Invariant Potentials Research Articles

Some exactly solvable $$\mathcal {PT}$$-invariant potentials with real spectra via the (extended) Nikiforov–Uvarov method

The (extended) Nikiforov–Uvarov method is employed to find exact solutions of the Schrodinger operator for three $$\mathcal {PT}$$-invariant potentials (periodic exponential, cotangent and $$\mathcal {PT}$$-symmetric harmonic plus centrifugal). It is shown that their corresponding Schrodinger operators can exhibit real energy eigenvalues. The results are compared with similar works but with different methods. The comparisons led to Rodrigues formulas of some functions of interest. The eigenfunctions of these examples are expressed in terms of Hankel functions, Romanovski polynomials and Heun functions. The method is proved to be felicitous and leads to closed energy formulas for the potentials under study.

On string-localized potentials and gauge fields

A recent idea, put forward by Mund, Rehren and Schroer, is discussed; it suggests that in gauge quantum field theory one can replace the point-localized gauge fields by string-localized vector potentials built from gauge invariant observables and a principle of string-independence. Based on a kinematical model, describing unmovable (static) fields carrying opposite charges, it is shown that these string-localized potentials cannot be used for the description of the gauge bridges between electrically charged fields. These bridges are needed in order to ensure the validity of Gauss' law. This observation does not preclude the existence of Poincar\'e invariant theories, describing the coupling of string-localized gauge invariant potentials to matter fields. But these potentials are not a full-fledged substitute for the gauge fields in ``usual'' quantum electrodynamics.

Modeling Flow and Pressure Fields in Porous Media with High Conductivity Flow Channels and Smart Placement of Branch Cuts for Variant and Invariant Complex Potentials

A long overdue distinction between so-called variant and invariant complex potentials is proposed here for the first time. Invariant complex potentials describe physical flows where a switch of the real and imaginary parts of the function will still describe the same type of physical flow (but only rotated by π/2). Such invariants can be formulated with Euler’s formula to depict the same flow for any arbitrary orientation with respect to the coordinate system used. In contrast, variant complex potentials, when swapping their real and imaginary parts, will result in two fundamentally different physical flows. Next, we show that the contour integrals of the real and imaginary part of simple variant and invariant complex potentials generally do not generate any discernable branch cut problems. However, complex potentials due to the multiple superpositions of simple flows, even when invariant, may involve many options for selecting the branch cut locations. Examples of such branch cut choices are given for so-called areal doublets and areal dipoles, which are powerful tools to describe the streamlines and pressure fields for flow in porous media with enhanced permeability flow channels. After a discussion of the branch cut solutions, applications to a series of synthetic and field examples with enhanced permeability flow channels are given with examples of the streamline and pressure field solutions.

Donor binding energies in a spherical core–shell quantum dot: parabolic and shifted parabolic shell potentials

The effects of parabolic and shifted parabolic potentials on centred hydrogenic impurity-related binding energies are studied. This is theoretically achieved by solving the Schrodinger equation within the effective mass approximation. The hydrogenic impurity situates in the core of a spatially constant electrostatic potential, surrounded by a shell material which has either an intrinsic parabolic potential or shifted parabolic potential. For a given outer radius of the shell, the parabolic potential enhances binding energies (with greater enhancement for small radii of the core), while the shifted parabolic potential dwindles binding energies for small cores, and increases the binding energies for larger cores. An increase in the spatially invariant potential in the core results in reduced binding energies, with appreciable decrement occurring in core–shell structures that have wider cores.

Modular symmetries and the swampland conjectures

Recent string theory tests of swampland ideas like the distance or the dS conjectures have been performed at weak coupling. Testing these ideas beyond the weak coupling regime remains challenging. We propose to exploit the modular symmetries of the moduli effective action to check swampland constraints beyond perturbation theory. As an example we study the case of heterotic 4d mathcal{N}=1 compactifications, whose non-perturbative effective action is known to be invariant under modular symmetries acting on the Kähler and complex structure moduli, in particular SL(2, Z) T-dualities (or subgroups thereof) for 4d heterotic or orbifold compactifications. Remarkably, in models with non-perturbative superpotentials, the corresponding duality invariant potentials diverge at points at infinite distance in moduli space. The divergence relates to towers of states becoming light, in agreement with the distance conjecture. We discuss specific examples of this behavior based on gaugino condensation in heterotic orbifolds. We show that these examples are dual to compactifications of type I’ or Horava-Witten theory, in which the SL(2, Z) acts on the complex structure of an underlying 2-torus, and the tower of light states correspond to D0-branes or M-theory KK modes. The non-perturbative examples explored point to potentials not leading to weak coupling at infinite distance, but rather diverging in the asymptotic corners of moduli space, dynamically forbidding the access to points with global symmetries. We perform a study of general modular invariant potentials and find that there are dS maxima and saddle points but no dS minima, and that all examples explored obey the refined dS conjecture.

