Narrow Results

We realize the number of bound states of a Schrödinger operator on ${ℝ}^n$ as an index pairing in all dimensions. Expanding on ideas of Guillopé and others, we use high-energy corrections to find representatives of the $K$-theory class of the scattering operator. These representatives allow us to compute the number of bound states using an integral formula involving heat kernel coefficients.

We study the semiclassical distribution of resonances of a $2\times 2$ matrix Schrödinger operator (1.1) near a fixed energy level where the underlying classical trajectories ${\mathrm{\Gamma }}_1$ of $P_1$ and ${\mathrm{\Gamma }}_2$ of $P_2$ are, respectively, periodic and non-trapping. The aim is to compute the imaginary part of the resonances appearing near the eigenvalues created by $P_1$ when ${\mathrm{\Gamma }}_1$ intersects ${\mathrm{\Gamma }}_2$ with finite contact order $m$. Recent results [M. Assal, S. Fujiié and K. Higuchi, Semiclassical resonance asymptotics for systems with degenerate crossings of classical trajectories, Int. Math. Res. Not.2024 (2024) 6879–6905] and [M. Assal, S. Fujiié and K. Higuchi, Transition of the semiclassical resonance widths across a tangential crossing energy-level, J. Math. Pures Appl.191 (2024) 103634] assert that the width of resonances is of polynomial order in the semiclassical parameter with exponent $1+2/(m+1)$, if the interaction $U=r_0(x)+ih{r}_1(x)D_x$ is elliptic at the crossing point. Here, we remove this ellipticity assumption and prove that the exponent is $1+2(k+1)/(m+1)$, where $k$ is a “vanishing order” of the interaction at the crossing points, suitably defined in terms of the vanishing orders of $r_0$ and $r_1$ depending on whether the crossing point is a turning point or not.

In this paper, we construct metastable states of atoms interacting with the quantized radiation field. These states emerge from the excited bound states of the non-interacting system. We prove that these states obey an exponential time-decay law. In detail, we show that their decay is given by an exponential function in time, predicted by Fermi's Golden Rule, plus a small remainder term. The latter is proportional to the (4+β)th power of the coupling constant and decays algebraically in time. As a result, though it is small, it dominates the decay for large times. A central point of the paper is that our remainder term is significantly smaller than the one previously obtained in [1] and as a result we are able to show that the time interval during which the Fermi's Golden Rule can be observed is significantly longer that the time interval obtained in [1]. This improvement is achieved by incorporating a part of the complex dilatation resonance states into our construction of the metastable states rather than using the unperturbed eigenstates (the excited bound states of the non-interacting system). Thus, the connection to resonance states allows us to introduce metastable states which qualify better in the description of unstable excited states of the interacting system.

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on ℝ with multiplicity space , , is constructed such that exactly the resonances (poles of the inverse of the Livšic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti)linear forms are calculated explicitly. Using the wave matrices for the wave (Möller) operators the corresponding eigen(anti)linear forms on the Schwartz space for the unperturbed Hamiltonian H₀ are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector λ → k/(ζ₀ - λ)^-1, ζ₀ resonance, , which is uniquely determined by restriction of to , where denotes the Hardy space of the upper half-plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup for t ≥ 0 of the Toeplitz type on . That is: Exactly those pre-Gamov vectors λ → k/(ζ - λ)^-1, ζ from the lower half-plane, , have an extension to a generalized eigenvector of H if ζ is a resonance and if k is from that subspace of which is uniquely determined by its corresponding Dirac type antilinear form.

The addendum refers mainly to Sec. 5 of the paper (Friedrichs model on the positive half line). The "Schwartz space framework" is omitted because it is dispensable for the results. Improvements of the proofs are indicated. A supplement presents in the special case G₀ := ℂ\(-∞,0] a surprising implication: the scattering matrix has only simple poles and its "main part" is a linear combination of all Gamov vectors.

We consider the selfadjoint operator H = H₀ + V, where H₀ is the free semi-classical Dirac operator on ℝ³. 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 complex distortions of ℝ³. 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-classical resonances of H and a local trace formula are obtained. In particular, if V is an electro-magnetic potential, we deduce a Weyl-type asymptotics of the spectral shift function. As a by-product, we obtain an upper bound O(h^-2) for the number of resonances close to non-critical energy levels in domains of width h and a Breit–Wigner approximation formula for the derivative of the spectral shift function.

We study the evolution of a one-dimensional model atom with δ-function binding potential, subjected to a dipole radiation field E(t)x with E(t) a 2π/ω-periodic real-valued function. We prove that when E(t) is a trigonometric polynomial, complete ionization occurs, i.e. the probability of finding the electron in any fixed region goes to zero as t → ∞.

For ψ(x, t = 0) compactly supported and general periodic fields, we decompose ψ(x, t) into uniquely defined resonance terms and a remainder. Each resonance is 2π/ω periodic in time and behaves like the exponentially growing Green's function near x = ±∞. The remainder is given by an asymptotic power series in t^-1/2 with coefficients varying with x.

The operator-theoretic renormalization group (RG) methods are powerful analytic tools to explore spectral properties of field-theoretical models such as quantum electrodynamics (QED) with non-relativistic matter. In this paper, these methods are extended and simplified. In a companion paper, our variant of operator-theoretic RG methods is applied to establishing the limiting absorption principle in non-relativistic QED near the ground state energy.

