We review the analytic results for the phase shifts delta_{l}(k) in non-relativistic scattering from a spherical well. The conditions for the existence of resonances are established in terms of time-delays. Resonances are shown to exist for p-waves (and higher angular momenta) but not for s-waves. These resonances occur when the potential is not quite strong enough to support a bound p-wave of zero energy. We then examine relativistic scattering by spherical wells and barriers in the Dirac equation. In contrast to the non-relativistic situation, s-waves are now seen to possess resonances in scattering from both wells and barriers. When s-wave resonances occur for scattering from a well, the potential is not quite strong enough to support a zero momentum s-wave solution at E = m. Resonances resulting from scattering from a barrier can be explained in terms of the `crossing' theorem linking s-wave scattering from barriers to p-wave scattering from wells. A numerical procedure to extract phase shifts for general short range potentials is introduced and illustrated by considering relativistic scattering from a Gaussian potential well and barrier.
We show that the energy spectrum of the one-dimensional Dirac equation in the presence of a spatial confining point interaction exhibits a resonant behavior when one includes a weak electric field. After solving the Dirac equation in terms of parabolic cylinder functions and showing explicitly how the resonant behavior depends on the sign and strength of the electric field, we derive an approximate expression for the value of the resonance energy in terms of the electric field and delta interaction strength.
We prove that the class of resonances of Dirac operators on the half-line with compactly supported potentials is closed with respect to $\ell^1$ perturbations. We also prove that the potential depends continuously on such perturbations. We show that similar results hold true for the Jost functions and Hermite-Biehler functions associated with Dirac operators.
We consider the massless Dirac operator $H=α\cdot D+Q(x)$ on the Hilbert space $L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$, where $Q(x)$ is a $4\times4$ Hermitian matrix valued function which suitably decays at infinity. We show that the the zero resonance is absent for $H$, extending recent results of Saitō-Umeda and Zhong-Gao.
It is shown that a Dirac particle of mass $m$ and arbitrarily small momentum will tunnel without reflection through a potential barrier $V=U_c(x)$ of finite range provided that the potential well $V=-U_c(x)$ supports a bound state of energy $E=-m.$ This is called a supercritical potential well.
We consider massless Dirac operators on the half-line with compactly supported potentials. We solve the inverse problems in terms of Jost function and scattering matrix (including characterization). We study resonances as zeros of Jost function and prove that a potential is uniquely determined by its resonances. Moreover, we prove the following: 1) resonances are free parameters and a potential continuously depends on a resonance, 2) the forbidden domain for resonances is estimated, 3) asymptotics of resonance counting function is determined, 4) these results are applied to canonical systems.
We consider the 1D massless Dirac operator on the real line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: 1) asymptotics of counting function, 2) estimates on the resonances and the forbidden domain, 3) the trace formula in terms of resonances.
We consider massless Dirac operators on the real line with compactly supported potentials. We solve two inverse problems (including characterization): in terms of zeros of reflection coefficient and in terms of poles of reflection coefficients (i.e. resonances). We prove that a potential is uniquely determined by zeros of reflection coefficients and there exist distinct potentials with the same resonances. We describe the set of "isoresonance potentials". Moreover, we prove the following: 1) a zero of the reflection coefficient can be arbitrarily shifted, such that we obtain the sequence of ze…
We have solved exactly the two-component Dirac equation in the presence of a spatially one-dimensional Hulthén potential, and presented the Dirac spinors of scattering states in terms of hypergeometric functions. We have calculated the reflection and transmission coefficients by the matching conditions on the wavefunctions, and investigated the condition for the existence of transmission resonances. Furthermore, we have demonstrated how the transmission-resonance condition depends on the shape of the potential.
The relativistic problem of spin- fermions subject to vector hyperbolic (kink-like) potential tanh (kx) is investigated by using the parametric Nikiforov-Uvarov method. The energy eigenvalue equation and the corresponding normalized wave functions are obtained in terms of the Jacobi polynomials for x>0 and x<0 cases.