Construction of solvable non-central potential using vector superpotential: a new approach

We introduce here vector superpotential which is useful to find general potential form using supersymmetric quantum mechanics (SUSY QM) approach. Using the vector superpotential, we reconstruct different solved central potentials and non-central shape variant or shape invariant potentials. To construct and reconstruct the central potentials and non-central potentials, appropriate choice of radial part of the superpotential as well as angular and azimuthal parts of the superpotential is required. Our main aim is to construct angular and azimuthal parts of potential directly from the appropriate choice of vector superpotential. We have reconstructed few potentials which are solved by different researchers using different methods of solution and also constructed new non-central potentials.

Prediction of many-electron wavefunctions using atomic potentials: extended basis sets and molecular dissociation.

A one-electron Schrödinger equation based on special one-electron potentials for atoms is shown to exist that produces orbitals for an arbitrary molecule that are sufficiently accurate to be used without modification to construct single- and multi-determinant wavefunctions. The exact Hamiltonian is used to calculate the energy variationally and to generate configuration interaction expansions. Earlier work on equilibrium geometries is extended to larger basis sets and molecular dissociation. For a test set of molecules representing different bonding environments, a single set of invariant atomic potentials gives wavefunctions with energies that deviate from configuration interaction energies based on SCF orbitals by less than 0.04 eV per bond or valence electron pair. On a single diagonalization of the Fock matrix, the corresponding errors are reduced 0.01 eV. Atomization energies are also in good agreement with CI values based on canonical SCF orbitals. Configuration interaction applications to single bond dissociations of water and glycine, and multiple bond dissociations of ethylene and oxygen produce dissociation energy curves in close agreement with CI calculations based on canonical SCF orbitals for the entire range of internuclear distances.

Single Image Depth Estimation With Normal Guided Scale Invariant Deep Convolutional Fields

Estimating scene depth from a single image can be widely applied to understand 3D environments due to the easy access of the images captured by consumer-level cameras. Previous works exploit conditional random fields (CRFs) to estimate image depth, where neighboring pixels (superpixels) with similar appearances are constrained to share the same depth. However, the depth may vary significantly in the slanted surface, thus leading to severe estimation errors. In order to eliminate those errors, we propose a superpixel-based normal guided scale invariant deep convolutional field by encouraging the neighboring superpixels with similar appearance to lie on the same 3D plane of the scene. In doing so, a depth-normal multitask CNN is introduced to produce the superpixel-wise depth and surface normal predictions simultaneously. To correct the errors of the roughly estimated superpiexl-wise depth, we develop a normal guided scale invariant CRF (NGSI-CRF). NGSI-CRF consists of a scale invariant unary potential that is able to measure the relative depth between superpixels as well as the absolute depth of superpixels, and a normal guided pairwise potential that constrains spatial relationships between superpixels in accordance with the 3D layout of the scene. In other words, the normal guided pairwise potential is designed to smooth the depth prediction without deteriorating the 3D structure of the depth prediction. The superpixel-wise depth maps estimated by NGSI-CRF will be fed into a pixel-wise refinement module to produce a smooth fine-grained depth prediction. Furthermore, we derive a closed-form solution for the maximum a posteriori (MAP) inference of NGSI-CRF. Thus, our proposed network can be efficiently trained in an end-to-end manner. We conduct our experiments on various datasets, such as NYU-D2, KITTI, and Make 3D. As demonstrated in the experimental results, our method achieves superior performance in both indoor and outdoor scenes.

Gauge Theory's Expansion in the Electro-Magnetic Field

In the special relativity theory, study the gauge theory in the Electro-magnetic field theory.Using that the Electro-magnetic potential is 4-vector, treat invariant potential. And the Electro-Magnetic field theory�s the gauge theory expand.

Exact Solutions of Schrodinger Equation in Cylindrical Coordinates for Double Ring-Shaped Coulomb Oscillator Potential Using SUSY QM Method

Schrodinger equation for a doubled ring-shaped coulomb oscillator potential is investigated using supersymmetric quantum mechanics approach. The three dimensional Schrodinger equation for a doubled ring-shaped coulomb oscillator potential in cylindrical coordinates is separated into three parts that contains one dimensional Schrodinger type equation which are solved using supersymmetric operator and the idea of shape invariant potential. The energy spectrum and the total wave functions are obtained.

Inertial spontaneous symmetry breaking and quantum scale invariance

Weyl invariant theories of scalars and gravity can generate all mass scales spontaneously, initiated by a dynamical process of ``inertial spontaneous symmetry breaking'' that does not involve a potential. This is dictated by the structure of the Weyl current, ${K}_{\ensuremath{\mu}}$, and a cosmological phase during which the Universe expands and the Einstein-Hilbert effective action is formed. Maintaining exact Weyl invariance in the renormalized quantum theory is straightforward when renormalization conditions are referred back to the VEV's of fields in the action of the theory, which implies a conserved Weyl current. We do not require scale invariant regulators. We illustrate the computation of a Weyl invariant Coleman-Weinberg potential.

