Diagonalization Procedure Research Articles

Hallmarks of tunneling dynamics with broken reflective symmetry

We study features of tunneling dynamics in an exactly-solvable model of N=4 supersymmetric quantum mechanics with a multi-well potential and with broken reflective symmetry. Quantum systems with a phenomenological potential of this type demonstrate the phenomenon of partial localization of under-barrier states, possibly resulting in the appearance of the so-called “resonant” tunneling, or the phenomenon of coherent tunneling destruction, referring to the complete localization. Taking the partial localization and the coherent tunneling destruction as basic examples, we indicate main advantages of using isospectral exactly-solvable Hamiltonians in studies quantum mechanical systems with two- and three-well potentials. They, in particular, are: having enough freedom of changing the potential shape in a wide range, that allows one to choose an exactly-solvable model close to characteristics of the phenomenological one; ability of changing the number of local minima and symmetry characteristics of the potential (symmetric or deformed) without changing the main part of the spectrum; engaging a smart basis of states, that dramatically decreases the dimensionality of matrices used in the diagonalization procedure of the corresponding spectral problem.

Applications of rational difference equations to spectral graph theory

We study a general class of recurrence relations that appear in the application of a matrix diagonalization procedure. We find a general closed formula and determine the analytical properties of the solutions. We finally apply these findings in several problems involving eigenvalues of graphs.

Interatomic interaction within a 4He−4He dimer with mobile nuclei

Schrodinger equation for system of two mobile spinless nucleus with Z0 = 2 and four electrons (4He2-dimer) has been solved in the framework of exact diagonalization approach. The spectrum has been obtained with basis of 1134 functions built as direct product of one-particle hydrogen-like functions which are solutions of the exactly solved Sturm-Liouville problem. The asymptotic (Van der Waals) behavior of interaction between two 4He atoms in the limit R0→∞ has been analyzed and discussed. The contribution to the interaction energy from four electronic spins on the 4He–4He bond has been calculated the exact diagonalization procedure.

Time dependent variation perturbation calculation of two-photon transition probability and hyperfine shift in hydrogen atom under plasma environment

A new theoretical approach has been adopted to calculate the non-relativistic parity conserved two-photon transition probability (TPTP) from ground to excited S and D states along with the hyperfine splitting (HFS) of ground and excited S states of H atom within principal quantum number n≤4 under classical and quantum plasma environment represented by screened potential model. The methodology involves direct calculation of TPTP via an intermediate virtual state obtained from time dependent variation perturbation formalism. The standard formula with sum over states representing the TPTP has been replaced by its variational equivalent obtained from currently established procedure for two-photon excitation via fourth order time dependent variation perturbation theory. Unlike previous methodologies the current formalism yields a compact representation of the physical nature of the intermediate virtual state having well defined profile. The HFS has been evaluated under first order perturbation theory. The plasma embedded ground state energy is obtained from standard diagonalization procedure. Finite Slater type basis sets have been used for the ground and all the excited states. The variational coefficients for the excited states are determined from optimization of an appropriate time averaged variational functional. Results for free H atom agree well with existing estimates for all the states. The results for the plasma embedded states follow smooth and interesting pattern.

Parity of the neutron consistent with neutron-antineutron oscillations

In the analysis of neutron-antineutron oscillations, it has been recently argued in the literature that the use of the $i\gamma^{0}$ parity $n^{p}(t,-\vec{x})=i\gamma^{0}n(t,-\vec{x})$ which is consistent with the Majorana condition is mandatory and that the ordinary parity transformation of the neutron field $n^{p}(t,-\vec{x}) = \gamma^{0}n(t,-\vec{x})$ has a difficulty. We show that a careful treatment of the ordinary parity transformation of the neutron works in the analysis of neutron-antineutron oscillations. Technically, the CP symmetry in the mass diagonalization procedure is important and the two parity transformations, $i\gamma^{0}$ parity and $\gamma^{0}$ parity, are compensated for by the Pauli-G\"ursey transformation. Our analysis shows that either choice of the parity gives the correct results of neutron-antineutron oscillations if carefully treated.

A tale of three diagonalizations

In addition to the diagonalization of a normal matrix by a unitary similarity transformation, there are two other types of diagonalization procedures that sometimes arise in quantum theory applications — the singular value decomposition and the Autonne–Takagi factorization. In this pedagogical review, each of these diagonalization procedures is performed for the most general [Formula: see text] matrices for which the corresponding diagonalization is possible, and explicit analytical results are provided in each of the three cases.

