arXiv subjects

C. R. Handy

Publications and source records attributed to C. R. Handy.

7 recordsLinked to original sources

arXiv2005

(Quasi)-Convexification of Barta's (Multi-Extrema) Bounding Theorem

There has been renewed interest in the exploitation of Barta's configuration space theorem (BCST, (1937)) which bounds the ground state energy. Mouchet's (2005) BCST analysis is based on gradient optimization (GO). However, it overlooks significant difficulties: (i) appearance of multi-extrema; (ii) inefficiency of GO for stiff (singular perturbation/strong coupling) problems; (iii) the nonexistence of a systematic procedure for arbitrarily improving the bounds. These deficiencies can be corrected by transforming BCST into a moments' representation equivalent, and exploiting a generalization of the Eigenvalue Moment Method (EMM), within the context of the well known Generalized Eigenvalue Problem (GEP), as developed here.

math-ph↗

arXiv2002

Comment on the article by Bender et al, J. Phys. A 35, L467 (2002)

The recent Letter by Bender, Berry, and Mandilara (2002, BBM) presents some interesting symmetry arguments which enable one to transform non-hermitian, PT invariant, (complex) polynomial potential hamiltonians, into secular equation representations with real coefficients. This achievement is claimed to ``demistify'' why these systems can admit real and/or complex eigenenergies. Two approximations underly their arguments, and weaken the implied significance of their work.

math-ph↗

arXiv2001

Extension of a Spectral Bounding Method to Complex Rotated Hamiltonians, with Application to $p^2-ix^3$

We show that a recently developed method for generating bounds for the discrete energy states of the non-hermitian $-ix^3$ potential (Handy 2001) is applicable to complex rotated versions of the Hamiltonian. This has important implications for extension of the method in the analysis of resonant states, Regge poles, and general bound states in the complex plane (Bender and Boettcher (1998)).

math-ph↗

arXiv2001

Multiscale Reference Function Analysis of the ${\cal P}{\cal T}$ Symmetry Breaking Solutions for the $P^2+iX^3+iαX$ Hamiltonian

The recent work of Delabaere and Trinh (2000 J. Phys. A 33 8771) discovered the existence of ${\cal P}{\cal T}$-symmetry breaking, complex energy, $L^2$ solutions for the one dimensional Hamiltonian, $P^2+iX^3+iαX$, in the asymptotic limit, $α\to -\infty$. Their asymptotic analysis produced questionable results for moderate values of $α$. We can easily confirm the existence of ${\cal P}{\cal T}$-symmetry breaking solutions, by explicitly computing the low lying states, for $|α| < O (10)$. Our analysis makes use of the Multiscale Reference Function (MRF) approach, developed by Tymczak et al (1998 Phys. Rev. Lett. 80 3678; 1998 Phys. Rev. A 58, 2708). The MRF results can be validated by comparing them with the converging eigenenergy bounds generated through the Eigenvalue Moment Method, as recently argued by Handy (2001a,b). Given the reliability of the MRF analysis, its fast numerical implementation, high accuracy, and theoretical simplicity, the present formalism defines an effective and efficient procedure for analyzing many related problems that have appeared in the recent literature.

math-ph↗

arXiv2001

Generating Converging Eigenenergy Bounds for the Discrete States of the -ix^3 Non-Hermitian Potential

Recent investigations by Bender and Boettcher (Phys. Rev. Lett 80, 5243 (1998)) and Mezincescu (J. Phys. A. 33, 4911 (2000)) have argued that the discrete spectrum of the non-hermitian potential $V(x) = -ix^3$ should be real. We give further evidence for this through a novel formulation which transforms the general one dimensional Schrodinger equation (with complex potential) into a fourth order linear differential equation for $|Ψ(x)|^2$. This permits the application of the Eigenvalue Moment Method, developed by Handy, Bessis, and coworkers (Phys. Rev. Lett. 55, 931 (1985);60, 253 (1988a,b)), yielding rapidly converging lower and upper bounds to the low lying discrete state energies. We adapt this formalism to the pure imaginary cubic potential, generating tight bounds for the first five discrete state energy levels.

math-ph↗

arXiv2001

Generating Converging Bounds to the (Complex) Discrete States of the $P^2 + iX^3 + iαX$ Hamiltonian

The Eigenvalue Moment Method (EMM), Handy (2001), Handy and Wang (2001)) is applied to the $H_α\equiv P^2 + iX^3 + iαX$ Hamiltonian, enabling the algebraic/numerical generation of converging bounds to the complex energies of the $L^2$ states, as argued (through asymptotic methods) by Delabaere and Trinh (J. Phys. A: Math. Gen. {\bf 33} 8771 (2000)).

math-ph↗

arXiv1997

A Nonperturbative Perspective on Inner Product Quantization: Highly Accurate Solutions to the Schr{ö}dinger Equation

We devise a new and highly accurate quantization procedure for the inner product representation, both in configuration and momentum space. Utilizing the representation $Ψ(ξ) = \sum_{i}a_i[E]ξ^i R_β(ξ)$, for an appropriate reference function, $R_β(ξ)$, we demonstrate that the (convergent) zeroes of the coefficient functions, $a_i[E] = 0$, approximate the exact bound/resonance state energies with increasing accuracy as $i \to \infty$. The validity of the approach is shown to be based on an extension of the Hill determinant quantization procedure. Our method has been applied, with remarkable success, to various quantum mechanical problems.

quant-ph↗