Unphysical Sheet Research Articles

On the location of resonances of exponentially decaying Sturm–Liouville potentials

The paper is concerned with analytic continuation into the unphysical sheet for Sturm–Liouville potentials whose decay at infinity is no faster than O(e − ax ). A resonance-free region is found and the existence of a natural boundary is discussed. An example related to the Bessel equation is examined in some detail.

Resonances of a λ-rational Sturm–Liouville problem

We consider a family of self-adjoint 2 × 2-block operator matrices Ãϑ in the space L2(0, 1) ⊕ L2(0, 1), depending on the real parameter ϑ. If Ã0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for ϑ ≠ 0 this eigenvalue in general disappears, but the resolvent of Ãϑ has a pole on the unphysical sheet of the Riemann surface. Such a pole is called a resonance pole. The unphysical sheet arises from analytic continuation from the upper half-plane ℂ+ across the essential spectrum. Furthermore, the asymptotic behaviour of this resonance pole for small ϑ is investigated. The results are proved by considering a certain λ-rational Sturm-Liouville problem and its Titchmarsh–Weyl coefficient.

Resonances of a λ-rational Sturm–Liouville problem

We consider a family of self-adjoint 2 × 2-block operator matrices Ãϑ in the space L2(0, 1) ⊕ L2(0, 1), depending on the real parameter ϑ. If Ã0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for ϑ ≠ 0 this eigenvalue in general disappears, but the resolvent of Ãϑ has a pole on the unphysical sheet of the Riemann surface. Such a pole is called a resonance pole. The unphysical sheet arises from analytic continuation from the upper half-plane C+ across the essential spectrum. Furthermore, the asymptotic behaviour of this resonance pole for small ϑ is investigated. The results are proved by considering a certain λ-rational Sturm–Liouville problem and its Titchmarsh–Weyl coefficient.

Resonances and analytic continuation for exponentially decaying Sturm–Liouville potentials

The Sturm–Liouville equation on the half-line [0,∞) is considered with a potential which decays exponentially at infinity. It is known that Green's function has an analytic continuation into part of the unphysical sheet im λ <0 . Here it is shown that analytic continuation into the whole of this sheet is possible for certain types of decay. The methods are constructive, and they also provide a new and rigorous way of locating resonances.

Operator Interpretation of Resonances Arising in Spectral Problems for 2 × 2 Operator Matrices

AbstractWe consider operator matrices \documentclass{article}\pagestyle{empty}\begin{document}${\rm{H = }}\left( {{_{B_{10} }^{A{_0}}} {_{A_{1} }^{B{_{01}}}} } \right)$\end{document} with self ‐ adjoint entries Ai, i = 0,1, and bounded B10 = B10 acting in the orthogonal sum μ=μ0 ⊕ μ1 of Hilbert spaces μ0 and μ1. We are especially interested in the case where the spectrum of, say, A1 is partly or totally embedded into the continuous spectrum of A0 and the transfer function V1(z) = B10 where (χ ‐ A0)−1 B01, admits analytic continuation (as an operator ‐ valued function) through the cuts along branches of the continuous spectrum of the entry A0 into the unphysical sheet(s) of the spectral parameter plane. The values of χ in the unphysical sheets where M1−1(z) and consequently the resolvent (H ‐ z)−1 have poles are usually called resonances. A main goal of the present work is to find non ‐ selfadjoint operators whose spectra include the resonances as well as to study the completeness and basis properties of the resonance eigenvectors of M1(z) in μ1. To this end we first construct an operator ‐ valued function V1(Y) on the space of operators in μ1 possessing the property: V1(Y)ϕ1 = V1(z)ϕ1 for any eigenvector ϕ1 of Y corresponding to an eigenvalue z and then study the equation H1 = A1 + V1(H1). We prove the solvability of this equation even in the case where the spectra of A0 and A1 overlap. Using the fact that the root vectors of the solutions H1 are at the same time such vectors for M1(z), we prove completeness and even basis properties for the root vectors (including those for the resonances).

Confined coherence and analytic properties of Green’s functions

A simple model of noninteracting electrons with a separable one-body potential is used to discuss the possible pole structure of single particle Green's functions for fermions on unphysical sheets in the complex frequency plane as a function of the system parameters. The poles in the exact Green's function can cross the imaginary axis, in contrast to recent claims that such a behaviour is unphysical. As the Green's function of the model has the same functional form as an approximate Green's function of coupled Luttinger liquids no definite conclusions concerning the concept of "confined coherence" can be drawn from the locations of the poles of this Green's function.

Operator interpretation of the resonances generated by 2×2 matrix Hamiltonians

We consider the analytic continuation of the transfer function for a 2x2 matrix Hamiltonian into the unphysical sheets of the energy Riemann surface. We construct a family of non-selfadjoint operators which reproduce certain parts of the transfer-function spectrum including resonances situated on the unphysical sheets neighboring the physical sheet. On this basis, completeness and basis properties for the root vectors of the transfer function (including those for the resonances) are proved.

