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The usual derivations of the S and K matrices for two-particle reactions proceed through the Lippmann–Schwinger equation with formal definitions of the incoming and outgoing scattering states. Here we present a simpler alternative derivation that is carried out completely in the Hamiltonian representation, using a discrete basis of configurations for the scattering channels as well as the quasi-bound configurations of the combined fragments. We use matrix algebra to derive an explicit expression for the K matrix in terms of the Hamiltonian of the internal states of the compound system and the coupling between the channels and the internal states. The formula for the K matrix includes explicitly a real dispersive shift matrix to the internal Hamiltonian that is easily computed in the formalism. That expression is applied to derive the usual form of the S matrix as a sum over poles in the complex energy plane. Some extensions and limitations of the discrete-basis Hamiltonian formalism are discussed in the concluding remarks and in the Appendix.

Proteins are complex, heterogeneous macromolecules that exist as ensembles of interconverting states on a complex energy landscape. A complete, molecular-level understanding of their function requires experimental tools to characterize them with high spatial and temporal precision. Infrared (IR) spectroscopy has an inherently fast time scale that can capture all states and their dynamics with, in principle, bond-specific spatial resolution. Two-dimensional (2D) IR methods that provide richer information are becoming more routine but remain challenging to apply to proteins. Spectral congestion typically prevents selective investigation of native vibrations; however, the problem can be overcome by site-specific introduction of amino acid side chains that have vibrational groups with frequencies in the "transparent window" of protein spectra. This Perspective provides an overview of the history and recent progress in the development of transparent window 2D IR of proteins.

Motivated by recent interest in the zeroes of S-matrix entries in the complex energy plane, I use the Heidelberg model of resonance scattering to introduce the notion of Reflection Time Difference which is shown to play the same role for the {\it zeroes} as the Wigner time delay plays for the S- matrix {\it poles}., Comment: The text is based on the presentation at the 9th Workshop on Quantum Chaos and Localization Phenomena, May 25--26, 2019, Warsaw, Poland

Physical review letters 122(4), 041102 (2019). doi:10.1103/PhysRevLett.122.041102, Published by APS, College Park, Md.

In this study, we have investigated the elastic scattering of 9Be + 27Al reaction using the phenomenological and double-folding model potentials within the framework of the optical model at different incident energies. In the phenomenological and double-folding calculations, we have used new parameters to explain 9Be + 27Al reaction. In the framework of double-folding model, the nuclear matter densities of 9Be projectile and 27Al target have been folded with the complex energy depending on the effective M3Y interaction. The imaginary potentials have been taken as Woods–Saxon volume shape. The real radius r v and imaginary radius r w parameters have been fixed to 1.2 fm in all the calculations. The cross sections, the potential volume integrals and χ 2/N values have been obtained for nine energies.

The origins of life bring into stark relief the inadequacy of our current synthesis of thermodynamic, chemical, physical and information theory to predict the conditions under which complex, living states of organic matter can arise. Origins research has traditionally proceeded under an array of implicit or explicit guiding principles in lieu of a universal formalism for abiogenesis. Within the framework of a new guiding principle for prebiotic chemistry called subsumed complexity , organic compounds are viewed as by-products of energy transduction phenomena at different scales (subatomic, atomic, molecular and polymeric) that retain energy in the form of bonds that inhibit energy from reaching the ground state. There is evidence for an emergent level of complexity that is overlooked in most conceptualizations of abiogenesis that arises from populations of compounds formed from atomic energy input. We posit that different forms of energy input can exhibit different degrees of dissipation complexity within an identical chemical medium. By extension, the maximum capacity for organic chemical complexification across molecular and macromolecular scales subsumes, rather than emerges from, the underlying complexity of energy transduction processes that drive their production and modification. This article is part of the themed issue ‘Reconceptualizing the origins of life’.

