Sturm--Liouville and Dirac Operators

ISBN-13: 9789401056670 / Angielski / Miękka / 2012 / 350 str.

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.