Abstract equipartition of energy theorems

Let H be a self-adjoint operator on a complex Hilbert space $\text{H}$. The solution of the abstract Schrödinger equation $\text{idu}\text{dt}\text{= Hu}$ is given by u(t) = exp(?itH)u(0). The energy E = ∥u(t)∥² is independent of t. When does the energy break up into different kinds of energy E = ∑_{j = 1}^NE_j(t) which become asymptotically equipartitioned ? (That is, $\text{E}{}_{\mathrm{j}}\text{(t) →}\text{E}\text{N}\text{as}\text{t → ± ∞}$ for all j and all data u(0).) The “classical” case is the abstract wave equation $\text{d}{}^2\text{v}\text{dt}{}^2\text{+ A}{}^2\text{v = 0}\text{with}\text{A}$ self-adjoint on $\text{H}$₁. This becomes a Schrödinger equation in a Hilbert space $\text{H}$ (essentially $\text{H}$ is two copies of $\text{H}$₁), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C¹-algebra approach to quantum field theory and statistical mechanics. (2) Let A₁,…, A_N be commuting self-adjoint operators with N = 2 or 4. Then the equation $\text{Π}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{N}}\text{(}\text{d}\text{dt}\text{? iA}{}_{\mathrm{j}}\text{) u(t) = 0}$ admits equipartition of energy if and only if exp(it(A_j ? A_k)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k.     

The amount of non-uniqueness for factored equations with Euler-Poisson-Darboux factors

Of concern are factored Euler–Poisson–Darboux equations of the type ∏^N_j=1(d²/dt²+(ρ/t)d/dt+A_j)u(t)=0, where, for example, A_j=−c_jΔ, Δ being the Dirichlet Laplacian acting on L²(Ω), Ω⊂ℝⁿ, and 0<c₁<…<c_N. More generally −A_j can be the square of the generator of a (C₀) group on a Banach space. When the constant ρ is negative, the initial value problem for the factored EPD equation is ill-posed. Nevertheless, we determine how many initial conditions are necessary to guarantee uniqueness of a solution. This number jumps up as ρ crosses a negative integer from right to left. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Nevanlinna Functions,Perturbation Formulas,and Triplets of Hilbert Spaces

Let S be a closed symmetric operator with defect numbers (1,1) in a Hilbert space ?? and let A be a selfadjoint operator extension of S in ??. Then S is necessarily a graph restriction of A and the selfadjoint extensions of S can be considered as graph perturbations of A, cf. [8]. Only when S is not densely defined and, in particular, when S is bounded, 5 is given by a domain restriction of A and the graph perturbations reduce to rank one perturbations in the sense of [23]. This happens precisely when the Q - function of S and A belongs to the subclass No of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space ?? the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space ?? is given a one-dimensional extension, or the use of Hilbert space triplets associated with A is invoked. If the Q - function of S and A belongs to the subclass N₁ of Nevanlinna functions, then it is convenient to describe the selfadjoint extensions of S including its generalized Friedrichs extension (see [6]) by interpolating the original triplet, cf. [5]. For the case when A is semibounded, see also [4]. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension.     

A Canonical Diffraction Problem With Two Media

A Wiener–Hopf equation in L² being equivalent [5] to a boundary value problem (of the first kind) for a wave-scattering Sommerfeld half-plane Σ=ℝ₊×{0} which faces two different media Ω_-: x₂<0, Ω₊: x₂>0, as a special configuration in [3], is solved by canonical Weiner–Hopf factorization of its L²-regular scalar symbol γ_o=γ_o- γ_o+. The factors are calculated by solving a Riemann–Hilbert boundary value problem on the semi-infinite branch cuts of t_j(ξ):=(ξ²−k²_j)^1/2, k_j∈ℂ⁺⁺ for j=1,2: taken parallel to the imaginary axis. The procedure following this idea is known as the Wiener–Hopf–Hilbert(–Hurd) method [2] and requires the evaluation of elliptic-type integrals. Formula (3.7) seems not to be contained in tables of integrals.     