Understanding the dynamics of complex systems requires the investigation of their energy landscape. In particular, the flow of probability on such landscapes is a central feature in visualizing the time evolution of complex systems. To obtain such flows, and the concomitant stable states of the systems and the generalized barriers among them, the threshold algorithm has been developed. Here, we describe the methodology of this approach starting from the fundamental concepts in complex energy landscapes and present recent new developments, the threshold-minimization algorithm and the molecular dynamics threshold algorithm. For applications of these new algorithms, we draw on landscape studies of three disaccharide molecules: lactose, maltose, and sucrose.

We construct the scattering matrix for a two-dimensional model of a Cl atom scattering from an OH dimer. We show that the scattering matrix can be written in terms of a non-Hermitian Hamiltonian whose complex energy eigenvalues can be used to compute Wigner-Smith delay times for the Cl-OH scattering process. We compute the delay times for a range of energies, and show that the scattering states with the longest delay times are strongly influenced by unstable periodic orbits in the classical dynamics.

Two-dimensional PT-symmetric quantum-mechanical systems with the complex cubic potential V_{12}=x^2+y^2+igxy^2 and the complex Henon-Heiles potential V_{HH}=x^2+y^2+ig(xy^2-x^3/3) are investigated. Using numerical and perturbative methods, energy spectra are obtained to high levels. Although both potentials respect the PT symmetry, the complex energy eigenvalues appear when level crossing happens between same parity eigenstates., 9 pages, 4 figures. Submitted as a conference proceeding of PHHQP 8

We propose a new solvable one-dimensional complex PT-symmetric potential as $V(x)= ig~ \mbox{sgn}(x)~ |1-\exp(2|x|/a)|$ and study the spectrum of $H=-d^2/dx^2+V(x)$. For smaller values of $a,g, Comment: 12 pages, 5 Figures, Ref.[21] newly added, Appendix removed

We study the average density of resonances, $$, in a semi-infinite disordered chain coupled to a perfect lead. The function $$ is defined in the complex energy plane and the distance $y$ from the real axes determines the resonance width. We concentrate on strong disorder and derive the asymptotic behavior of $$ in the limit of small $y$., Comment: latex, 1 eps figure, 9 pages; v2 - final version, published in the JPhysA Special Issue Dedicated to the Physics of Non-Hermitian Operators

We calculate the tipping time of a quantum rod that has a height several times that of the edge length of its square base. We use an expression for the tipping time that has heuristic value, and gives the average time at which, upon measurement, the initially balanced rod is found to tip. We use two methods to calculate the tipping time. One method is to examine the "late time" behaviour of the quantum state of the center of mass of the rod by using an equation that has the form of the time-independent Schrödinger equation except that it involves a "complex energy." The other method uses energy resonances in the eigenstates of the Hamiltonian to determine the tipping time. We use the well-known Wentzel–Kramers–Brillouin approximation to calculate the energy eigenstates. With these methods, we obtain expressions for the tipping time that are valid for very long tipping times. PACS Nos.: 03.65.–w, 03.65.Xp

A new construction of quantum mechanical Gamow states is presented. The physical nature of these states is revealed without introducing the notion of complex energy. In the presented aproach the time reversal is a linear transformation instead of the customary antilinear one. Also the Fourier integral solution of the free nonrelativistic wave equation contains both positive and negative frequencies.

A formalism to evaluate the resonant states produced by two particles moving outside a closed shell core is presented. The two-particle states are calculated by using a single-particle representation consisting of bound states, Gamow resonances and scattering states in the complex energy plane (Berggren representation). Two representative cases are analysed corresponding to whether the Fermi level is below or above the continuum threshold. It is found that long-lived resonances are mostly determined by either bound states or by narrow Gamow resonances. However, they are significantly affected by wide resonances and the continuum background itself.

The folding trajectories of a 16-residue β-hairpin are studied using the activation–relaxation technique with a generic energy model. From more than 70 trajectories, three folding pathways emerge. All involve a simultaneous optimization of the complete hydrophobic and hydrogen bonding interactions. The first two follow closely those observed by previous theoretical studies, while the third can be described as a reptation move of one strand of the β-sheet with respect to the other. This reptation move indicates that non-native interactions can play a dominant role in the folding of secondary structures.

