We show that the Riccati form of the Schrodinger equation can be reformulated in terms of two linear equations depending on an arbitrary function G. When $G$ and the potential are polynomials, the solutions of these two equations are entire functions (L and K) and the zeroes of K are identical to those of the wave function. Requiring such a zero at a large but finite value of the argument yields the low energy eigenstates with exponentially small errors. Judicious choice of G can improve dramatically the numerical treatment. The method yields many significant digits with modest computer means.
Anharmonic oscillators with the sextic and decatic potentials are studied employing the refinable interpolating scale functions. This method yields highly accurate values of both energy eigenvalues and eigenfunctions for the sextic and decatic oscillator without constraining the potential parameters. Convergence of the solutions in the present method is noticed to be very fast.
We construct a time-dependent double well potential as an exact spectral equivalent to the explicitly time-dependent negative quartic oscillator with a time-dependent mass term. Defining the unstable anharmonic oscillator Hamiltonian on a contour in the lower-half complex plane, the resulting time-dependent non-Hermitian Hamiltonian is first mapped by an exact solution of the time-dependent Dyson equation to a time-dependent Hermitian Hamiltonian defined on the real axis. When unitary transformed, scaled and Fourier transformed we obtain a time-dependent double well potential bounded from belo…
It is already known that the quantum quartic single-well anharmonic oscillator $V_{ao}(x)=x^2+g^2 x^4$ and double-well anharmonic oscillator $V_{dw}(x)= x^2(1 - gx)^2$ are essentially one-parametric, their eigenstates depend on a combination $(g^2 \hbar)$. Hence, these problems are reduced to study the potentials $V_{ao}=u^2+u^4$ and $V_{dw}=u^2(1-u)^2$, respectively. It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction $Ψ_{ao}(u)$, obtained recently, see JPA 54 (2021) 295204 [1] and Arxiv 2102.04623 [2], and then forming the function $Ψ_{dw}(u)=Ψ…
Eigenvalues are defined for any element of an algebra of observables and do not require a representation in terms of wave functions or density matrices. A systematic algebraic derivation based on moments is presented here for the harmonic oscillator, together with a perturbative treatment of anharmonic systems. In this process, a collection of inequalities is uncovered which amount to uncertainty relations for higher-order moments saturated by the harmonic-oscillator excited states. Similar saturation properties hold for anharmonic systems order by order in perturbation theory. The new method,…
Improved Convergence Proof of the Delta Expansion and Order Dependent Mappings
本文改进和推广了非谐振子能量本征值的δ展开-阶数相关映射（变分扰动展开）收敛性的最近严格证明。对于单阱非谐振子，证明了$g\in[0,\infty]$的收敛一致性。收敛性证明也扩展到复值的$g$躺在一个广泛的领域的黎曼曲面的$E(g)$。通过尺度关系à la Symanzik，这证明了强耦合区（其中标准微扰级数是非Borel可和的）中双阱的δ展开的收敛性，以及某些亚稳势中的复"能量本征值“的收敛性。δ展开式收敛的充分条件以三个定理的形式总结出来，这些定理适用于广泛的量子力学和高维场论系统。
In this comment we show that the eigenvalues of a quartic anharmonic oscillator obtained recently by means of the asymptotic iteration method may not be as accurate as the authors claim them to be.
The state-of-the-art quantum computing hardware has entered the noisy intermediate-scale quantum (NISQ) era. Having been constrained by the limited number of qubits and shallow circuit depth, NISQ devices have nevertheless demonstrated the potential of applications on various subjects. One example is the variational quantum eigensolver (VQE) that was first introduced for computing ground states. Although VQE has now been extended to the study of excited states, the algorithms previously proposed involve a recursive optimization scheme which requires many extra operations with significantly dee…
