Sturm--Liouville and Dirac Operators

ISBN-13: 9780792309925 / Angielski / Twarda / 1990 / 350 str.

Boris Moiseevich Levitan; B. M. Levitan; I. S. Sargsjan

Opis

'Et moi, ... si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point allC: .' human. race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'ttre of this series.

Spis treści

one. Sturm-Liouville operators.- 1 Spectral theory in the regular case.- 1.1 Basic properties of the operator.- 1.2 Asymptotic behaviour of the eigenvalues and eigenfunctions.- 1.3 Sturm theory on the zeros of solutions.- 1.4 The periodic and the semi-periodic problem.- 1.5 Proof of the expansion theorem by the method of integral equations.- 1.6 Proof of the expansion theorem in the periodic case.- 1.7 Proof of the expansion theorem by the method of contour integration.- 2 Spectral theory in the singular case.- 2.1 The Parseval equation on the half-line.- 2.2 The limit-circle and limit-point cases.- 2.3 Integral representation of the resolvent.- 2.4 The Weyl-Titchmarsh function.- 2.5 Proof of the Parseval equation in the case of the whole line.- 2.6 Floquet (Bloch) solutions.- 2.7 Eigenfunction expansion in the case of a periodic potential.- 3 The study of the spectrum.- 3.1 Discrete, or point, spectrum.- 3.2 The spectrum in the case of a summable potential.- 3.3 Transformation of the basic equation.- 3.4 The study of the spectrum as q(x) ? -?.- 4 The distribution of the eigenvalues.- 4.1 The integral equation for Green’s function.- 4.2 The first derivative of the function G(x, ?; ?).- 4.3 The second derivative of the function G(x, ?; ?).- 4.4 Further properties of the function G(x, ?; ?).- 4.5 Differentiation of Green’s function with respect to its parameter.- 4.6 Asymptotic distribution of the eigenvalues.- 4.7 Eigenfunction expansions with unbounded potential.- 5 Sharpening the asymptotic behaviour of the eigenvalues and the trace formulas.- 5.1 Asymptotic formulas for special solutions.- 5.2 Asymptotic formulas for the eigenvalues.- 5.3 Calculation of the sums Sk(t).- 5.4 Another trace regularization—auxiliary lemmas.- 5.5 The regularized trace formula for the periodic problem.- 5.6 The regularized first trace formula in the case of separated boundary conditions.- 6 Inverse problems.- 6.1 Definition and simplest properties of transformation operators.- 6.2 Transformation operators with boundary condition at x = 0.- 6.3 Derivation of the basic integral equation.- 6.4 Solvability of the basic integral equation.- 6.5 Derivation of the differential equation.- 6.6 Derivation of the Parseval equation.- 6.7 Generalization of the basic integral equation.- 6.8 The case of the zero boundary condition.- 6.9 Reconstructing the classical problem.- 6.10 Inverse periodic problem.- 6.11 Determination of the regular operator from two spectra.- two. One-dimensional Dirac operators.- 7 Spectral theory in the regular case.- 7.1 Definition of the operator—basic properties.- 7.2 Asymptotic formulas for the eigenvalues and for the vector-valued eigenfunctions.- 7.3 Proof of the expansion theorem by the method of integral equations.- 7.4 Periodic and semi-periodic problems.- 7.5 Trace calculation.- 8 Spectral theory in the singular case.- 8.1 Proof of the Parseval equation on the half-line.- 8.2 The limit-circle and the limit-point cases.- 8.3 Integral representation of the resolvent. The formulas for the functions p(?) and m(z).- 8.4 Proof of the expansion theorem in the case of the whole line.- 8.5 Floquet (Bloch) solutions.- 8.6 The self-adjointness of the Dirac systems.- 9 The study of the spectrum.- 9.1 The spectrum in the case of summable coefficients.- 9.2 Transformation of the basic system.- 9.3 The case of a pure point spectrum.- 9.4 Other cases.- 10 The solution of the Cauchy problem for the nonstationary Dirac system.- 10.1 Derivation of the formula for the solution of the Cauchy problem.- 10.2 The Goursat problem for the solution kernel of the Cauchy problem.- 10.3 The transformation matrix operator.- 10.4 Solution of the mixed problem on the half-line.- 10.5 Solution of the problem (1.1), (1.2) for t < 0.- 10.6 Asymptotic behaviour of the spectral function.- 10.7 Sharpening the expansion theorem.- 11 The distribution of the eigenvalues.- 11.1 The integral equation for Green’s matrix function.- 11.2 Asymptotic behaviour of the matrix as ? ? ?.- 11.3 Other properties of the matrix G(x, ? ?).- 11.4 Derivation of the bilateral asymptotic formula.- 12 The inverse problem on the half-line, from the spectral function.- 12.1 Stating the problem. Auxiliary propositions.- 12.2 Derivation of the basic integral equation.- 12.3 Solvability of the basic integral equation.- 12.4 Derivation of the differential equation.- 12.5 Derivation of the Parseval equation.- References.- Name Index.