The delocalization phenomenon was discovered by Hatano et al. and Miller et al. for a class of non-Hermitian quantum-mechanical problems. We show that the delocalization is only one example of many possible critical phenomena which are associated with the self-orthogonality of an eigenstate of the non-Hermitian Hamiltonian. It is shown that in this class of problems the self-orthogonality occurs at the series of branch points in the complex energy plane that serve as gates for the particle to hop from one Bloch energy band to another one.

Intrinsically disordered proteins (IDPs) and intrinsically disordered regions (IDRs) of proteins are very common and instrumental for cellular signaling. Recently, a number of studies have investigated the kinetic binding mechanisms of IDPs and IDRs. These results allow us to draw conclusions about the energy landscape for the coupled binding and folding of disordered proteins. The association rate constants of IDPs cover a wide range (10(5)-10(9) M-1 s(-1)) and are largely governed by long-range charge-charge interactions, similarly to interactions between well-folded proteins. Off-rate constants also differ significantly among IDPs (with half-lives of up to several minutes) but are usually around 0.1-1000 s(-1), allowing for rapid dissociation of complexes. Likewise, affinities span from pM to mu M suggesting that the low-affinity high-specificity concept for IDPs is not straightforward. Overall, it appears that binding precedes global folding although secondary structure elements such as helices may form before the protein-protein interaction. Short IDPs bind in apparent two-state reactions whereas larger IDPs often display complex multi-step binding reactions. While the two extreme cases of two-step binding (conformational selection and induced fit) or their combination into a square mechanism is an attractive model in theory, it is too simplistic in practice. Experiment and simulation suggest a more complex energy landscape in which IDPs bind targets through a combination of conformational selection before binding (e. g., secondary structure formation) and induced fit after binding (global folding and formation of short-range intermolecular interactions).

Some aspects of resonances in nuclear physics are discussed. Their properties are studied and it is shown how to obtain them numerically in realistic situations for experimental detection. Particular attention is given to resonances in non-spherical potentials that represent deformed exotic nuclei, and their trajectories in the complex energy plane are followed.

We study a wide class of solvable [Formula: see text] symmetric potentials in order to identify conditions under which these potentials have regular solutions with complex energy. Besides confirming previous findings for two potentials, most of our results are new. We demonstrate that the occurrence of conjugate energy pairs is a natural phenomenon for these potentials and that the present method can readily be extended to further potential classes.

We study the properties of an impurity of mass $M$ moving through a spatially homogeneous three-dimensional fully polarized Fermi gas of particles of mass $m$. In the weakly attractive limit, where the effective coupling constant $g\to0^-$ and perturbation theory can be used, both for a broad and a narrow Feshbach resonance, we obtain an explicit analytical expression for the complex energy $\Delta E(\KK)$ of the moving impurity up to order two included in $g$. This also gives access to its longitudinal and transverse effective masses $m_\parallel^*(\KK)$, $m_\perp^*(\KK)$, as functions of the impurity wave vector $\KK$. Depending on the modulus of $\KK$ and on the impurity-to-fermion mass ratio $M/m$ we identify four regions separated by singularities in derivatives with respect to $\KK$ of the second-order term of $\Delta E(\KK)$, and we discuss the physical origin of these regions. Remarkably, the second-order term of $m_\parallel^*(\KK)$ presents points of non-differentiability, replaced by a logarithmic divergence for $M=m$, when $\KK$ is on the Fermi surface of the fermions. We also discuss the third-order contribution and relevance for cold atom experiments., Comment: 6 pages, 4 figures; final version, including a finite temperature calculation

This paper revisits earlier work on complex classical mechanics in which it was argued that when the energy of a classical particle in an analytic potential is real, the particle trajectories are closed and periodic, but that when the energy is complex, the classical trajectories are open. Here it is shown that there is a discrete set of eigencurves in the complex-energy plane for which the particle trajectories are closed and periodic., Comment: 12 pages, 9 figures