We give an expansion of the solution of the evolution equation for the massless Dirac fields in the outer region of de Sitter-Reissner-Nordström black hole in terms of resonances. By means of this method we describe the decay of local energy for compactly supported data. The proof uses the cut-off resolvent estimates for the semi-classical Schrödinger operators from Bony and Häfner, 2008. The method extends to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.
The zero modes and zero resonances of the Dirac operator $H=α\cdot D + Q(x)$ are discussed, where $α= (α_1, α_2, α_3)$ is the triple of $4 \times 4$ Dirac matrices, $ D=\frac{1}{i} \nabla_x$, and $Q(x)=\big(q_{jk} (x) \big)$ is a $4\times 4$ Hermitian matrix-valued function with $| q_{jk}(x) | \le C < x >^{-ρ} $, $ρ>1$. We shall show that every zero mode $f(x)$ is continuous on ${\mathbb R}^3$ and decays at infinity with the decay rate $|x|^{-2}$. Also, we shall show that $H$ has no zero resonance if $ρ> 3/2$.
The shifted-i expansion technique (SLET) is extended to solve for Dirac particle trapped in spherically symmetric scalar and/or 4-vector potentials. A parameter λ=0,1 is introduced in such a way that one can obtain the Klein-Gordon (KG) bound states from Dirac bound states. The 4-vector Coulomb, the scalar linear, and the equally mixed scalar and 4-vector power-law potentials are used in KG and Dirac equations. Exact numerical results are obtained for the 4-vector Coulomb potential in both KG and Dirac equations. Highly accurate and fast converging results are obtained for the scalar linear an…
We consider the perturbations $H := H_{0} + V$ and $D := D_{0} + V$ of the free 3D Hamiltonians $H_{0}$ of Pauli and $D_{0}$ of Dirac with non-constant magnetic field, and $V$ is a electric potential which decays super-exponentially with respect to the variable along the magnetic field. We show that in appropriate Banach spaces, the resolvents of $H$ and $D$ defined on the upper half-plane admit meromorphic extensions. We define the resonances of $H$ and $D$ as the poles of these meromorphic extensions. We study the distribution of resonances of $H$ close to the origin 0 and that of $D$ close …
We consider a 3-dimensional Dirac operator H_0 with non-constant magnetic field of constant direction, perturbed by a sign-definite matrix-valued potential V decaying fast enough at infinity. Then we determine asymptotics, as the energy goes to +m and -m, of the spectral shift function for the pair (H_0,H_0+V). We obtain, as a by-product, a generalised version of Levinson's Theorem relating the eigenvalues asymptotics of H_0+V near +m and -m to the scattering phase shift for the pair (H_0,H_0+V).
Approximate scattering and bound state solutions of the one-dimensional effective-mass Dirac equation with the Woods-Saxon potential are obtained in terms of the hypergeometric-type functions. Transmission and reflection coefficients are calculated by using behavior of the wave functions at infinity. The same analysis is done for the constant mass case. It is also pointed out that our results are in agreement with those obtained in literature. Meanwhile, an analytic expression is obtained for the transmission resonance and observed that the expressions for bound states and resonances are equal…
We consider the self-adjoint operator $H=H_0+V$, where $H_0$ is the free semi-classical Dirac operator on $R^3$. We suppose that the smooth matrix-valued potential $V=O(<x>^{-δ}), δ>0,$ has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator $H$ by a complex distortions of $R^{3}$.We establish an upper bound $O(h^{-3})$ for the number of resonances in any compact domain. For $δ>3$, a representation of the derivative of the spectral shift function $ξ(λ,h)$ related to the semi…
We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr-Newman-de Sitter black holes. The resonances split in a way similar to the Zeeman effect. The method is based on the extension to Dirac operators of techniques applied by Dyatlov 2011, 2012 to the (uncharged) Kerr-de Sitter black holes. We show that the mass of the Dirac field does not have effect on the two leading terms in the expansions of resonances.
