Eigenvalue Moment Method Research Articles

Orthonormal polynomial projection quantization: an algebraic eigenenergy bounding method

The ability to generate tight eigenenergy bounds for low dimension bosonic or ferminonic, hermitian or non-hermitian, Schrödinger operator problems is an important objective in the computation of quantum systems. Very few methods can simultaneously generate lower and upper bounds. One of these is the Eigenvalue Moment Method (EMM) originally introduced by Handy and Besssis, exploiting the use of semidefinite programming/nonlinear-convex optimization (SDP) techniques as applied to the positivity properties of the multidimensional bosonic ground state for a large class of important physical systems (i.e. those admitting a moments’ representation). A recent breakthrough has been achieved through another, simpler, moment representation based quantization formalism, the Orthonormal Polynomial Projection Quantization Bounding Method (OPPQ-BM). It is purely algebraic and does not require any SDP analysis. We discuss its essential structure in the context of several one dimensional examples.

A moments' analysis of quasi-exactly solvable systems: a new perspective on the sextic potential

A quantum system is quasi-exactly solvable (QES) when a subset of its eigenvalues can be obtained in closed form, or they are the roots of closed form expressions. In one dimension, QES states assume the form , involving a known positive reference function and a polynomial P(x), so that the Taylor expansion of truncates at a finite order. For non-QES states this truncation procedure, for suitable reference functions, can provide approximate values for the eigenvalues. It corresponds to the Hill determinant method, which is known to be unstable when assumes the controlling asymptotic form of the physical state. For this reason, it cannot be used to simultaneously generate exact QES states while approximating non-QES states. To address these limitations, the orthogonal polynomial projection quantization (OPPQ) method was developed (Handy and Vrinceanu 2013 J. Phys. A: Math. Theor. 46 135202; 2013 J. Phys. B: Atom. Mol. Opt. Phys. 46 115002). It is demonstrated in this paper that OPPQ can directly and transparently provide, at the same time, both algebraic solutions for QES states and convergent, numerically stable, approximations for non-QES states. Within the OPPQ analysis for the wavefunction representation , the Bender–Dunne energy orthogonal polynomials correspond, exactly (up to a numerical factor), to the energy dependent power moments ν(p) = ∫dx xpΦ(x). Within this perspective, the existence of QES states is associated with an anomalous kink behavior in the order of the finite difference moment equation corresponding to the ν's, suggesting a change in the number of degrees of freedom. This was first noted through the implementation of the eigenvalue moment method, the first application of semidefinite programming analysis to quantum operators (Handy and Bessis 1985 Phys. Rev. Lett. 55 931). This moments' perspective also reveals additional properties for the non-QES states of the same symmetry as the QES states: their lower order ν-moments must be zero. We demonstrate our results in the context of the two sextic potentials: Vsa(x) = x6 + mx2 + bx4 and Vss(x) = x6 + mx2 + b/x2.

Orthogonal polynomial projection quantization: a new Hill determinant method

Accurate energy eigenvalues are obtained by simply projecting the unknown bound state wave function on, essentially, arbitrary sets of orthogonal polynomials, and setting a subset of these to zero. The projection integrals are represented in terms of the power moments of the wave function, obtained recursively by transforming Schrödinger’s equation into a moment equation. Because unbounded wave functions do not have power moments, all solutions are guaranteed to be L2, resulting in a more robust, rapidly converging and stable method when compared with configuration space Hill determinant methods. More importantly, our approach permits the use of arbitrary, nonanalytic, positive reference functions, including those that manifest the true asymptotic behavior of the discrete states. These advantages are not usually possible with the standard Hill approach. The formulation presented here can be applied to any problem for which Schrödinger’s equation can be transformed into a moment equation. In this regard it is related to the L2 quantization prescription developed by Tymczak et al (1998 Phys. Rev. Lett. 80 3674; 1998 Phys. Rev. A 58 2708) corresponding to the Hill determinant method in momentum space, and to the eigenvalue moment method (Handy and Bessis 1985 Phys. Rev. Lett. 55 931; Handy et al 1988 Phys. Rev. Lett. 60 253), the first use of semidefinite programming related analysis in quantum physics. The latter method can be used to generate the ground state, whose orthogonal polynomials can serve to generate even more rapidly converging estimates for the excited discrete state energies. We demonstrate the power of this new approach on the sextic and quartic anharmonic oscillators, as well as on recently studied one- and two-dimensional pseudo-Hermitian Hamiltonians.

(Quasi)-convexification of Barta's (multi-extrema) bounding theorem:

There has been renewed interest in the exploitation of Barta's configuration space theorem (BCST) (Barta 1937 C. R. Acad. Sci. Paris 204 472) which bounds the ground-state energy, , by using any Φ lying within the space of positive, bounded, and sufficiently smooth functions, . Mouchet's (Mouchet 2005 J. Phys. A: Math. Gen. 38 1039) 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 within . 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. EMM is an alternative eigenenergy bounding, variational procedure, overlooked by Mouchet, which also exploits the positivity of the desired physical solution. Furthermore, it is applicable to Hermitian and non-Hermitian systems with complex-number quantization parameters (Handy and Bessis 1985 Phys. Rev. Lett. 55 931, Handy et al 1988 Phys. Rev. Lett. 60 253, Handy 2001 J. Phys. A: Math. Gen. 34 5065, Handy et al 2002 J. Phys. A: Math. Gen. 35 6359). Our analysis exploits various quasi-convexity/concavity theorems common to the GEP representation. We outline the general theory, and present some illustrative examples.