In-medium T matrix in case of pairing instability

The diagonalization technique for two-body propagator in nuclear matter is generalized by inclusion of the two-hole continuum. In the approach, the total effective two-body Hamiltonian is introduced which allows us to find the complete propagator as well as the reaction $T$ matrix at many energies and relative momenta simultaneously from a single diagonalization procedure for the Hamiltonian matrix. Explicit treatment of bound and continuum states allows us to determine forward- and backward-propagating parts of the $T$ matrix in a closed form which simplifies strongly the evaluation of single-particle and single-hole self-energies. Special attention is paid to the bound-state contribution to the reaction matrix in case of pairing instability when complex conjugated eigenvalues of the effective Hamiltonian appear. A generalization of the approach to calculation of pairing gaps in case of nonzero center-of-mass momentum of the pair is also presented.

Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

<abstract><p>In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $ L^p-L^q $ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.</p></abstract>

Linear Prediction-Based Covariance Matrix Reconstruction for Robust Adaptive Beamforming

In this letter, a novel reconstruction-based adaptive beamformer is proposed, which uses linear prediction to generate virtual sensor data and extend array aperture. To overcome suppression failure of reconstruction-based adaptive beamformer, a double-side array extending algorithm is proposed for uniform linear array, where the virtual array data can be obtained by using the data from real array. Then, we estimate the spatial spectrum of extended array data with higher resolution by using a modified diagonal loading-type procedure, which is decided by the condition numbers of the sample covariance matrix (SCM) and the extended SCM, and the interference-plus-noise covariance matrix (INCM) of extended array is estimated. Without any optimization procedure, steering vector (SV) of extended array is corrected as the eigenvector corresponding to the dominant eigenvalue of the extended SCM. Numerical Simulations demonstrated that the proposed beamformer can use both deep nulls and low side-lobes to ensure better interference suppression capability to other beamformers in the case of sensor position errors.

Neutrino masses and mixing angles in a model with six Higgs triplets and A4 symmetry

We have considered a model \cite{ma-weg}, where masses and a mixing pattern for neutrinos are governed by six Higgs triplets and $A_4$ symmetry. In this model we have applied a certain diagonalisation procedure through which we have shown that neutrino masses can have both normal or inverted hierarchy. We have also shown that current neutrino oscillation data can be explained in this model.

Inclusive probability of particle creation on classical backgrounds

The quantum theories of boson and fermion fields with quadratic nonstationary Hamiltoanians are rigorously constructed. The representation of the algebra of observables is given by the Hamiltonian diagonalization procedure. The sufficient conditions for the existence of unitary dynamics at finite times are formulated and the explicit formula for the matrix elements of the evolution operator is derived. In particular, this gives the well-defined expression for the one-loop effective action. The ultraviolet and infrared divergencies are regularized by the energy cutoff in the Hamiltonian of the theory. The possible infinite particle production is regulated by the corresponding counterdiabatic terms. The explicit formulas for the average number of particles N_D recorded by the detector and for the probability w(D) to record a particle by the detector are derived. It is proved that these quantities allow for no-regularization limit and, in this limit, N_D is finite and w(D)in [0,1). As an example, the theory of a neutral boson field with stationary quadratic part of the Hamiltonian and nonstationary source is considered. The average number of particles produced by this source from the vacuum during a finite time evolution and the inclusive probability to record a created particle are obtained. The infrared and ultraviolet asymptotics of the average density of created particles are derived. As a particular case, quantum electrodynamics with a classical current is considered. The ultraviolet and infrared asymptotics of the average number of photons are derived. The asymptotics of the average number of photons produced by the adiabatically driven current is found.

