Sturm-Liouville and Dirac operators / by B. M. Levitan and I. S. Sargsjan.
Idioma: Inglés Lenguaje original: Ruso Series Mathematics and its applications (Kluwer Academic Publishers)Soviet series: 59.Editor: Dordrecht : Kluwer Academic Publishers, c1991Descripción: xi, 350 p. ; 25 cmISBN: 0792309928Títulos uniformes: Operatory Shturma-Liuvillia i Diraka. Inglés Tema(s): Sturm-Liouville equation | Dirac equationOtra clasificación: 34L40 (34B24 34Lxx 47A55 47E05)Part one. Sturm-Liouville operators [1] 1 Spectral theory in the regular case [3] 1.1 Basic properties of the operator [3] 1.2 Asymptotic behaviour of the eigenvalues and eigenfunctions [5] 1.3 Sturm theory on the zeros of solutions [12] 1.4 The periodic and the semi-periodic problem [16] 1.5 Proof of the expansion theorem by the method of integral equations [24] 1.6 Proof of the expansion theorem in the periodic case [31] 1.7 Proof of the expansion theorem by the method of contour integration [33] 2 Spectral theory in the singular case [38] 2.1 The Parseval equation on the half-line [38] 2-2 The limit-circle and limit-point cases [44] 2.3 Integral representation of the resolvent [49] 2.4 The Weyl-Titchmarsh function [55] 2.5 Proof of the Parseval equation in the case of the whole line [59] 2.6 Floquet (Bloch) solutions [66] 2.7 Eigenfunction expansion in the case of a periodic potential [69] 3 The study of the spectrum [74] 3.1 Discrete, or point, spectrum [74] 3.2 The spectrum in the case of a summable potential [77] 3.3 Transformation of the basic equation [84] 3.4 The study of the spectrum as g(x) —> - ∞ [86] 4 The distribution of the eigenvalues [92] 4.1 The integral equation for Green’s function [92] 4.2 The first derivative of the function G(χ,η;μ) [97] 4.3 The second derivative of the function G(χ,η;μ) [99] 4.4 Further properties of the function G(χ,η;μ) [101] 4.5 Differentiation of Green’s function with respect to its parameter [103] 4.6 Asymptotic distribution of the eigenvalues [108] 4.7 Eigenfunction expansions with unbounded potential [114] 5 Sharpening the asymptotic behaviour of the eigenvalues and the trace formulas [118] 5.1 Asymptotic formulas for special solutions [118] 5.2 Asymptotic formulas for the eigenvalues [119] 5.3 Calculation of the sums Sk(t) [122] 5.4 Another trace regularization—auxiliary lemmas [124] 5.5 The regularized trace’ formula for the periodic problem [129] 5.6 The regularized first trace formula in the case of separated boundary conditions [133] 6 Inverse problems [139] 6.1 Definition and simplest properties of transformation operators [139] 6.2 Transformation operators with boundary condition at x = 0 [140] 6.3 Derivation of the basic integral equation [145] 6.4 Solvability of the basic integral equation [150] 6.5 Derivation of the differential equation [153] 6.6 Derivation of the Parseval equation [155] 6.7 Generalization of the basic integral equation [160] 6.8 The case of the zero boundary condition [162] 6.9 Reconstructing the classical problem [164] 6.10 Inverse periodic problem [168] 6.11 Determination of the regular operator from two spectra [171] Part two. One-dimensional Dirac operators [183] 7 Spectral theory in the regular case [185] 7.1 Definition of the operator—basic properties [185] 7.2 Asymptotic formulas for the eigenvalues and for the vector-valued eigenfunctions [188] 7.3 Proof of the expansion theorem by the method of integral equations [193] 7.4 Periodic and semi-periodic problems [201] 7.5 Trace calculation [207] 8 Spectral theory in the singular case [213] 8.1 Proof of the Parseval equation on the half-line [213] 8.2 The limit-circle and the limit-point cases [217] 8.3 Integral representation of the resolvent. The formulas for the functions p(λ) and m(z) [222] 8.4 Proof of the expansion theorem in the case of the whole line [228] 8.5 Floquet (Bloch) solutions [234] 8.6 The self-adjointness of the Dirac systems [236] 9 The study of the spectrum [243] 9.1 The spectrum in the case of summable coefficients [243] 9.2 Transformation of the basic system [247] 9.3 The case of a pure point spectrum [251] 9.4 Other cases [256] 10 The solution of the Cauchy problem for the nonstationary Dirac system [259] 10.1 Derivation of the formula for the solution of the Cauchy problem [259] 10.2 The Goursat problem for the solution kernel of the Cauchy problem [263] 10.3 The transformation matrix operator [265] 10.4 Solution of the mixed problem on the half-line [272] 10.5 Solution of the problem (1.1), (1.2) for t < 0 [274] 10.6 Asymptotic behaviour of the spectral function [277] 10.7 Sharpening the expansion theorem [288] 11 The distribution of the eigenvalues [294] 11.1 The integral equation for Green’s matrix function [294] 11.2 Asymptotic behaviour of the matrix G'μ(χ,ξ;iμ) as μ —> ∞ [303] 11.3 Other properties of the matrix G(x,ξ;λ) [310] 11.4 Derivation of the bilateral asymptotic formula [316] 12 The inverse problem on the half-line, from the spectral function [324] 12.1 Stating the problem. Auxiliary propositions [324] 12.2 Derivation of the basic integral equation [327] 12.3 Solvability of the basic integral equation [331] 12.4 Derivation of the differential equation [333] 12.5 Derivation of the Parseval equation [336]
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Traducción de: Operatory Shturma-Liuvillia i Diraka. 1988.