Representations for the Three‐Body T‐Matrix, Scattering Matrices and Resolvent in Unphysical Energy Sheets

AbstractExplicit representations for the Faddeev components of the three‐body T‐matrix continued analytically into unphysical sheets of the energy Riemann surface are formulated. According to the representations, the T‐matrix in unphysical sheets is explicitly expressed in terms of its components taken in the physical sheet only. The representations for the T‐matrix are then used to construct similar representations for the analytic continuation of the three‐body scattering matrices and the resolvent. Domains on unphysical sheets are described where the representations obtained can be applied. A method for finding three‐body resonances based on the Faddeev differential equations is proposed.

Resonance effects in low-energy electron scattering by HBr

Earlier ab initio calculations for electron scattering by HBr are analysed to obtain the behaviour of the S-matrix poles in the complex momentum plane as a function of the internuclear distance. This analysis shows the existence of a pole corresponding to a bound state at large internuclear distances as well as a pole corresponding to a shape resonance at small internuclear distances. It is found that the bound-state pole does not evolve into the shape resonance pole as the internuclear distance decreases but instead splits into two mirror image poles which move onto unphysical Riemann sheets of the complex momentum plane confirming model-potential calculations by Herzenberg and Saha. The physical implications of the mirror image poles and the shape resonance pole are also discussed.

Summation of the eigenvalue perturbation series by multi-valued Pade approximants: application to resonance problems and double wells

Quadratic Pade approximants are used to obtain energy levels both for the anharmonic oscillator x2/2- lambda x4 and for the double well -x2/2+ lambda x4. In the first case, the complex-valued energy of the resonances is reproduced by summation of the real terms of the perturbation series. The second case is treated formally as an anharmonic oscillator with a purely imaginary frequency. We use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function. Then, we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants. In this way we can reconstruct the real energy by summation of the complex series. Such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory (around the minima of the double well potential) incorrect.

Rigorous QCD-potential for the $$t\bar t$$ -system at threshold

Recent evidence for the top mass in the region of 160 GeV for the first time provides an opportunity to use the full power of relativistic quantum field theoretical methods, available also for weakly bound systems. Because of the large decay width Γ of the top quark individual energy-levels in “toponium” will be unobservable. However, the potential for the\(t\bar t\) system, based on a systematic expansion in powers of the strong coupling constant α s can be rigorously derived from QCD and plays a central role in the threshold region. It is essential that the neglect of nonperturbative (confining) effects is fully justified here for the first time to a large accuracy, also justbecause of the large Γ. The different contributions to that potential are computed from real level corrections near the bound state poles of the\(t\bar t\)-system which for Γ≠0 move into the unphysical sheet of the complex energy plane. Thus, in order to obtain the different contributions to that potential we may use the level corrections at that (complex) pole. Within the relevant level shifts we especially emphasize the corrections of orderO(α 4 s m t ) and numerically comparable ones to that order also from electroweak interactions which may become important as well.

Longitudinal critical angle singularities and their effect on V(Z) of the line-focus-beam acoustic microscope

In order to investigate a mechanism which causes a velocity difference between the longitudinal wave and leaky surface skimming compressional wave (LSSCW) observed in a line-focus-beam acoustic microscope, the analytic property of an acoustic reflection coefficient and its effect on a V(z) analysis were studied. A pole hidden in the unphysical Riemann sheet close to the longitudinal branch point is found to be responsible for the abrupt phase change at the longitudinal critical angle. This, together with an effect of a dominant Rayleigh wave pole, affects the V(z) measurement of the LSSCW. A method to estimate the longitudinal and shear wave velocities is discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Two-dimensional analytic theory for toroidal Alfvén eigenmodes

A two-dimensional analysis of the toroidal Alfvén eigenmodes (TAE) is presented, based on an integrodifferential equation describing the shear Alfvén perturbation of a toroidal plasma equilibrium in terms of coupling among the toroidal Alfvén continua with the usual gap structure. Using a method similar to the Van Kampen–Case analysis for the Vlasov equation, exact analytic expressions are derived for the dispersion function and the two-dimensional eigenmode structure. The dispersion function is expressed in terms of Cauchy-type integrals, which explicitly expresses the global character of TAE modes and facilitates the calculation of their damping. The continuum-damped TAE modes are shown to be, in general, not true eigenmodes of the toroidal plasma equilibrium, but rather resonances corresponding to zeros of the analytic continuation of the dispersion function onto unphysical sheets of its Riemann surface. Approximate but explicit expressions for the dispersion relation and the eigenfunction are also obtained in the limit of vanishing inverse aspect ratio.

