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Register a new userMaximal velocity of photons in non-relativistic QED
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We consider the problem of propagation of photons in the quantum theory of non-relativistic matter coupled to electromagnetic radiation, which is, presently, the only consistent quantum theory of matter and radiation. Assuming that the matter system is in a localized state (i.e for energies below the ionization threshold), we show that the probability to find photons at time t at the distance greater than ct, where c is the speed of light, vanishes as t goes to infinity as an inverse power of t.
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We consider an atom interacting with the quantized electromagnetic field in the standard model of non-relativistic QED. The nucleus is supposed to be fixed. We prove smoothness of the resolvent and local decay of the photon dynamics for quantum states in a spectral interval I just above the ground state energy. Our results are uniform with respect to I. Their proofs are based on abstract Mourres theory, a Mourre inequality established in [FGS1], Hardy-type estimates in Fock space, and a low-energy dyadic decomposition.
We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({mathbb R^2})$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $Lambda_q$, $q in {mathbb Z}_+$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${rm Op}^{rm w}({mathcal V})$ with real-valued Weyl symbol ${mathcal V}$, such that ${rm Op}^{rm w}({mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{{mathcal V}} = H_0 + {rm Op}^{rm w}({mathcal V})$. First, we construct symbols ${mathcal V}$, possessing a suitable symmetry, such that the operator $H_{mathcal V}$ admits an explicit eigenbasis in $L^2({mathbb R^2})$, and calculate the corresponding eigenvalues. Moreover, for ${mathcal V}$ which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of $H_{mathcal V}$ adjoining any given $Lambda_q$. We find that the effective Hamiltonian in this context is the Toeplitz operator ${mathcal T}_q({mathcal V}) = p_q {rm Op}^{rm w}({mathcal V}) p_q$, where $p_q$ is the orthogonal projection onto ${rm Ker}(H_0 - Lambda_q I)$, and investigate its spectral asymptotics.
We consider some compact non-selfadjoint perturbations of fibered one-dimensional discrete Schrodinger operators. We show that the perturbed operator exhibits finite discrete spectrum under suitable- regularity conditions.
We study the spectrum of the linear operator $L = - partial_{theta} - epsilon partial_{theta} (sin theta partial_{theta})$ subject to the periodic boundary conditions on $theta in [-pi,pi]$. We prove that the operator is closed in $L^2([-pi,pi])$ with the domain in $H^1_{rm per}([-pi,pi])$ for $|epsilon| < 2$, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in $H^1_{rm per}([-pi,pi])$.
Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2(mathbb{R}^2; dt dx)$ of the type $D_A = (d/dt) + A$, where $A = int^{oplus}_{mathbb{R}} dt , A(t)$, and the family of self-adjoint operators $A(t)$ in $L^2(mathbb{R}; dx)$ is explicitly given by $A(t) = - i (d/dx) + theta(t) phi(cdot)$, $t in mathbb{R}$. Here $phi: mathbb{R} to mathbb{R}$ has to be integrable on $mathbb{R}$ and $theta: mathbb{R} to mathbb{R}$ tends to zero as $t to - infty$ and to $1$ as $t to + infty$. In particular, $A(t)$ has asymptotes in the norm resolvent sense $A_- = - i (d/dx)$, $A_+ = - i (d/dx) + phi(cdot)$ as $t to mp infty$. Since $D_A$ violates the relative trace class condition introduced in [9], we now employ a new approach based on an approximation technique. The approximants do fit the framework of [9] and lead to the following results: Introducing $H_1 = {D_A}^* D_A$, $H_2 = D_A {D_A}^*$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$ W_r(D_A) = lim_{lambda to 0} (- lambda) {rm tr}_{L^2(mathbb{R}^2; dtdx)}((H_1 - lambda I)^{-1} - (H_2 - lambda I)^{-1}). $$ In the concrete example at hand, we prove $$ W_r(D_A) = xi(0_+; H_2, H_1) = xi(0; A_+, A_-) = 1/(2 pi) int_{mathbb{R}} dx , phi(x). $$ Here $xi(, cdot , ; S_2, S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $xi(lambda; H_2, H_1) = 0$, $lambda < 0$.
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