Trace class

Abstract: The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(\ensuremath{\alpha})$ of the $P$ representation, the Wigner distribution $W(\ensuremath{\alpha})$, and the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function $P(\ensuremath{\alpha})$ of the $P$ representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function $P(\ensuremath{\alpha})$ as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. The Wigner distribution $W(\ensuremath{\alpha})$ is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the $P$ representation. A parametrized integral expansion of the density operator is introduced in which the weight function $W(\ensuremath{\alpha},s)$ may be identified with the weight function $P(\ensuremath{\alpha})$ of the $P$ representation, with the Wigner distribution $W(\ensuremath{\alpha})$, and with the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when the order parameter $s$ assumes the values $s=+1, 0, \ensuremath{-}1$, respectively. The function $W(\ensuremath{\alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $\ensuremath{\delta}$ function defined in the preceding paper. This operator is in the trace class for $\mathrm{Res}l0$, has bounded eigenvalues for $\mathrm{Res}=0$, and has infinite eigenvalues for $s=1$. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ are exhibited as the order parameter $s$ is varied continuously from $s=\ensuremath{-}1$, corresponding to the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, to $s=+1$, corresponding to the function $P(\ensuremath{\alpha})$. Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the $P$ representation is appropriate.

Abstract: I Basics onClosed Operators.- 1 Closed Operators and Adjoint Operators.- 2 Spectrum of Closed Operators.- 3 Some Classes of Unbounded Operators.- II Spectral Theory.- 4 Spectral Measures and Spectral Integrals.- 5 Spectral Decomposition of Selfadjoint and Normal Operators.- III Special Topics.- 6 One-Parameter Groups and Semigroups of Operators.- 7 Miscellaneous.- IV Petirbations of Selfadjointness and of Spectra of Selfadjoint Operators.- 8 Perturbations of Selfadjoint Operators.- 9 Trace Class Perturbations of Spectra of Selfadjoint Operators.- V Forms and Operators.- 10 Semibounded Forms and Selfadjoint Operators.- 11 Sectorial Forms and m-Sectorial Operators.- 12 Discrete Spectrum of Selfadjoint Operators.- VI Selfadjoint Extention Theory of Symmetric Operators.- 13 Selfajoint Extensions: Cayley Transform and Krein Transform.- 14 Selfadjoint Extensions: Boundary Triplets.- 15 Sturm-Liouville Operators.- One-Dimensional Moment Problem.

Abstract: The main subject of this book is applications of methods of scattering theory to differential operators, primarily the Schrodinger operator There are two different trends in scattering theory for differential operators The first one relies on the abstract scattering theory The second one is almost independent of it In this approach the abstract theory is replaced by a concrete investigation of the corresponding differential equation In this book both of these trends are presented The first half of this book begins with the summary of the main results of the general scattering theory of the previous book by the author, ""Mathematical Scattering Theory: General Theory"", American Mathematical Society, 1992 The next three chapters illustrate basic theorems of abstract scattering theory, presenting, in particular their applications to scattering theory of perturbations of differential operators with constant coefficients and to the analysis of the trace class method In the second half of the book direct methods of scattering theory for differential operators are presented After considering the one-dimensional case, the author returns to the multi-dimensional problem and discusses various analytical methods and tools appropriate for the analysis of differential operators, including, among others, high- and low-energy asymptotics of the Green function, the scattering matrix, ray and eikonal explansions The book is based on graduate courses taught by the author at Saint-Petersburg (Russia) and Rennes (France) Universities and is oriented towards a reader interested in studying deep aspects of scattering theory (for example, a graduate student in mathematical physics)

Abstract: Recent theory of infinite dimensional Riccati equations is applied to the linear-quadratic optimal control problem for hereditary differential systems, and it is shown that, for most such problems, the operator solutions of the Riccati equations are of trace class (i.e., nuclear). With special attention to trace-norm convergence, an abstract approximation theory is developed and applied to a particular approximation scheme. Numerical examples are given.Problems on both finite and infinite time intervals are studied. For both the hereditary system and the approximating systems in the infinite time problem, characteristic equations are derived for the closed-loop eigenvalues, and formulas for the corresponding eigenvectors are given.