SturmLiouville and Dirac Operators [Paperback]

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 Greens 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 Greens 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 regularizationauxiliary 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 TranslÓÆ