Within the framework of the J-matrix method of scattering, a given multichannel potential is modelled by its restriction on a subspace spanned by a certain set of L2 functions. The scattering S-matrix is found exactly for this model potential at the Harris eigenvalues which result from the diagonalization of the scattering Hamiltonian in the same subspace. These values are then analytically continued in the complex energy plane. The poles of the S-matrix are then identified with the complex resonance energies. The associated widths for each resonance are then extracted from the residues of the S-matrix elements at the designated energy.

Calculation of the electron radial Green's function leads to solution of a system of ordinary differential equations for three functions: F, G, which represent the regular solution of the Dirac equation for r → 0, and W− (the antiWronskian) —a function regular everywhere: W− = 1 for r = 0; W− ≈ 1/r as r →∞. An algorithm is formulated for calculating boundary values of W− as r → 0 and for estimation purposes at asymptotically large values of r.

The nudged-elastic band (NEB) method is modified with concomitant two climbing images (C2-NEB) to find a transition state (TS) in complex energy landscapes, such as those with serpentine minimal energy path (MEP). If a single climbing image (C1-NEB) successfully finds the TS, C2-NEB finds it with higher stability and accuracy. However, C2-NEB is suitable for more complex cases, where C1-NEB misses the TS because the MEP and NEB directions near the saddle point are different. Generally, C2-NEB not only finds the TS but guarantees that the climbing images approach it from the opposite sides along the MEP, and it estimates accuracy from the three images: the highest-energy one and its climbing neighbors. C2-NEB is suitable for fixed-cell NEB and the generalized solid-state NEB (SS-NEB)., Comment: 3 pages, 4 figures

We present an extension of Hohenberg-Kohn-Sham density functional theory to the domain of complex local potentials and complex electron densities. The approach is applicable to resonance (Siegert) [Phys. Rev. 56, 750 (1939)] states and other scattering and transport problems that can be described by a normalized state of a Hamiltonian containing a complex local potential. Such Hamiltonians are non-Hermitian and their eigenvalues are in general complex, the imaginary part being inversely proportional to the lifetime of the system. The one-to-one correspondence between complex local potentials nu and complex electron densities rho is established provided that the complex variables are sufficiently close to real local potentials and densities of nondegenerate ground states. We show that the exchange-correlation functionals, contributing to the complex energy, are determined through analytic continuation of their ground-state-theory counterparts. This implies that the exchange-correlation effects on the lifetime of a resonance are, under appropriate conditions, already determined by the functionals of the ground-state theory.

The Raman amplitude when extended into the complex energy plane can have zeros in the right-half plane. Minimum phase approaches to invert the Raman excitation profile (REP) to recover the Raman amplitude are valid only for the case where the Raman amplitude does not admit zeros in the right-half plane. A minimum phase algorithm applied to a REP gives a unique amplitude, but this amplitude and its corresponding cross-correlation function are wrong if the REP corresponds to a nonminimum phase problem. The minimum phase treatment therefore does not carry all possible physical information for a nonminimum phase REP.

Basin-hopping (BH) or Monte-Carlo minimization is so far the most reliable algorithms in chemical physics to search for the lowest-energy structure of atomic clusters and macromolecular systems. BH transforms the complex energy landscape into a collection of basins, and explores them by hopping, which is achieved by random Monte-Carlo moves and acceptance/rejection using the Metropolis criterion. In this report, we introduce the jumping process in addition to the hopping process in BH. Jumping are invoked when the hopping stagnates by reaching the local optima, and are achieved using the Monte-Carlo move at the temperature T = ∞ without rejection. Our basin-hopping with occasional jumping (BHOJ) algorithm is applied to the Lennard–Jones clusters of several notoriously difficult sizes. It was found that the probability of locating the true global optima using BHOJ is significantly higher than the original BH.

