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Let X be a compact Kähler manifold of complex dimension k ≥ 2 and f : X → X a holomorphic correspondence with simple action on cohomology such that f - 1 is also a holomorphic correspondence. We prove that the sequence of normalized pull-backs of a non-pluripolar current under iterates of f converges to the Green current associated with f. [ABSTRACT FROM AUTHOR]

Supersymmetric quantum mechanics (SUSYQM) plays an important role in solving the Schrödinger equation, and it is also important to find more superpotentials that can be solved accurately. On the basis of studying the characteristics of existing superpotentials, the authors find a missing superpotential and put forward a method by combining two functions to construct all existing solvable superpotentials and prove the existence of the missing superpotential. First, based on the idea of SUSYQM, this paper studies the shape invariance of the partner potentials with the form of two functions and obtains the energy spectrum in many different cases. Second, according to the results of solving differential equations satisfied by two functions, the authors not only construct most existing solvable superpotentials successfully but also generate a missing solvable superpotential. Third, for the missing solvable superpotential, some discussions are made, such as the corresponding partner potentials, energy spectrum, ground state wave function, and excited state wave functions. Lastly, the summary is made, and the prospects are projected. [ABSTRACT FROM AUTHOR]

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In this research, the singularity of the central shape-invariant potentials, which have a singularity of the inverse-square power α/r², has been investigated. It has been shown that in quantum mechanics, for α ≥ 3/4, the eigenvalue problem is well-defined and, as a result, the energy spectrum can be determined. In the transition region, for - 1/4 ≤ α < 3/4, both regular and irregular wave functions are square integrable and therefore acceptable, but the boundary conditions for determining the eigenvalues and eigenfunctions are not sufficient and there is no a specific predetermined mechanism for choosing a linear combination of wave functions. For α < -1/4, the particle is drawn to the singularity, and therefore, there is no any ground state with finite energy. It has also been shown using supersymmetric quantum mechanics that the inverse-square potential is the result of the singular inverse superpotential β/r. Supersymmetric quantum mechanics provides a mechanism that, without any additional constraints, the less singular wave function is chosen and the potential is placed in the transition region for - 3/2 < β < 1/2. [ABSTRACT FROM AUTHOR]

We extend the scope of the unified factorization method to the solution of conditionally and unconditionally exactly solvable models of quantum mechanics, proposed in a previous paper [R.R. Nigmatullin, A.A. Khamzin, D. Baleanu, Results in Physics 41 (2022) 105945]. The possibilities of applying the unified approach in the factorization method are demonstrated by calculating the energy spectrum of a potential constructed in the form of a second-order polynomial in many of the linearly independent functions. We analyze the solutions in detail when the potential is constructed from two linearly independent functions. We show that in the general case, such kinds of potentials are conditionally exactly solvable. To verify the novel approach, we consider several known potentials. We show that the shape of the energy spectrum is invariant to the number of functions from which the potential is formed and is determined by the type of differential equations that the potential-generating functions obey. [ABSTRACT FROM AUTHOR]

Supersymmetric quantum mechanics is an effective method to solve the exact solution of the Schrödinger equation. This paper studies the solution of the Schrödinger equation with the partner potentials generated by the superpotential (Bcscαx-Acotαx) with two parameters (A and B). Firstly, the shape invariance of the partner potentials generated by the superpotential is obtained. The parametric additivity of shape invariance satisfies a special additivity characteristic: the two-parameter cross-additivity(A→B+α2,B→A+α2), which is completely different from the general additivity characteristic(A→A+α2,B→B+α2). Secondly, we discuss the case that belongs to two-parameter cross additive shape invariance in detail, and find that this two-parameter cross-additivity resulted in partial states missing. The existing energy spectrum and eigenfunctions of the Schrödinger equation with this new parametric transformation are worked out. Thirdly, we discuss the Shape invariance of the partner potentials generated again by the two parameters with cross additive characteristics through the potential algebra method. Lastly, the conclusions and discussions are made.

In contrast to the conventional one-parameter class of isospectral deformation using translation, we calculate the two- and three-parameter classes of isospectral deformation of the well-known reflectionless potential by utilizing a far different approach of scaling methodology. Subsequently, using these results, we find that this more general class of deformations is not unique but instead subsume in the same class of conventional one-parameter translational deformation. We also provide a theoretical foundation for how these two incredibly different approaches converge at the same destination. Finally, we show that the most generic class of potentials, obtained by scaling deformation, are solutions of the nonlinear KdV equation. • We found the two- and three-parameter dependent isospectral deformation of the reflectionless potential using scaling methodology. • We have illustrated that the scaling deformation ends up in translation deformation and explicitly pointed out how these two methods converge. • We have also shown that the generated two- and three-parameter families of potential are the solutions to the non-linear KdV equation. [ABSTRACT FROM AUTHOR]

• This paper solves the Schrödinger equation with the partner potentials generated by the superpotential (B Csc αx - A Cot αx). • The additivity shape invariance with two parameters is discussed in detail. • The shape invariance of the partner potentials of the two parameters shows cross-additivity characteristics which is entirely different from the general additivity characteristics. • Through the potential algebra method, we discuss again the shape invariance of the partner potentials generated by the two parameters with cross-additivity characteristics. • We obtain the energy eigenvalue and the recursion relation of the wave function with even energy level number. Supersymmetric quantum mechanics is an effective method to solve the exact solution of the Schrödinger equation. This paper studies the solution of the Schrödinger equation with the partner potentials generated by the superpotential (B csc α x - A cot α x) with two parameters (A and B). Firstly, the shape invariance of the partner potentials generated by the superpotential is obtained. The parametric additivity of shape invariance satisfies a special additivity characteristic: the two-parameter cross-additivity (A → B + α 2 , B → A + α 2) , which is completely different from the general additivity characteristic (A → A + α 2 , B → B + α 2). Secondly, we discuss the case that belongs to two-parameter cross additive shape invariance in detail, and find that this two-parameter cross-additivity resulted in partial states missing. The existing energy spectrum and eigenfunctions of the Schrödinger equation with this new parametric transformation are worked out. Thirdly, we discuss the Shape invariance of the partner potentials generated again by the two parameters with cross additive characteristics through the potential algebra method. Lastly, the conclusions and discussions are made. [ABSTRACT FROM AUTHOR]