Spectral theory for the wave equation in two adjacent wedges

Consider the two adjacent rectangular wedges K₁, K₂ with common edge in the upper halfspace of ℝ³ and the operator A (=−Laplacian multiplied by different constant coefficients a₁, a₂ in K₁, K₂, respectively) acting on a subspace of ∏²_j=1L₂(K_j). This subspace should consist of those sufficiently regular functions u=(u₁,u₂) satisfying the homogeneous Dirichlet boundary condition on the bottom of the upper halfspace. Moreover, the coincidence of u₁ and u₂ along the interface of the two wedges is prescribed as well as a transmission condition relating their first one-sided derivatives. We interpret the corresponding wave equation with A defining its spatial part as a simple model for wave propagation in two adjacent media with different material constants. In this paper it is shown (by Friedrichs' extension) that A is selfadjoint in a suitable Hilbert space. Applying the Fourier (-sine) transformations we reduce our problem with singularities along the z-axis to a non-singular Klein–Gordon equation in one space dimension with potential step. The resolvent, the limiting absorption principle and expansion in generalized eigenfunctions of A are derived (by Plancherel theory) from the corresponding results concerning the latter equation in one space dimension. An application of the spectral theorem for unbounded selfadjoint operators on Hilbert spaces yields the solution of the time dependent problem with prescribed initial data. The paper is concluded by a discussion of the relation between the physical geometry of the problem and its spectral properties. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On Generalized Friedrichs and Krein‐von Neumann Extensions and Canonical Systems

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so‐called Q‐function from the Nevanlinna class N . In this note we show to which subclasses N _γ of N the Q‐functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q‐functions of the generalized Friedrichs and Krein‐von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh‐Weyl coefficient of the two‐dimensional canonical system J_y′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation T_min with defect numbers (1,1) in the Hilbert space L²_H, and Q is equal to the Q‐function with respect to the extension corresponding to the boundary condition y₁(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q‐functions and show under which conditions the generalized Friedrichs or Krein‐von Neumann extension exists.     

Sudakov-type minoration for gaussian chaos processes

Consider a setA of symmetricn×n matricesa=(a _i,j) _i,j≤n . Consider an independent sequence (g _i) _i≤n of standard normal random variables, and letM=Esup_a∈A|Σ_i,j⪯na_i,jg_ig_j|. Denote byN ₂(A, α) (resp.N _t(A, α)) the smallest number of balls of radiusα for thel ₂ norm ofR ^{n 2} (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN ₂(A, α))^1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN _t≤K(δ)M. Work partially supported by an N.S.F. grant.     

Dissipators for symmetric quasi-free dynamical semigroups on the CAR algebra

Let L be a negative self-adjoint bounded operator on a Hilbert space $\text{H}$, and p a projection on $\text{H}$ with pLp trace class, and let {T_t: t ? 0} be the extension of {e^tL: t ? 0} to a strongly continuous semigroup of completely positive quasi-free unital maps of Fock type on the fermion algebra A$\text{H}$ built over $\text{H}$. Then it is shown that there exists a strongly continuous self-adjoint contraction semigroup {G_t: t ? 0} on the Hilbert space of the GNS decomposition of the quasi-free state gw_p such that in the representation of that state: T_t ? G_t(·)G_t, t ?0.     

Some results,concerning the variation of eigenvalues,when these depend on a real parameter

In this article we deal with a Hamiltonial of the form H(v) = Ho + A(v) where Ho is a self-adjoint bounded or unbounded operator on a Hilbert space and A(v) is a bounded self-adjoint perturbation depending on a real parameter v. In quantum mechanics a variety of results has been obtained by taking formally the derivative of the eigenvectors and eigenvalues of H(v).The differentiability of the eigenvectors and eigenvalues has been rigorously proved under several assumptions. Among these assumptions is the assumption that the eigenvalues are simple and the assumption that the perturbation A(v) is a uniformly bounded self-adjoint operator. A part of this article is dealing with examples, which show that these two assumptions are essential. The rest of this article is devoted to different applications concerning asymptotic relations of eigenvalues and a result for the solutions of the equation d_y/d_t= M(t)y in an abstract infinite dimensional Hilbert space, where iM(t)(1²=-1) is self-adjoint for every t in an interval. This result finds a succesful application to the theory of Toda and Langmuir lattices.     