Local scale invariant Kaluza-Klein reduction

We perform the 4-dimensional Kaluza-Klein (KK) reduction of the 5-dimensional locally scale invariant Weyl-Dirac gravity. While compactification unavoidably introduces an explicit length scale into the theory, it does it in such a way that the KK radius can be integrated out from the low energy regime, leaving the KK vacuum to still enjoy local scale invariance at the classical level. Imitating a $U(1)\times\tilde{U}(1)$ gauge theory, the emerging 4D theory is characterized by a kinetic Maxwell-Weyl mixing whose diagonalization procedure is carried out in detail. In particular, we identify the unique linear combination which defines the 4D Weyl vector, and fully classify the 4D scalar sector. The later consists of (using Weyl language) a co-scalar and two in-scalars. The analysis is performed for a general KK $m$-ansatz, parametrized by the power $m$ of the scalar field which factorizes the 4D metric. The no-ghost requirement, for example, is met provided $-\frac{1}{2}\leq m \leq 0$. An $m$-dependent dictionary is then established between the original 5D Brans-Dicke parameter $\omega_5$ and the resulting 4D $\omega_4$. The critical $\omega_5=-\frac{4}{3}$ is consistently mapped into critical $\omega_4 = -\frac{3}{2}$. The KK reduced Maxwell-Weyl kinetic mixing cannot be scaled away as it is mediated by a 4D in-scalar (residing within the 5D Weyl vector). The mixing is explicitly demonstrated within the Einstein frame for the special physically motivated choice of $m=-\frac{1}{3}$. For instance, a super critical Brans-Dicke parameter induces a tiny positive contribution to the original (if introduced via the 5-dimensional scalar potential) cosmological constant. Finally, some no-scale quantum cosmological aspects are studied at the universal mini-superspace level.

Existence of Solutions to Shallow Lake Type Optimal Control Problems

We consider the optimal control problem associated with a general version of the well known Shallow Lake model, and we prove the existence of an optimum in the class of all positive, locally integrable functions with finite discounted integral. Any direct proof seems to be missing in the literature. In order to represent properly the concrete optimization problem, locally unbounded controls must be admitted. In an infinite horizon setting, the non-compactness of the control space can make the existence problem quite hard to treat, because of the lack of good a priori estimates. To face the technical difficulties, we develop an original method, which is in a way opposite to the classical control theoretic approach used to solve finite horizon Mayer or Bolza problems. Synthetically, our method is based on the following scheme: (i) a couple of uniform localization lemmas providing, for any given finite time interval, a maximizing sequence of controls, which is uniformly essentially bounded in that interval; (ii) a special diagonal procedure dealing with sequences not extracted one from the other; (iii) a “standard” diagonal procedure. Further, this approach does not require any concavity assumption on the state equation. The optimum turns out to be locally bounded by construction.

Studies on the nonlinear optical properties of two-step GaAs/Ga1−xAlxAs quantum well

In this paper, the numerical computation for the absorption coefficient and the relative refractive index change, considering the third order correction nonlinear optical properties, is reported. This study was performed for a symmetric two-step GaAs/Ga1−xAlxAs quantum well, subjected to a constant electric field applied along the growth direction z, and an in-plane constant magnetic field B. We also consider the intense laser field effect, characterized through the laser-dressing parameter α0. The electronic structure computation was obtained by working under the effective mass approximation and the Schödinger equation was solved by diagonalization procedure. The optical properties are calculated by using the well-established compact density matrix formalism expressions for the nonlinear optical properties of interest. In general, we found that the structural parameters, as the step-like potential or the central barrier, permit the resonant peak and the amplitude design. We also found that the system becomes more sensitive to electric than to magnetic field, and finally that the intense, non-resonant, laser field can strongly change the optical properties of interest. Our results indicate that the implementation of the step-like potential profile, experimentally feasible, enhance the optical properties of interest, that falls within the THz electromagnetic range, and can be used to design a photodetector, or even can be used for quantum cascade lasers design.

Excitation of interfacial waves via surface–interfacial wave interactions

We consider interactions between surface and interfacial waves in a two-layer system. Our approach is based on the Hamiltonian structure of the equations of motion, and includes the general procedure for diagonalization of the quadratic part of the Hamiltonian. Such diagonalization allows us to derive the interaction cross-section between surface and interfacial waves and to derive the coupled kinetic equations describing spectral energy transfers in this system. Our kinetic equation allows resonant and near-resonant interactions. We find that the energy transfers are dominated by the class III resonances of Alam (J. Fluid Mech., vol. 691, 2012, pp. 267–278). We apply our formalism to calculate the rate of growth for interfacial waves for different values of wind velocity. Using our kinetic equation, we also consider the energy transfer from wind-generated surface waves to interfacial waves for the case when the spectrum of the surface waves is given by the JONSWAP spectrum and interfacial waves are initially absent. We find that such energy transfer can occur along a time scale of hours; there is a range of wind speeds for the most effective energy transfer at approximately the wind speed corresponding to white capping of the sea. Furthermore, interfacial waves oblique to the direction of the wind are also generated.

