Gabriel Gonzalez
The problem of a particle localized in a ultra-short potential in one dimension is considered. By proposing a general solution to Schrodinger;s equation we show that the energy spectra and the probability of the particle have definite bounds for an arbitrary ultra-short potential. These results are relevant for the confinement of particles in nanodevices.
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We calculate accurate eigenvalues of the Schrödinger equation with the potential $V(r)=V_{0}r^α$, $α\geq -1$, $V_{0}α>0$. We resort to the Riccati-Padé method that is based on a rational approximation to the logarithmic derivative of the wavefunction. This approach applies when $α$ is a rational number.
We compare the Wronskian method (WM) and the Schrödinger eigenvalue march or canonical function method (SEM--CFM) for the calculation of the energies and eigenfunctions of the Schrödinger equation. The Wronskians between linearly independent solutions of the Schrödinger equation provide a rigorous basis for some of the assumptions of the SEM-CFM, like, for example, the concept of "saturation". We compare the performance of both approaches on a simple one-dimensional model and suggest that taking into account the asymptotic behaviour of the wavefunction (as is already done in the WM) may make t…
The Schrödinger equation in $D$-dimensions for the Manning-Rosen potential with the centrifugal term is solved approximately to obtain bound states eigensolutions (eigenvalues and eigenfunctions). The Nikiforov-Uvarov(NU) method is used in the calculations. We present numerical calculations of energy eigenvalues to two- and four-dimensional systems for arbitrary quantum numbers $n$ and $l$ with three different values of the potential parameter $α.$ It is shown that because of the interdimensional degeneracy of eigenvalues, we can also reproduce eigenvalues of a upper/lower dimensional sytem fr…
On the exact solutions of a one-dimensional Schrödinger equation with a rational potential
本文研究具有有理势的一维条件可解薛定谔方程，通过精确本征函数的节点建立了孤立精确本征值与哈密顿算符实际本征值之间的联系。
The polynomial solution of the N-dimensional space Schrodinger equation for a special case of Mie potential is obtained for any arbitrary $% l-state. The exact bound-state energy eigenvalues and the corresponding eigenfunctions are calculated for diatomic molecular systems in the Mie-type potential. Keywords: Mie potential, Schrodinger equation, Eigenvalue, Eigenfunction, Diatomic molecules
Exact solution of Schrodinger equation for the pseudoharmonic potential is obtained for an arbitrary angular momentum. The energy eigenvalues and corresponding eigenfunctions are calculated by Nikiforov-Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The energy eigenvalues are calculated numerically for some values of l and n with n<5 for some diatomic molecules.
The Canonical Function Method (CFM) is a powerful accurate and fast method that solves the Schrödinger equation for the eigenvalues directly without having to evaluate the eigenfunctions. Its versatility allows to solve several types of problems and in this work it is applied to the solution of several 1D potential problems, the 3D Hydrogen atom and the Morse potential.
A novel method is proposed to determine an analytical expression for eigenfunctions and numerical result for eigenvalues of the Schrödinger type equations, within the context of Taylor expansion of a function. Optimal truncation of the Taylor series gives a best possible analytical expression for eigenfunctions and numerical result for eigenvalues.
Diverging eigenvalues in domain truncations of Schrödinger operators with complex potentials are analyzed and their asymptotic formulas are obtained. Our approach also yields asymptotic formulas for diverging eigenvalues in the strong coupling regime for the imaginary part of the potential.
We prove the reality of the perturbed eigenvalues of some PT symmetric Hamiltonians of physical interest by means of stability methods. In particular we study 2-dimensional generalized harmonic oscillators with polynomial perturbation and the one-dimensional $x^2(ix)^ε$ for $-1<ε<0$.
It is well known that a suggestive relation exists that links Schrödinger's equation (SE) to the information-optimizing principle based on Fisher's information measure (FIM). The connection entails the existence of a Legendre transform structure underlying the SE. Here we show that appeal to this structure leads to a first order differential equation for the SE's eigenvalues that, in certain cases, can be used to obtain the eigenvalues without explicitly solving SE. Complying with the above mentioned equation constitutes a necessary condition to be satisfied by an energy eigenvalue. We show th…
We give a rigorous deduction of the eigenvalue problem of the nonlinear Schrödinger equation (NLS) at Dirac Points for potential of honeycomb lattice symmetry. Based on a bootstrap method, we observe the bifurcation of the eigenfunctions into eight distinct modes from the two-dimensional degenerated eigenspace of the regressive linear Schrödinger equation. We give the existence, the way of construction, uniqueness in $H^2$ space and the $C^\infty$ continuity of these eigenfunctions.
For integers $m\geq 3$, we study the non-self-adjoint eigenvalue problems $-u^{\prime\prime}(x)+(x^m+P(x))u(x)=E u(x)$, $0\leq x<+\infty$, with the boundary conditions $u(+\infty)=0$ and $αu(0)+βu^{\prime}(0)=0$ for some $α, β\in\C$ with $|α|+|β|\not=0$, where $P(x)=a_1 x^{m-1}+a_2 x^{m-2}+...+a_{m-1} x$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $E_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues.
For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problem $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=λu(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\fracπ{2}\pm \frac{(\ell+1)π}{m+2}$ in the complex plane, where $P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $λ_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expan…
We study the eigenvalues of the non-self adjoint problem $-y^{\prime\prime}+V(x)y=E y$ on the half-line $0\leq x<+\infty$ under the Robin boundary condition at $x=0$, where $V$ is a monic polynomial of degree $\geq 3$. We obtain a Bohr-Sommerfeld-like asymptotic formula for $E_n$ that depends on the boundary conditions. Consequently, we solve certain inverse spectral problems, recovering the potential $V$ and boundary condition from the first $(m+2)$ terms of the asymptotic formula.
Path integral analysis of Schrödinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states
本研究将传统路径积分方法推广到复平面角域边界条件的薛定谔型本征值问题，定义复化围道上的泛函积分，明确了实时间量子隧穿与欧几里得瞬子方法所得衰变率的一一对应关系。
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