Lieb-Thirring bound and generalized weak time operators associated with Schrodinger operators (Mathematical Aspects of Quantum Fields and Related Topics)
This is a short version of [Hir15].A weak time operator $T$ associated with a given self-adjoint operator $H$ is a symmetric operator such that $(H\phi, T\psi)-$ $(T\phi, H\psi)=-i(\phi, \psi)$ for $\phi,$ $\psi\in D$ with some domain $D$ .In this paper we generalize weak time operators as a densely defined symmetric quadratic form, and a generalized weak time operator $T_{H}$ associated with a Schr\"odinger operator of the form $H=-\Delta/2+V$ on $\mathscr{H}=L^{2}(\mathbb{R}^{d})$ is constructed.It is assumed that the quadratic moment of the negative eigenvalues $\{E_{j}\}_{j=1}^{\infty}$ of $H$ is finite, i.e., $\sum_{j=1}^{\infty}E_{j}^{2}<\infty$ .This is ensured by the Lieb-Thirring inequality.Then we can construct $T_{H}$ ) : $\mathscr{H}\cross \mathscr{H}arrow \mathbb{C}$ such that $T_{H}(H\phi, \psi)-T_{H}(\phi, H\psi)=-i(\phi, \psi)$ for all $\phi,$ $\psi\in \mathscr{D}$ with some domain $\mathscr{D}.$
This is a short version of [Hir15].A weak time operator T associated with a given self-adjoint operator H is a symmetric operator such that (Hϕ,Tψ)−(Tϕ,Hψ)=−i(ϕ,ψ) for ϕ,ψ∈D with some domain D .In this paper we generalize weak time operators as a densely defined symmetric quadratic form, and a generalized weak time operator TH associated with a Schr"odinger operator of the form H=−Δ/2+V on H=L2(Rd) is constructed.It is assumed that the quadratic moment of the negative eigenvalues {Ej}j=1∞ of H is finite, i.e., ∑j=1∞Ej2<∞ .This is ensured by the Lieb-Thirring inequality.Then we can construct TH ) : H\crossHarrowC such that TH(Hϕ,ψ)−TH(ϕ,Hψ)=−i(ϕ,ψ) for all ϕ,ψ∈D with some domain D.
Let H be a self-adjoint operator on a Hilbert space [Formula: see text], T be a symmetric operator on [Formula: see text] and K(t)(t ∈ ℝ) be a bounded self-adjoint operator on [Formula: see text]. We say that (T, H, K) obeys the generalized weak Weyl relation (GWWR) if e-itHD(T) ⊂ D(T) for all t ∈ ℝ and Te-itHψ = e-itH(T+K(t))ψ, ∀ψ ∈ D(T) (D(T) denotes the domain of T). In the context of quantum mechanics where H is the Hamiltonian of a quantum system, we call T a generalized time operator of H. We first investigate, in an abstract framework, mathematical structures and properties of triples (T, H, K) obeying the GWWR. These include the absolute continuity of the spectrum of H restricted to a closed subspace of [Formula: see text], an uncertainty relation between H and T (a "time-energy uncertainty relation"), the decay property of transition probabilities |〈ψ,e-itHϕ〉|2 as |t| → ∞ for all vectors ψ and ϕ in a subspace of [Formula: see text], where 〈·,·〉 denotes the inner product of [Formula: see text]. We describe methods to construct various examples of triples (T, H, K) obeying the GWWR. In particular, we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces and fermion Fock spaces) which may have applications to quantum field models with interactions.
