Melin [Me85] and Marchenko [Ma86]. It is known that one can construct V uniquely from either the left scattering data fR; f j g; fc l;j gg or the right scattering data fL; f j g; fc r;j gg: Here j for j = 1; N are N distinct positive numbers such that Gamma 2 j represent the bound state energies for V; and c l;j ....

....c l;j and c r;j are positive constants called bound state norming constants; c l;j and c r;j are the reciprocals of the norms of the eigenfunctions f l (i j ; Delta) and f r (i j ; Delta) respectively. Among the characterization conditions listed in Theorem 6.1 of [Me85] and Theorem 3.5. 1 of [Ma86] is the condition (5.1) lim k 0 k T (k) R(k) 1] 0; k 2 R; and it plays an important role in the reconstruction of V from the scattering data. Condition (5.1) provides a way of proving the characterization theorem without using the continuity of R(k) and T (k) at k = 0; which was not known ....

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V. A. Marchenko, Sturm-Liouville operators and applications, Birkhauser, Basel, 1986.

.... we will show in Section 2, the expansion in Theorem 2 lets us identify d( with a Jost function, the value of the Jost solution at x = 0 (this is not a new result; it is due to Jost Pais [12] But Levin [17] has a Laplace transform formula for the Jost solution as used extensively by Marchenko [18] and that allows one to prove Theorem 4 from Theorem 2 (the estimate we use to show a 2 supp(t) is in Marchenko s book) Our final results are on a different subject Theorem 6. In a half line problem (Cases 2, 3 or 4) suppose h has n bound states 0 1 Delta Delta Delta n . Then each ....

V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

....Schrodinger operators, especially in one dimension, is the question of inverse theory: How does one go from spectral or scattering information to the potential. There is a huge literature, including three books I would like to refer the reader to: Chadan Sabatier [45] Levitan [176] and Marchenko [190]. I will only touch some noteworthy ideas here. In one dimension, a key role is played by the Weyl m function and the associated spectral measure, dae. Given a potential V so that H is selfadjoint with u(0) 0 boundary conditions, for each z with Im z 0, there is a solution u(x; z) of Gammau ....

....scattering for short range potentials since dae on [0; 1) is determined by scattering data. Scattering data also determine the positions of the negative eigenvalues. One needs to supplement that with the weight of the pure points at these negative eigenvalues known as norming constants. Marchenko [190, 189] has an approach to inverse scattering related to the Gel fand Levitan approach by using a different representation than (6.8) When R 1 0 xjV (x)j dx 1, Levin [174] has proven that in Im k 0, there is a solution (x; k) of Gamma 00 V = k 2 given by (x; k) e ixk Z 1 x ....

V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

....momentum or fixed energy) the problem of reconstruction of a potential is considered from a phase shift given either on a half line fk 0; l is fixedg or on a half line fl 0; k is fixedg. There are many of books and papers dealing with the theoretical and numerical aspects of these problems (see [2, 8, 14, 15, 16, 19, 20, 31]) Detailed bibliography and many applications to physical problems can be found in [8, 12, 17, 18, 31] At the same time there are very few results concerning other statements of the problem. However the analysis of such new statements is interesting for mathematicians and physicists as well. The ....

Marchenko, V.A. [1986] Sturm-Liouville Operators and Applications, Birkhauser, Basel.

....end of Section 8. The estimate (6. 8) in the case of non Dirichlet boundary conditions at x = 0 , appears to be due to Marchenko [26] Since it is a fundamental ingredient in the inverse spectral problem, it generated considerable attention; see, for instance, 12] 18] 19] 20] 22] 27] [28], Sect. 2.4. The case of a Dirichlet boundary at x = 0 was studied in detail by Levitan [20] These authors, in addition to studying the spectral asymptotics of ae( as # Gamma1, were also particularly interested in the asymptotics of ae( and 1 and established Theorem 6.1 (and (A.9) In ....

....0 ; z) dx 0 ; we claim that A(ff) Gamma2 y L(2ff; y) fi fi fi fi y=0 : 9.11) We will first proceed formally without worrying about regularity conditions. Detailed discussions of transformation operators can be found, for instance, in [11] 21] Ch. 1, 22] 24] 25] 26] [28], Ch. 1, 31] Ch. VIII, 34] 35] and, in the particular case of scattering theory, in [2] Chs. I and V, 8] and [29] Let dae( be the spectral measure for Gamma d 2 dx 2 q(x) and dae (0) Gamma1 [0;1) p d (9.12) the spectral measure for Gamma d 2 dx 2 (both ....

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V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

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Marchenko V. Sturm-Liouville operators and applications, Kiev, Naukova Dumka, 1977. 7

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Marchenko V.: Sturm-Liouville operator and applications. Basel: Birkhauser 1986.

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Marchenko V. Sturm-Liouville operators and applications, Kiev, Naukova Dumka, 1977.

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Marchenko V.: Sturm-Liouville operator and applications. Basel: Birkhauser 1986.

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V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

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V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

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V. A. Marchenko, Sturm-Liouville Operators and Applications (Birkhauser, Basel,

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V. A. Marchenko, Sturm-Liouville operators and applications (Birkhauser, Basel, 1986).

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