The flow equation approach investigated by Wegner et al. is applied to an unbounded Hamiltonian system with a generalization. We show that a well-known quantized complex energy eigenvalues which is related to decay widths can be given with this approach., Comment: Revtex, 4 pages, 2 figures. To appear in Physics Letters A

Spin dependent transport in one-dimensional (1D) three-terminal rings is investigated in the presence of the Rashba spin-orbit coupling (RSOC). We focus on zero-conductance resonances and spin polarizations. For these purposes, the transmission functions are derived analytically. The total conductances are analyzed in the complex energy plane with a focus on the zero-pole structure characteristic of transmission (anti)resonances. The spin polarizations in symmetrically and asymmetrically coupled three-terminal rings are studied as a function of the incident electron energy. It is found that in the absence of the RSOC there are three kinds of conductance zeros. In the presence of the RSOC, the zeros of the first and the third kinds are lifted, while some of the second kind persist. The lifting of the conductance zeros is related to the breaking of the spin-reversal symmetry, and the lifted conductance zeros evolve into spin polarization zeros.

Finding the ground state of a system with a complex energy landscape is important for many physical problems including protein folding, spin glasses, chemical clusters, and neural networks. Such problems are usually solved by heuristic search methods whose efficacy is judged by empirical performance on selected examples. We present a proof that, within the large class of algorithms that simulate a random walk on the landscape, threshold accepting is the best possible strategy. In particular, it can perform better than simulated annealing and Tsallis statistics. Our proof is the first example of a provably optimal strategy in this area.

By adding a negative imaginary potential of variable strength eta to the Hamiltonian, the resonance state of a system can be found as complex energy stabilized points in the eta-trajectories of the eigenvalues. One problem that arises in practical calculations is the appearance of nonphysical complex energy stabilized points. A new method for separating the physical from the nonphysical complex energy stabilized points is proposed. The method is best illustrated with strongly correlated two-electron systems.

We present a new method for analytic continuation of real variational eigenvalues to calculate the complex energy of a collisional or photofragmentation resonance. The method is tested successfully against accurate quantum-mechanical results for model problems.

A global parametrisation is used to fit data on a discrepancy function formed from the K-p forward scattering amplitude. No specific form of the K-p amplitude on its unphysical cut is required for the construction of the discrepancy-function data. The class of admissible functions is restricted by imposing constraints on the number and positions of the zeros of the discrepancy function. When analytically continued to the unphysical cut, the fit is found to be compatible with typical K-matrix parametrisations of the K-p scattering amplitude. Predictions are given for the real parts of the K+or-p forward scattering amplitudes, for the effective KNY coupling constant and for the zeros of the amplitude in the complex energy plane.

A practical non-perturbative approach is presented for multiphoton ionization of atoms induced by circularly polarized radiation. By use of co-ordinate rotation transformation and L 2 -discretization of the atomic continuum, the complex energy spectrum of a stationary quesienergy operator can be located and multiphoton ionization rates determined as a function of time and arbitrary photon intensity. The theory is applied to the two-photon ionization of the H atom in intense fields.

By adding a small amount of CO2 to mixtures of N2/H2O, N2/H2S, and N2/CH4, the rate at which nitrogen is vibrationally de‐excited by these collision partners at room temperature has been determined. A general theory has been developed which is capable of relating rate constants to fluorescence decay times for a complex energy transfer scheme. Some of the problems involved in deducing rate constants from relaxation times are addressed and it is shown that meaningful rate constants can be determined. The rates of N2* (ν=1) de‐excitation by H2O, H2S, and CH4 were found to be 1.0±0.2×105 sec−1 atm−1, 1.1±0.2×106 sec−1 atm−1, and 2.0±0.4×105 sec−1 atm−1, respectively. The rates of CO2* (ν3) de‐excitation by H2O, H2S, and CH4 were also measured in the course of the study.

The uniform semiclassical method has been used to calculate complex energy eigenvalues for a Lennard-Jones (12, 6) potential and a cubic anharmonic oscillator. Good agreement with accurate quantum results is obtained.

