In this thesis spectral inequalities and trace formulae for discrete and continuous differential operators are discussed. We first investigate spectral inequalities for Jacobi operators with matrix-valued potentials and present a new, direct proof of a sharp inequality corresponding to a Lieb–Thirring inequality for the power 3/2 using the commutation method. For the special case of a discrete Schrödinger operator we also prove new inequalities for higher powers of the eigenvalues and the potential and compare our results to previously established bounds. We then approximate a Schrödinger operator on L^2(\R) by Jacobi operators on \ell^2(\Z) and use the established inequalities to provide new proofs of sharp Lieb–Thirring inequalities for the powers \gamma=1/2 and \gamma=3/2. By means of interpolation we derive spectral inequalities for Jacobi operators that yield (non-sharp) Lieb–Thirring constants on the real line for powers 1/2