Incluye referencias bibliográficas (p. 341-345) e índices.
Part one. Sturm-Liouville operators [1] --
1 Spectral theory in the regular case [3] --
1.1 Basic properties of the operator [3] --
1.2 Asymptotic behaviour of the eigenvalues and eigenfunctions [5] --
1.3 Sturm theory on the zeros of solutions [12] --
1.4 The periodic and the semi-periodic problem [16] --
1.5 Proof of the expansion theorem by the method of integral equations [24] --
1.6 Proof of the expansion theorem in the periodic case [31] --
1.7 Proof of the expansion theorem by the method of contour integration [33] --
2 Spectral theory in the singular case [38] --
2.1 The Parseval equation on the half-line [38] --
2-2 The limit-circle and limit-point cases [44] --
2.3 Integral representation of the resolvent [49] --
2.4 The Weyl-Titchmarsh function [55] --
2.5 Proof of the Parseval equation in the case of the whole line [59] --
2.6 Floquet (Bloch) solutions [66] --
2.7 Eigenfunction expansion in the case of a periodic potential [69] --
3 The study of the spectrum [74] --
3.1 Discrete, or point, spectrum [74] --
3.2 The spectrum in the case of a summable potential [77] --
3.3 Transformation of the basic equation [84] --
3.4 The study of the spectrum as g(x) —> - ∞ [86] --
4 The distribution of the eigenvalues [92] --
4.1 The integral equation for Green’s function [92] --
4.2 The first derivative of the function G(χ,η;μ) [97] --
4.3 The second derivative of the function G(χ,η;μ) [99] --
4.4 Further properties of the function G(χ,η;μ) [101] --
4.5 Differentiation of Green’s function with respect to its parameter [103] --
4.6 Asymptotic distribution of the eigenvalues [108] --
4.7 Eigenfunction expansions with unbounded potential [114] --
5 Sharpening the asymptotic behaviour of the eigenvalues and the trace formulas [118] --
5.1 Asymptotic formulas for special solutions [118] --
5.2 Asymptotic formulas for the eigenvalues [119] --
5.3 Calculation of the sums Sk(t) [122] --
5.4 Another trace regularization—auxiliary lemmas [124] --
5.5 The regularized trace’ formula for the periodic problem [129] --
5.6 The regularized first trace formula in the case of separated boundary conditions [133] --
6 Inverse problems [139] --
6.1 Definition and simplest properties of transformation operators [139] --
6.2 Transformation operators with boundary condition at x = 0 [140] --
6.3 Derivation of the basic integral equation [145] --
6.4 Solvability of the basic integral equation [150] --
6.5 Derivation of the differential equation [153] --
6.6 Derivation of the Parseval equation [155] --
6.7 Generalization of the basic integral equation [160] --
6.8 The case of the zero boundary condition [162] --
6.9 Reconstructing the classical problem [164] --
6.10 Inverse periodic problem [168] --
6.11 Determination of the regular operator from two spectra [171] --
Part two. One-dimensional Dirac operators [183] --
7 Spectral theory in the regular case [185] --
7.1 Definition of the operator—basic properties [185] --
7.2 Asymptotic formulas for the eigenvalues and for the vector-valued eigenfunctions [188] --
7.3 Proof of the expansion theorem by the method of integral equations [193] --
7.4 Periodic and semi-periodic problems [201] --
7.5 Trace calculation [207] --
8 Spectral theory in the singular case [213] --
8.1 Proof of the Parseval equation on the half-line [213] --
8.2 The limit-circle and the limit-point cases [217] --
8.3 Integral representation of the resolvent. The formulas for the functions p(λ) and m(z) [222] --
8.4 Proof of the expansion theorem in the case of the whole line [228] --
8.5 Floquet (Bloch) solutions [234] --
8.6 The self-adjointness of the Dirac systems [236] --
9 The study of the spectrum [243] --
9.1 The spectrum in the case of summable coefficients [243] --
9.2 Transformation of the basic system [247] --
9.3 The case of a pure point spectrum [251] --
9.4 Other cases [256] --
10 The solution of the Cauchy problem for the nonstationary Dirac system [259] --
10.1 Derivation of the formula for the solution of the Cauchy problem [259] --
10.2 The Goursat problem for the solution kernel of the Cauchy problem [263] --
10.3 The transformation matrix operator [265] --
10.4 Solution of the mixed problem on the half-line [272] --
10.5 Solution of the problem (1.1), (1.2) for t < 0 [274] --
10.6 Asymptotic behaviour of the spectral function [277] --
10.7 Sharpening the expansion theorem [288] --
11 The distribution of the eigenvalues [294] --
11.1 The integral equation for Green’s matrix function [294] --
11.2 Asymptotic behaviour of the matrix G'μ(χ,ξ;iμ) as μ —> ∞ [303] --
11.3 Other properties of the matrix G(x,ξ;λ) [310] --
11.4 Derivation of the bilateral asymptotic formula [316] --
12 The inverse problem on the half-line, from the spectral function [324] --
12.1 Stating the problem. Auxiliary propositions [324] --
12.2 Derivation of the basic integral equation [327] --
12.3 Solvability of the basic integral equation [331] --
12.4 Derivation of the differential equation [333] --
12.5 Derivation of the Parseval equation [336] --
MR, 92i:34119
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