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We study the spectral problem in deformed supersymmetric quantum mechanics with polynomial superpotential by using the exact WKB method and the TBA equations. We apply the ODE/IM correspondence to the Schrödinger equation with an effective potential deformed by integrating out the fermions, which admits a continuous deformation parameter. We find that the TBA equations are described by the ℤ4-extended ones. For cubic superpotential corresponding to the symmetric double-well potential, the TBA system splits into the two D3-type TBA equations. We investigate in detail this example based on the TBA equations and their analytic continuation as well as the massless limit. We find that the energy spectrum obtained from the exact quantization condition is in good agreement with the diagonalization approach of the Hamiltonian. [ABSTRACT FROM AUTHOR]

To the existing list of alternative formulations of quantum mechanics, a new version of the non-Hermitian interaction picture is added. What is new is that, in contrast to the more conventional non-Hermitian model-building recipes, the primary information about the observable phenomena is provided not only by the Hamiltonian but also by an additional operator with a real spectrum (say, R (t) ) representing another observable. In the language of physics, the information carried by R (t) ≠ R † (t) opens the possibility of reaching the exceptional-point degeneracy of the real eigenvalues, i.e., a specific quantum phase transition. In parallel, the unitarity of the system remains guaranteed, as usual, via a time-dependent inner-product metric Θ (t) . From the point of view of mathematics, the control of evolution is provided by a pair of conjugate Schrödiner equations. This opens the possibility od an innovative dyadic representation of pure states, by which the direct use of Θ (t) is made redundant. The implementation of the formalism is illustrated via a schematic cosmological toy model in which the canonical quantization leads to the necessity of working with two conjugate Wheeler-DeWitt equations. From the point of view of physics, the "kinematical input" operator R (t) may represent either the radius of a homogeneous and isotropic expanding empty Universe or, if you wish, its Hubble radius, or the scale factor a (t) emerging in the popular Lemaitre-Friedmann-Robertson-Walker classical solutions, with the exceptional-point singularity of the spectrum of R (t) mimicking the birth of the Universe ("Big Bang") at t = 0 . [ABSTRACT FROM AUTHOR]

Following our previous study of the Bohr–Sommerfeld (BS) quantization condition for one-dimensional case [J. C. del Valle and A. V. Turbiner, Int. J. Mod. Phys. A36, 2150221 (2021)], we extend it to d -dimensional power-like radial potentials. The BS quantization condition for S -states of the d -dimensional radial Schrödinger equation is proposed. Based on numerical results obtained for the spectra of power-like potentials, V (r) = r m with m ∈ [ − 1 , ∞) , the correctness of the proposed BS quantization condition is established for various dimensions d. It is demonstrated that by introducing the WKB correction γ into the right-hand side of the BS quantization condition leads to the so-called exact WKB quantization condition, which reproduces the exact energies, while γ remains always very small. For m = 2 (any integer d) and for m = − 1 (at d = 2) the WKB correction γ = 0 : for S states the BS spectra coincides with the exact ones. Concrete calculations for physically important cases of linear, cubic, quartic and sextic oscillators, as well as Coulomb and logarithmic potentials in dimensions d = 2 , 3 , 6 are presented. Radial quartic anharmonic oscillator is considered briefly. [ABSTRACT FROM AUTHOR]

Using the thin-layer quantization, we formulate the problem of a Schrödinger particle constrained to move along a coordinate surface of the bi-spherical coordinate system. In three-dimensional space, the free Schrödinger equation is not separable in this coordinate system. However, when we consider the equation for a particle constrained to a given surface, there are only two degrees of freedom. One has to introduce a geometrical potential to attach the particle to the surface. This well-known potential has two contributions: one from Gauss' curvature and the other from the mean curvature. The Schrödinger equation leads to a general Heun equation. We solve it exactly and present the eigenfunctions and plots of the probability densities, and, as an application of this methodology, we study the problem of an electric charge propagating along these coordinate surfaces in the presence of a uniform magnetic field. [ABSTRACT FROM AUTHOR]

An approach of the proper quantization rule to find exact solution of radial Schrödinger equation for non-polynomial potentials, which are quasi exactly solvable, is developed in this work. Using this rule, the determination of the energy spectrum En for non-pollynomialy potentials is somewhat impossible and limited, which provides their energy at ground state only. To overcome this difficulty, we devised in this approach the potential into two potentials: an exact potential Ve(r) and a non-linear extension Va(r), the potential will be expressed as:V (r) = Ve(r)+Va(r), then calculate the energy level εn and the ground state ε0 energy for exactly solvable potential Ve(r) using the proper quantization rule, from which an analytical expression of energie En for non-polynomial potential V (r) related with the ground level energy E0 is found easily. We conclude that the study of non-exactly potential remain to the study of the first exactly solvable potential. [ABSTRACT FROM AUTHOR]

The system under study is the Λ -Kantowski–Sachs universe. Its canonical quantization is provided based on a recently developed method: the singular minisuperspace Lagrangian describing the system, is reduced to a regular (by inserting into the dynamical equations the lapse dictated by the quadratic constraint) possessing an explicit (though arbitrary) time dependence; thus a time-covariant Schrödinger equation arises. Additionally, an invariant (under transformations t = f (t ~) ) decay probability is defined and thus "observers" which correspond to different gauge choices obtain, by default, the same results. The time of decay for a Gaussian wave packet localized around the point a = 0 (where a the radial scale factor) is calculated to be of the order ∼ 10 - 42 - 10 - 41 s . The acquired value is near the end of the Planck era (when comparing to a FLRW universe), during which the quantum effects are most prominent. Some of the results are compared to those obtained by following the well known canonical quantization of cosmological systems, i.e. the solutions of the Wheeler–DeWitt equation. [ABSTRACT FROM AUTHOR]

Simple Summary: We clarify the question of clock transformations and trajectories in quantum cosmology in a vacuum Bianchi I minisuperspace. We consider a simple cosmological model consisting of an empty Bianchi I Universe, whose Hamiltonian we deparametrise to provide a natural clock variable. The model thus effectively describes an isotropic universe with an induced clock given by the shear. By quantising this model, we obtain various different possible bouncing trajectories (semiquantum expectation values on coherent states or obtained by the de Broglie–Bohm formulation) and explicit their clock dependence, specifically emphasising the question of symmetry across the bounce. [ABSTRACT FROM AUTHOR]

• Present a new quantization rule to energy spectrum. • Its final form expressed by original parameters appearing in general form of SE. • An analytical formula can be found without difficulty. • A few special cases are discussed. In this paper, a new quantization rule to calculate energy spectrum is presented for the bound state problem in non-relativistic quantum mechanics based on the parametric Nikiforov-Uvarov (NU) method. Moreover, it is observed that its final form is conveniently expressed in terms of the original parameters that appear in the general form of the Schrödinger equation. As a result, an analytical formula to the bound state problem is written out explicitly in a straightforward manner without difficulty. The fool-proof nature of the method will appeal to the theorist who requires an analytical solution on the fly. A few special cases arising from this formula are also provided to demonstrate its accuracy. [ABSTRACT FROM AUTHOR]