Wigner model for Klein tunneling in graphene

Quantum corrections to the electron-hole motion in graphene are investigated by applying the multiband Wigner approach. A diagonalization procedure regarding the pseudo-spin degree of freedom is proposed. The resulting formulation of the equation of motion reveals to be particularly close to the classical description of the particle motion. This formal analogy offers a framework in which quantum phenomena can be described with a classical language and the question of the quantum-classical correspondence can be directly investigated. Some numerical results are presented showing the ability of our quasi-diagonal Wigner model to reproduce Klein tunneling in graphene under the application of an external electric field.

Application of the Bohr Hamiltonian with a double-well sextic potential to collective states in Mo isotopes

A diagonalization procedure for an SO(5) invariant Bohr Hamiltonian with a sextic potential for the β shape variable, having simultaneous spherical and deformed minima, is presented. The double-well potential allows for the study of shape coexistence and mixing phenomena from a geometrical perspective. The spectral and dynamical features of the model are investigated for different shapes of the potential amended with the centrifugal contribution from the kinetic term and subjected to the condition of degenerate minima. A generalization of the model is applied for the description of the low-lying energy spectra and shape mixing evolution in Mo nuclei. For each isotope, a shape transition from small to large deformation is found to occur between distinct states with various degrees of intensity and shape mixing.

Convergence analysis of a periodic-like waveform relaxation method for initial-value problems via the diagonalization technique

We present in this paper a time parallel algorithm for $${\dot{u}}=f(t,u)$$ with initial-value $$u(0)=u_0$$ , by using the waveform relaxation (WR) technique, and the diagonalization technique. With a suitable parameter $$\alpha $$ , the WR technique generates a functional sequence $$\{u^k(t)\}$$ via the dynamic iterations $${\dot{u}}^k=f(t,u^k)$$ , $$u^k(0)=\alpha u^k(T)-\alpha u^{k-1}(T)+u_0$$ , and at convergence we get $$u^{\infty }(t)=u(t)$$ . Each WR iterate represents a periodic-like differential equation, which is very suitable for applying the diagonalization technique yielding direct parallel-in-time computation. The parameter $$\alpha $$ controls both the roundoff error arising from the diagonalization procedure and the convergence factor of the WR iterations, and we perform a detailed analysis for the influence of the parameter $$\alpha $$ on the method. We show that the roundoff error is proportional to $$\epsilon (2N+1)\max \{|\alpha |^2, |\alpha |^{-2}\}$$ ( $$N=T/\varDelta t$$ and $$\epsilon $$ is the machine precision), and the convergence factor can be bounded by $$|\alpha |e^{-TL}/(1-|\alpha |e^{-TL})$$ , where $$L\ge 0$$ is the one-sided Lipschitz constant of f. We also perform a convergence analysis at the discrete level and the effect of temporal discretizations is explored. Our analysis includes the heat and wave equations as special cases. Numerical results are given to support our findings.

Very weak solutions to the two-dimensional Monge-Ampére equation

In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Amp\'ere equation with H\"older-continuous first derivatives of exponent $\beta<1/5$. Our approach is based on combining the approach of Lewicka-Pakzad \cite{Lewicka:2015gz} with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by the second author with De Lellis and Inauen in \cite{DeLellis:2015wm} for the isometric immersion problem.

Application of the homotopy analysis method to multispecies reactive transport equations with general initial conditions

Multispecies solute transport through soils is characterized by the dynamic processes of convection, diffusion, decay and a series of chemical reactions, posing difficulties when seeking to derive analytical solutions related to subsurface hydrology. At present, most analytical solutions for multiple-species contaminant transport problems are based on the Laplace transform or Fourier transform technique, a decomposition strategy such as ingenious transform format and matrix diagonalization procedure, classical integral methods, the generalized integral transform technique, etc. These methods have been limited to some cases with simple initial conditions, identical retardation coefficients for all species, and complicated mathematical procedures, which restrict their use for cases under complex initial conditions. To cope with these limitations, the homotopy analysis method (HAM) is implemented to solve multispecies reactive transport models with more general, smooth initial conditions. Applying an auxiliary linear and nonlinear operator, a zero-order deformation equation is derived to obtain an approximate solution with high accuracy (different cases with various initial and boundary conditions). The semi-analytical solutions are compared with exact analytical solutions from the literature; good agreement between the resulting simulation and analytical solutions has been achieved. This illustrates that HAM is one of the most effective methods for solving the multispecies reactive transport models with more general and smooth initial conditions.