The supersymmetric WKB formalism is not exact for all additive shape invariant potentials

Following the verification of the conjecture made by Comtet et al that the supersymmetry-inspired semiclassical method known as SWKB is exact for the conventional additive shape invariant potentials, it was widely believed that SWKB yields exact results for all additive shape invariant potentials. In this paper we present a concrete example of an additive shape invariant potential for which the SWKB method fails to produce exact results.

Harmonic oscillator Wigner function extension to exceptional polynomials

In this paper, we construct isospectral Hamiltonians without shape invariant potentials for a harmonic oscillator Wigner function on a real line. In this case, we actually remove the ground state of the second Hamiltonian, which forms a special case, $$m=0$$ , of an exceptional Laguerre differential equation with solutions $$\{L_n^{-2}\}_{n=2}^\infty $$ as eigenfunctions form a complete orthogonal set in the Hilbert space.

Willmore-Like Tori in Killing Submersions

The first variation formula and Euler-Lagrange equations for Willmore-like surfaces in Riemannian 3-spaces with potential are computed and, then, applied to the study of invariant Willmore-like tori with invariant potential in the total space of a Killing submersion. A connection with generalized elastica in the base surface of the Killing submersion is found, which is exploited to analyze Willmore tori in Killing submersions and to construct foliations of Killing submersions made up of Willmore tori with constant mean curvature.

The adapted hyper-Kähler structure on the crown domain

Let $$\Xi $$ be the crown domain associated with a non-compact irreducible Hermitian symmetric space G / K. We give an explicit description of the unique G-invariant adapted hyper-Kahler structure on $$\Xi $$ , i.e., compatible with the adapted complex structure $$J_\mathrm{ad}$$ and with the G-invariant Kahler structure of G / K. We also compute invariant potentials of the involved Kahler metrics and the associated moment maps.

Deformed shape invariance symmetry and potentials in curved space with two known eigenstates

We consider two families of extensions of the oscillator in a d-dimensional constant-curvature space and analyze them in a deformed supersymmetric framework, wherein the starting oscillator is known to exhibit a deformed shape invariance property. We show that the first two members of each extension family are also endowed with such a property, provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build potentials with known ground and first excited states. To extend such results to any members of the two families, we devise a general method wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function W+(r) [and its accompanying function W−(r)].

RATIONALLY EXTENDED SHAPE INVARIANT POTENTIALS IN ARBITRARY D DIMENSIONS ASSOCIATED WITH EXCEPTIONAL Xm POLYNOMIALS

Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of <em>X<sub>m</sub></em> Laguerre or <em>X<sub>m</sub></em> Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

Exact diagonalization of cubic lattice models in commensurate Abelian magnetic fluxes and translational invariant non-Abelian potentials

We present a general analytical formalism to determine the energy spectrum of a quantum particle in a cubic lattice subject to translationally invariant commensurate magnetic fluxes and in the presence of a general space-independent non-Abelian gauge potential. We first review and analyze the case of purely Abelian potentials, showing also that the so-called Hasegawa gauge yields a decomposition of the Hamiltonian into sub-matrices having minimal dimension. Explicit expressions for such matrices are derived, also for general anisotropic fluxes. Later on, we show that the introduction of a translational invariant non-Abelian coupling for multi-component spinors does not affect the dimension of the minimal Hamiltonian blocks, nor the dimension of the magnetic Brillouin zone. General formulas are presented for the case and explicit examples are investigated involving π and magnetic fluxes. Finally, we numerically study the effect of random flux perturbations.

Dirac bi-spinor entanglement under local noise and its simulation by Jaynes-Cummings interactions

A description of the effects of the local noise on the quantum entanglement constraining the internal degrees of freedom of Dirac bi-spinor structures driven by arbitrary Poincaré invariant potentials is proposed. Given that the Dirac equation dynamics including external potentials can be simulated by a suitable four level trapped ion setup, quantum entanglement of two-qubit ionic states with quantum numbers related to the total angular momentum and to its projection onto the direction of the external magnetic field (used for lift the ions degeneracy), are recovered by means of a suitable ansatz. This formalism allows the inclusion of noise effects, which leads to disentanglement in the four level trapped ion quantum system. Our results indicate the role of interactions in bi-spinor entanglement, as well as the description of disentanglement in ionic states under local noises. For a state prepared initially in one of the ionic levels, local noise induces entanglement sudden death followed by sudden revivals driven by the noiseless dynamics of the state. Residual quantum correlations are observed in the intervals where such state is separable. Schrödinger cat and Werner states partially loose their initial entanglement content due to the interaction with the noisy environment but presenting entanglement oscillations without sudden death. Because Dirac equation describes low energy excitations of mono layer and bi-layer graphene, the formalism can also be applied to compute, for instance, electron-hole or electron/electron entanglement in various circumstances.

A short note on “Group theoretic approach to rationally extended shape invariant potentials” [Ann. Phys. 359 (2015) 46–54

It is proved the equivalence of the compatibility condition of Ramos (2011, 2012) with a condition found in Yadav et al. (2015). The link of Shape Invariance with the existence of a Potential Algebra is reinforced for the rationally extended Shape Invariant potentials. Some examples on X1 and Xℓ Jacobi and Laguerre cases are given.