A stability result for abstract evolution problems

Consider an abstract evolution problem in a Hilbert space H (1) where A(t) is a linear, closed, densely defined operator in H with domain independent of t ≥ 0 and G(t,u) is a nonlinear operator such that ‖G(t,u)‖a(t) ‖u‖^p, p = const > 1, ‖f(t)‖ ≤ b(t). We allow the spectrum of A(t) to be in the right half‐plane Re(λ) < λ₀(t), λ₀(t) > 0, but assume that lim_{t → ∞}λ₀(t) = 0. Under suitable assumptions on a(t) and b(t), the boundedness of ‖u(t)‖ as t → ∞ is proved. If f(t) = 0, the Lyapunov stability of the zero solution to problem (1) with u₀ = 0 is established. For f ≠ 0, sufficient conditions for the Lyapunov stability are given. The novel point in our study of the stability of the solutions to abstract evolution equations is the possibility for the linear operator A(t) to have spectrum in the half‐plane Re(λ) < λ₀(t) with λ₀(t) > 0 and lim_{t → ∞}λ₀(t) = 0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright © 2012 John Wiley & Sons, Ltd.     

The Friedrichs Extension of the Aharonov–Bohm Hamiltonian on a Disc

We show that the Aharonov–Bohm Hamiltonian considered on a disc has a four-parameter family of self-adjoint extensions. Among the in- finitely many self-adjoint extensions, we determine to which parameters the Friedrichs extension H^F corresponds and its lowest eigenvalue is found. Moreover, we note that the diamagnetic inequality holds for H^F.     

Sampling of Band-Limited Vectors

Given a self-adjoint, positive definite operator on a Hilbert space the concept of band-limited vectors (with a given band-width) is developed, using the spectral decomposition of that operator. By means of this concept sufficient conditions on collections of linear functionals {j_n}\{\varphi_{\nu}\} are derived which imply that all band limited vectors in a given class are uniquely determined resp.can be reconstructed in a stable way from the set of discrete values {j_n(f)}\{\varphi_{\nu}(f)\}.     

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript> by the use of finite submatrices

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l ²(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J _n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].     

Stieltjes Moment Problems and the Friedrichs Extension of a Positive Definite Operator

For an indeterminate Stieltjes moment sequence the multiplication operator Mp(x) = xp(x) is positive definite and has self-adjoint extensions. Exactly one of these extensions has the same lower bound as M, the so-called Friedrichs extension. The spectral measure of this extension gives a certain solution to the moment problem and we identify the corresponding parameter value in the Nevanlinna parametrization of all solutions to the moment problem. In the case where σ is indeterminate in the sense of Stieltjes, relations between the (Nevanlinna matrices of) entire functions associated with the measures t^kdσ(t) are derived. The growth of these entire functions is also investigated.     

Growth estimates for solutions to initial-boundary value problems in viscoelasticity

For the abstract Volterra integro-differential equation u_tt ? Nu + ∝_?∞^t K(t ? τ) u(τ) dτ = 0 in Hilbert space, with prescribed past history u(τ) = U(τ), ? ∞ < τ < 0, and associated initial data u(0) = f, u_t(0) = g, we establish conditions on K(t), ? ∞ < t < + ∞ which yield various growth estimates for solutions u(t), belonging to a certain uniformly bounded class, as well as lower bounds for the rate of decay of solutions. Our results are interpreted in terms of solutions to a class of initial-boundary value problems in isothermal linear viscoelasticity.     

Anti‐periodic solutions for evolution equations with mappings in the class (S+)

In this paper, we study the existence of anti‐periodic solutions for the first order evolution equation in a Hilbert space H, where G : H → ? is an even function such that ?G is a mapping of class (S₊) and f : ? → ? satisfies f(t + T) = –f(t) for any t ∈ ? with f(·) ∈ L²(0, T; H). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the evaluation at (<Emphasis Type="Italic">j</Emphasis>, <Emphasis Type="Italic">j</Emphasis><Superscript>2</Superscript>) of the Tutte polynomial of a ternary matroid

F. Jaeger has shown that up to a ± sign the evaluation at (j, j ²) of the Tutte polynomial of a ternary matroid can be expressed in terms of the dimension of the bicycle space of a representation over GF(3). We give a short algebraic proof of this result, which moreover yields the exact value of ±, a problem left open in Jaeger's paper. It follows that the computation of t(j, j ²) is of polynomial complexity for a ternary matroid.E. Gioan: C.N.R.S., MontpellierM. Las Vergnas: C.N.R.S., Paris     

Advective estimates

Consider the partial differential equation: w_t + bw + a · ?w = g in ?ⁿ × [0, T] (x₁ <0) and if the function a ₁ is positive, then, in the left half-space, the solution w is an O (ν) for small Viscosity ν.