The problem of determining the phases of the diffracted waves when only their intensities can be measured is a question that has received a great deal of attention in the case of X-ray diffraction but has so far been largely ignored in low-energy electron diffraction. By applying a technique similar to the one employed in the Kramers-Kronig analysis of optical reflectivity data it is found that under certain circumstances a good approximation to the phase can be found. The error involved depends on the location of the zeros of the reflectivity in the complex energy plane. In a test calculation for Ni (100) it is found that these zeros are favourably placed for the clean, unrelaxed surface but that any departure from this leads to a serious deterioration in the accuracy of the phase determination.

In the hydrogenic Stark problem, the comparison equation is used to study poles of the S -matrix close to the positive real axis in the complex energy plane. simple WKB-type formulae which determine parameters of the resonances are derived. Comparisons with the full quantum mechanical calculations are presented.

In this paper, we report the results of the ATA calculations of the band structures and the Fermi surface of V 1- x Mo x ( x =0.8, 0.5 and 0.2), Cr 0.5 Mo 0.5 and V 0.5 Nb 0.5 . In order to test the reliability of the ATA framework, the complex energy bands and the electronic density of states in V 0.5 Mo 0.5 and Nb 0.5 Mo 0.5 are calculated by the CPA. Any marked difference between the ATA and the CPA results are not found for these systems.

Using the coordinate rotation method with Pekeris trial functions, we find evidence that a resonance occurs in e/sup +/-H S-wave scattering at a complex energy (-0.257374-i0.000067) Ry. No lower resonance is found in the calculation.

The Prufer phase function method is applied to calculation of complex energy eigenvalues of a cubic anharmonic oscillator model potential. The method yields accurate resonance energies and widths and an unambiguous prescription for the quantization path on which the solution has the usual oscillatory property familiar with the bound state wave functions.

The resonance behaviour of a system of three neutrons is considered. For the quantum numbers (T. S, L) = (2/3, 1/2, 1−) and s-wave Yamaguchi two-particle interaction the dominating pole pair of the S-matrix is determined. To find the pole positions and the asymptotic normalization parameters contour deformation technique has been applied to solve the homogeneous Faddeev equation on the unphysical sheets of the S-matrix. For the positions and the asymptotic normalization constants of the dominating pole pair the values zp=(−88·0∓i27·9) keV and C p 2 = −0.05±i0·1 are obtained. Although the pole pair is close to the real axis in the complex energy plane it is not expected to cause an observable experimental effect due to the smallness of the asymptotic normalization constant.

An attempt is made to clarify the fundamental physical basis of the intrinsic characteristics of unstable elementary particles and to replace the conventional, purely phenomenological, description in terms of decaying states and complex energy levels, by definitions which are consistent with general requirements of relativistic quantum mechanics like Hermiticity, unitarity, and causality.

We theoretically study the elastic deformation of a fluid membrane induced by an adhering spherical colloidal particle within the framework of a Helfrich energy. Based on a full optimization of the membrane shape we find a continuous binding and a discontinuous envelopment transition, the latter displaying a potentially substantial energy barrier. A small gradient approximation permits membrane shape and complex energy to be calculated analytically. While this only leads to a good representation of the complex geometry for very small degrees of wrapping, it still gives the correct phase boundaries in the regime of low tension., Comment: 5 pages, 5 figures, RevTeX style

A relation between the number of bound elementary excitations of an atomic Bose-Einstein condensate and the phase shift of elastically scattered atoms is derived. Within the Bogoliubov model of a weakly interacting Bose gas this relation is exact and generalizes Levinson's theorem. Specific features of the Bogoliubov model such as complex energy and continuum bound states are discussed and a numerical example is given.

We have calculated the effect of externally applied longitudinal microwave phonons on the complex energy gap of superconductors. The resulting change in the tunneling current of superconducting junctions accounts for some discrepancies between experiment and previous theory.