We derive the relativistic Hamiltonian of hydrogen atom in dynamical noncommutative spaces (DNCS or τ-space). Using this Hamiltonian we calculate the energy shift of the ground state and as well the [2P]_(1/2), [2S]_(1/2) levels. In all cases the energy shift depend on the dynamical noncommutative parameter τ. Using the accuracy of the energy measurement we obtain an upper bound for τ. We also study the lamb shift in DNCS. Both the levels [2P]_(1/2) and [2S]_(1/2) receive corrections due to dynamical noncommutativity of space which is in contrast to the non-dynamical noncommutative spaces (NDN…
The properties of a prescription for the inner products of the resonance (Gamow states), scattering (Dirac kets), and bound states for 1-dimensional quantum barriers are worked out. The divergent asypmtotic behaviour of the Gamow states is regularized using a Gaussian convergence factor first introduced by Zel'dovich. With this prescription, most of these states (with discrete complex energies) are found to be orthogonal to each other, to the bound states, and to the Dirac kets, except when they are neighbors, in which case the inner product is divergent. Therefore, as it happens for the conti…
We study the global symmetries of naive lattices Dirac operators in QCD-like theories in any dimension larger than two. In particular we investigate how the chosen number of lattice sites in each direction affects the global symmetries of the Dirac operator. These symmetries are important since they do not only determine the infra-red spectrum of the Dirac operator but also the symmetry breaking pattern and, thus, the lightest pseudo-scalar mesons. We perform the symmetry analysis and discuss the possible zero modes and the degree of degeneracy of the lattice Dirac operators. Moreover we expli…
We consider the first order periodic systems perturbed by a $2N\ts 2N$ matrix-valued periodic potential on the real line. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has branch points, which we call resonances. We prove the existence of real or complex resonances. We determine the asymptotics of the periodic, anti-p…
We consider a Dirac operator with a dislocation potential on the real line. The dislocation potential is a fixed periodic potential on the negative half-line and the same potential but shifted by real parameter $t$ on the positive half-line. Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each non-empty gap. Its resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface. We prove that it has only two simple poles on each open gap: on the first sheet (an eigenvalue) or on the second sheet (a resonance). Th…
This paper is a continuation of my previous work on absolutely continuous and singular spectral shift functions, where it was in particular proved that the singular part of the spectral shift function is an a.e. integer-valued function. It was also shown that the singular spectral shift function is a locally constant function of the coupling constant $r,$ with possible jumps only at resonance points. Main result of this paper asserts that the jump of the singular spectral shift function at a resonance point is equal to the so-called resonance index, --- a new (to the best of my knowledge) noti…
We investigate $L^1\to L^\infty$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\frac12}$ decay rate, which may be improved to $t^{-\frac12-γ}$ for any $0\leq γ<\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natura…
We consider an open (scattering) quantum system under the action of a perturbation of its closed counterpart. It is demonstrated that the resulting shift of resonance widths is a sensitive indicator of the non-orthogonality of resonance wavefunctions, being zero only if those were orthogonal. Focusing further on chaotic systems, we employ random matrix theory to introduce a new type of parametric statistics in open systems, and derive the distribution of the resonance width shifts in the regime of weak coupling to the continuum.
Our goal is to extend the theory of the spectral shift function to the case where only the difference of some powers of the resolvents of self-adjoint operators belongs to the trace class. As an example, we consider a couple of Dirac operators.
The quasi-normal modes for black holes are the resonances for the scattering of incoming waves by black holes. Here we consider scattering of massless uncharged Dirac fields propagating in the outer region of de Sitter-Reissner-Nordstr{ö}m black hole, which is spherically symmetric charged exact solution of the Einstein-Maxwell equations. Using the spherical symmetry of the equation and restricting to a fixed harmonic the problem is reduced to a scattering problem for the 1D massless Dirac operator on the line. The resonances for the problem are related to the resonances for a certain semiclas…