The mass and the width of the Higgs in the strongly interacting minimal standard model

By the use of the [1, 1] Padé approximanton to the one-loop amplitude for the W LW L → W LW L scattering in the strongly interacting regime of the minimal standard model we build up a new amplitude for this process which has all the required physical properties such as analyticity, unitarity, absence of ghosts, etc. We also show that the I = J = 0 partial wave amplitude has a simple pole in the unphysical sheet which can be interpreted as the physical Higgs particle. The mass of this Higgs particle is shown to be finite and in particular under 1.4 TeV in the limit λ( m H) going to infinity.

Movement and interplay of the bound state, resonance, and shadow poles of the scattering amplitude in multiphoton processes.

We examine, in a heuristic fashion, the analytic structure of the scattering amplitude for an electron interacting with an atomic potential in the presence of a radiation field. For each resonance pole of the amplitude there are infinitely many shadow poles lying on different unphysical energy sheets of the infinitely many sheeted Riemann surface. The pole having the dominant influence on the scattering amplitude is the one closest to the physical energy axis. All poles undergo significant movement when the field intensity is varied, and when a resonance pole which is dominant passes by a multiphoton ionization threshold a shadow pole usually moves closer to the physical energy axis and hence becomes the dominant pole. When this happens, an initially bound electron may jump eigenvalue curves, and in this way the electron can undergo a very large (ponderomotive) shift in its energy and still maintain the correct physical character of its wave function. The movement of resonance poles near thresholds has implications for the fate of autoionizing states and for population trapping. Consideration of threshold effects might also shed light on a puzzling result of a recent energy-shift measurement. We address the problem of how to determine, in numerical calculations, which sheet the energy eigenvalue is on so that the dominant pole can be identified. We illustrate some of our remarks by results of numerical calculations, and we also touch on the question of the completeness of a basis set in numerical calculations of resonance eigenvalues.

Low-energy electron scattering by HF

The R-matrix method is used to calculate integral and differential cross sections for low energy, vibrationally inelastic scattering of electrons by HF. Nuclear vibration is included using a non-adiabatic approximation. The authors find sharp peaks in the cross sections very close to each vibrational threshold. These can be associated with S-matrix poles lying on an unphysical energy sheet just below each threshold and can therefore be interpreted as nuclear-excited Feshbach resonances. The role of the long-range dipole interaction is also discussed. Differential cross sections are calculated and are found to be non-isotropic in the vicinity of the threshold peaks.

Extension of the minimal three-nucleon model to the unphysical sheet of energy.

In the framework of the three-nucleon zero-range theory, we have calculated the correlation between the energy of the virtual state of the triton and the doublet neutron-deuteron scattering length. Our results show that the Girard-Fuda plot for the energy of the virtual state as a function of the doublet n-d scattering length is reproduced by this model. We also observe that the plot remains essentially unchanged when the nucleon-nucleon singlet input is varied.

Pole structure of the J pi =3/2(+) resonance in 5He.

The S-matrix pole structure of the ${\mathrm{J}}^{\mathrm{\ensuremath{\pi}}}$${=(3/2}^{+}$ resonance in $^{5}\mathrm{He}$ is obtained from R-matrix parameters for the system. Two poles are found on different unphysical Riemann sheets. One is a conventional resonance that is primarily responsible for structure in the n-\ensuremath{\alpha} total cross section. The other is a ``shadow'' pole that contributes to the large values of the cross section for the reaction $^{3}\mathrm{H}$(d,n${)\mathrm{}}^{4}$He at low energies. It is the first experimental evidence for the existence of shadow poles in nuclear and particle physics. The two-pole structure of the resonance accounts for different peak positions and widths in the amplitudes for the n-\ensuremath{\alpha} total cross section and for the $^{3}\mathrm{H}$(d,n) reaction cross section.

Efimov states for three 4He atoms?

The system of three 4He atoms interacting through the intermolecular potential HFDHE2 is solved exactly within the Faddeev scheme. We find that the trimer has two bound states at E=−0.11 K and E=−0.16(−2) K, where the excited state shows the property of an Efimov state. It is overrun by the two-body fragmentation threshold for an artificially enhanced overall interaction strength. Uncertainties in the potential parameters, possibly combined with solvent effects, may allow a slightly weaker force, which supports a second Efimov state. In any case dynamical effects of the Efimov states will be the same whether it lies close to E=0 on the physical or unphysical sheet.

Discrete symmetry for bound and virtual levels

Proceeding from the analytic continuation of the Lippmann-Schwinger integral equation onto the unphysical energy sheet, a new simple formula for the residue of the t-matrix in the pole on the unphysical sheet is derived. The formula expresses the residue in terms of the partial scattering amplitude at the corresponding energy point on the physical sheet. For the Yukawa type potential a new symmetry theorem for bound and virtual levels is proved. The theorem makes it possible to indicate the points on level trajectories which are mirror-symmetrical with respect to zero momentum.