We investigate the existence of resonances for two-center Coulomb systems with arbitrary charges in two dimensions, defining them in terms of generalized complex eigenvalues of a non-selfadjoint deformation of the two-center Schrödinger operator. We construct the resolvent kernels of the operators and prove that they can be extended analytically to the second Riemann sheet. The resonances are then analyzed by means of perturbation theory and numerical methods.

We consider a non-self-adjoint $H$ acting on a complex Hilbert space. We suppose that $H$ is of the form $H=H_0+CWC$ where $C$ is a bounded, positive definite and relatively compact with respect to $H_0$, and $W$ is bounded. We suppose that $C{(H_0-z)}^{-1}C$ is uniformly bounded in $z\in ℂ\setminus ℝ$. We define the regularized wave operators associated to $H$ and $H_0$ by $W_{\pm }(H,H_0):={\mathrm{\text{s-lim}}}_{t\to \infty }{e}^{\pm itH}{r}_{\mp }(H){\mathrm{\Pi }}_{\mathrm{\text{p}}}{(H^{\star })}^{\perp }{e}^{\mp it{H}_0}$ where ${\mathrm{\Pi }}_{\mathrm{\text{p}}}(H^{\star })$ is the projection onto the direct sum of all the generalized eigenspaces associated to eigenvalues of $H^{\star }$ and $r_{\mp }$ is a rational function that regularizes the “incoming/outgoing spectral singularities” of $H$. We prove the existence and study the properties of the regularized wave operators. In particular, we show that they are asymptotically complete if $H$ does not have any spectral singularity.

The 3α resonant states of ¹²C are investigated by taking into account the correct boundary condition for three-body resonant states. In order to show how the 3α resonant states having complex eigenvalues contribute to the real energy, we calculated the Continuum Level Density in the Complex Scaling Method.

We study signals for beyond standard model physics and consider the virtues of single photon signals or associated photons in the final states in identifying different scenarios of new physics models in a very efficient and novel way.

In the framework of a simple quantum field theory describing the decay of a scalar state into two (pseudo)scalar ones we study the pole(s) motion(s) of its propagator: besides the expected pole on the second Riemann sheet, we find — for a large enough coupling constant — a second, additional pole on the first Riemann sheet below threshold, which corresponds to a stable state. We then perform a numerical study for a hadronic system in which a scalar particle couples to pions. We investigate under which conditions a stable state below the two-pion threshold can emerge. In particular, we study the case in which this stable state has a mass of 38 MeV, which corresponds to the recently claimed novel scalar state E(38). Moreover, we also show that the resonance f₀(500) and the stable state E(38) could be two different manifestations of the same "object". Finally, we also estimate the order of magnitude of its coupling to photons.

In the framework of the modified potential cluster model (PCM), the possibility of describing the available experimental data for the total cross-sections for n¹¹B radiative capture at thermal and astrophysical energies were considered with taking into account the 21 keV and 430 keV resonances.

In this paper, we study analytical solutions for fermion localization in Randall–Sundrum (RS) models. We show that there exist special couplings between scalar fields and fermions giving us discrete massive localizable modes. Besides this we obtain resonances in some models by analytical methods.

This article summarizes three searches for diboson resonances in the all-hadronic final state using data collected at a center-of-mass energy of $\sqrt{\mathrm{\text{s}}}$ = 13 TeV with the CMS experiment at the CERN LHC. The boson decay products are contained in one large-radius jet, resulting in dijet final states which are resolved using jet substructure techniques. The analyses presented use 2.3 fb${}^{-1}$, 35.9 fb${}^{-1}$ and 77.3 fb${}^{-1}$ of data collected between 2015 and 2017. These include the first search for diboson resonances in data collected at a 13 TeV collision energy, the introduction of a new algorithm to tag vector bosons in the context of analyzing the data collected in 2016, and the development of a novel multidimensional fit improving on the sensitivity of the previous search method with up to 30%. The results presented here are the most sensitive to date of all diboson resonance searches in the dijet final state. An emphasis on improvements in technique for vector boson tagging is made.

The relative center-of-mass energy spread ${\sigma }_W∕W$ at $e^{+}{e}^{-}$ colliders is $\mathcal{𝒪}(10^{-3})$, which is much larger than the widths of narrow resonances $J∕\psi$, $\psi (2S)$, $\mathrm{\Upsilon }(1S)$, $\mathrm{\Upsilon }(2S)$, $\mathrm{\Upsilon }(3S)$ mesons, tauonium and some others. Its reduction would significantly increase the resonance production rates and open up great opportunities in the search for new physics. In this paper, we propose a new monochromatization method for colliders with a large crossing angle (which can provide a high luminosity). The contribution of the beam energy spread to ${\sigma }_W$ is canceled by introducing an appropriate energy–angle correlation at the interaction point; ${\sigma }_W∕W\sim (0.5-1)\times 10^{-5}$ appears possible.

We review the analytic results for the phase shifts δ_l(k) in nonrelativistic 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 nonrelativistic 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.

Considering the strong coupling to a field of two particles bound by a quark–quark-like potential, we calculate the energy and width of a hybrid state in a simplified model.

We have measured the reaction γp→η′p from threshold up to 2.6 GeV incident beam energy using the Crystal Barrel and TAPS detectors at the ELSA accelerator. At present the knowledge about the resonance contribution to this process is very limited. Our measurement is a significant improvement of the data basis. We will present preliminary results.