Some aspects of strong time operators in an abstract algebraic quantum mechanics (including quantum statistical mechanics) and in quantum field theory (QFT) are described.As for the facts presented in QFT, they have essentially been established in a previous paper (A.Arai, Rev. Math.Phys.17(2005), 10711109). §1. Introduction: Strong Time Operators and Their FundamentalProperties This paper is concerned with time operators in algebraic quantum mechanics (including quantum statistical mechanics) and quantum field theory.The concept of time operators itself is a general one and there are some types on them [1, 2, 5, 9].In the present paper we concentrate our attention on strong time operators as defined below and their realizations in algebraic quantum mechanics and quantum field theory.Let \ ma t h f r a k { H} be a complex Hilbert space.We denote the inner product and the norm of \ ma t h f r a k { H} by \l a ng l e \c dot , \ r a n g l e _ { \ m a t h f r a k { H } } (anti-linear in the first variable) and \ V e r t \ c d o t \ V e r t _ { \ ma t h f r a k { H} } respectively.If there is no danger of confusion, then the subscript \ ma t h f r a k { H} in \l a ng l e \c do t , \ r a n g l e _ { \ m a t h f r a k { H } } and \ operator A on a Hilbert space, \mat hrm{D}(A) (resp. Ran (A)) denotes the domain (resp. range) of A . IfA is closable, then we denote its closure by \ov e r l i ne { A} .Let H be a self-adjoint operator on \ ma t h f r a k { H} and T be a symmetric operator on \ ma t h f r a k { H} .If there exists a bounded self-adjoint operator C\neq 0 on \ ma t h f r a k { H} such that \mathrm{D}(C)=\mathf rak{H}, (1.1) e^{-i tH}\mathrm{D}(T)\subset \mathrm{D}(T) , \foral l t\i n \mathbb{R},
The main motivation behind the call for this special issue was to gather recent results, developments and open problems in quantum physics with non-Hermitian operators. There have been previous special issues in this journal [1, 2] and elsewhere on this subject. The intention of this issue is to reflect the current state of this rapidly-developing field. It has therefore been open to all contributions containing new results on non-Hermitian theories that are explicitly -symmetric and/or pseudo-Hermitian or quasi-Hermitian. In the last decade these types of systems have proved to be viable self-consistent physical theories with well defined unitary time-evolution and real spectra. As the large number of responses demonstrates, this is a rapidly evolving field of research. A consensus has been reached regarding most of the fundamental problems, and the general ideas and techniques are now readily being employed in many areas of physics. Nonetheless, this issue still contains some treatments of a more general nature regarding the spectral analysis of these models, in particular, the physics of the exceptional points, the breaking of the -symmetry, an interpretation of negative energies and the consistent implementation of the WKB analysis. This issue also contains a treatment of a scattering theory associated with these types of systems, weak measurements, coherent states, decoherence, unbounded metric operators and the inclusion of domain issues to obtain well defined self-adjoint theories. Contributions in the form of applications of the general ideas include: studies of classical shock-waves and tunnelling, supersymmetric models, spin chain models, models with ring structure, random matrix models, the Pauli equation, the nonlinear Schrödinger equation, quasi-exactly solvable models, integrable models such as the Calogero model, Bose–Einstein condensates, thermodynamics, nonlinear oligomers, quantum catastrophes, the Landau–Zener problem and pseudo-Fermions. Applications close to experimental realization are proposed in optics, including short light pulse models, waveguides and laser systems, and also in electronics. We hope that this issue will become a valuable reference and inspiration for the broader scientific community working in mathematical and theoretical physics. References [1] Fring A, Jones H F and Znojil M (ed) 2008 J. Phys. A: Math. Theor. 41 240301 [2] Geyer H, Heiss D and Znojil M (ed) 2006 J. Phys. A: Math. Gen. 39 9963
Let H be a self-adjoint operator (a Hamiltonian) on a complex Hilbert space $ \mathcal{H}$
We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation.
Some of the articles in this collection give up-to-date accounts of areas in mathematical physics to which Valentine Bargmann made pioneering contributions. The others treat a selection of the most interesting current topics in the field. The contributions include both reviews and original results. Contents: The Inverse r-Squared Force (Henry D. I. Abarbanel; Certain Hilbert Spaces of Analytic Functions Associated with the Heisenberg Group (Donald Babbitt); Lower Bound for the Ground State Energy of the Schrodinger Equation Using the Sharp Form of Young's Inequality (John F. Barnes, Herm Jan Brascamp, and Elliott II. Lieb); Alternative Theories of Gravitation (Peter G. Bergmann; )Generalized Wronskian Relations (F. Calogero); Old and New Approaches to the Inverse-Scattering Problem (Freeman J. Dyson); A Family of Optimal Conditions for the Absence of Bound States in a Potential (V. Glaser, A. Martin, H. Grosse, and W. Thirring); Spinning Tops in External Fields (Sergio Hojman and Tullio Regge); Measures on the Finite Dimensional Subspaces of a Hilbert Space (Res Jost); The Froissart Bound and Crossing Symmetry (N. N. Khuri); Intertwining Operators for SL(n,R) (A. W. Knapp and E. M. Stein); Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relations to Sobolev Inequalities (Elliott H. Lieb and Walter Thirriny); On the Number of Bound States of Two Body Schrodinger Operators (Barry Simon); Quantum Dynamics: From Automorphism to Hamiltonian (Barry Simon); Semiclassical Analysis Illuminates the Connection between Potential and Bound States and Scattering (John Archibald Wheeler); Instability Phenomena in the External Field Problem for Two Classes of Relativistic Wave Equations (A. S. Wightman) Originally published in 1976. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.