Derivation of Exact Eigenvalues and Eigenfunctions Based onthe Analytical Transfer Matrix Method

We extend the analytical transfer matrix method (ATMM) to calculateboth the energy eigenvalues and wavefunctions for the Morse potentialand the regulated Coulomb potential. Derivations of the exacteigenenergies and eigenfunctions are presented in detail by the ATMM.We compare our results with that obtained by relaxational approach andthe eigenvalue moment method, and it is shown that the ATMM can produceaccurate eigenvalues. The eigenfunctions by the ATMM are also proven tobe correct and meaningful.

Reply to Comment on ‘Excited states in the infinite quantum lens potential: conformalmapping and moment quantization methods’

The suitability of conformal transformation (CT) analysis, and the eigenvalue momentmethod (EMM), for determining the eigenenergies and eigenfunctions of a quantumparticle confined within a lens geometry, is reviewed and compared to the recentresults by Even and Loualiche (2003 J. Phys.: Condens. Matter 15 8465). It is shownthat CT and EMM define two accurate and versatile analytical/computationalmethods relevant to lens shaped regions of varying geometrical aspect ratios.

Excited states in the infinite quantum lens potential: conformal mapping and momentquantization methods

The conformal mapping method (CMM) and the eigenvalue moment method(EMM) are employed to study the eigenvalue problem defined by a free particle in aquantum lens geometry. The characteristics of the spectrum and the correspondingspatial properties of wavefunctions are studied for varying symmetry quantizationnumbers and lens parameter values. It is shown that the states belong to twoindependent Hilbert subspaces corresponding to even and odd azimuthal,m, quantum numbers. The CMM analysis is used to reduce the Dirichlet problem for theHelmholtz’s equation into a 2D problem defined by a region with semicirculargeometry. This approach allows one to obtain explicit analytical solutions in terms ofthe lens geometry. In the case of small geometry deformations (relative to thesemicircular case), the solutions can be found by a perturbative method. The exact andapproximate solutions are compared for different values of the lens parameters. Wecompare these results with those derived through the EMM approach, which,although more computationally expensive, can yield converging lower and upperbounds. The particular formulation developed here differs significantly from anearlier application of EMM to the ground state case (within each symmetry class).Accordingly, we develop the new formulation and apply it to a limited number of statesin order to confirm the results derived by the other, aforementioned, methods.

Generating bounds for the discrete state energy values of the infinite quantum lens potential

Moment based methods have produced efficient multiscale quantization algorithms for solving singular perturbation/strong coupling problems. One of these, the Eigenvalue Moment Method (EMM), developed by Handy et al (Phys. Rev. Lett.{\bf 55}, 931 (1985); ibid, {\bf 60}, 253 (1988b)), generates converging lower and upper bounds to a specific discrete state energy, once the signature property of the associated wavefunction is known. This method is particularly effective for multidimensional, bosonic ground state problems, since the corresponding wavefunction must be of uniform signature, and can be taken to be positive. Despite this, the vast majority of problems studied have been on unbounded domains. The important problem of an electron in an infinite quantum lens potential defines a challenging extension of EMM to systems defined on a compact domain. We investigate this here, and introduce novel modifications to the conventional EMM formalism that facilitate its adaptability to the required boundary conditions.

Extension of a spectral bounding method to the PT-invariant states of the −(iX)Nnon-Hermitian potential

The eigenvalue moment method (EMM) developed by Handy (J. Phys. A: Math. Gen. 2001 34 L271, J. Phys. A: Math. Gen. 2001 34 5065). Handy et al (J. Phys. A: Math. Gen. 2001 34 5593), and Handy and Xiao-Qian Wang (J. Phys. A: Math. Gen. 2001 34 8297), which generates converging lower and upper bounds to the (complex) discrete state energies, is extended to the case of discrete states with non-Real support. In particular, Bender and Boettcher (Phys. Rev. Lett. 1998 80 5243) have argued on the reality of the discrete state spectrum for the −(iX)N potential. For N (integer) ≥ 4, such PT-invariant solutions can only exist on appropriate complex contours. We develop and apply the necessary EMM formalism to such cases. In particular, the restriction of EMM to the anti-Stokes angles significantly increases the convergence rate of the bounds.

Generation of converging Regge-pole bounds: a new formulation of complex rotation quantization

We propose an unprecedented bounding theory which generates converging bounds to the Regge poles of rational fraction scattering potentials. This is made possible by the recent work of Handy ( 2001 J. Phys. A: Math. Gen. 34 L271) and Handy and Wang (2001 J. Phys. A: Math. Gen. 34 8297) which transforms the Schrödinger equation into an equivalent fourth-order, lineardifferential equation for the probability density. This new representation is better suited for numerical considerations, since the rapid oscillations of the Regge-pole wavefunction are factored out. More importantly, the moments of the probability density can be constrained (and thereby the underlying complex angular momentum parameter of the effective potential function) through appropriate moment problem theorems, as incorporated within the eigenvalue moment method of Handy and Bessis (1985 Phys. Rev. Lett. 55 931) and Handy et al (1988 Phys. Rev. Lett. 60 253).

Multiscale reference function analysis of the 𝒫𝒯 symmetry breaking solutions for theP2+ iX3+ iαXHamiltonian

The recent work of Delabaere and Trinh (Delabaere E and Trinh D T 2000 J. Phys. A: Math. Gen. 33 8771) discovered the existence of symmetry breaking, complex energy, L2 solutions for the one-dimensional Hamiltonian, P2 + iX 3 + iαX, in the asymptotic limit α→-∞. Their asymptotic analysis produced questionable results for moderate values of α. We can easily confirm the existence of 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 (Tymczak C J, Japaridze G S, Handy C R and Wang Xiao-Qian 1998a Phys. Rev. Lett. 80 3678; 1998b 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 analysing many related problems that have appeared in the recent literature.

Generating converging bounds to the (complex) discrete states of theP2+ iX3+ iαXHamiltonian

The Eigenvalue Moment Method (EMM), Handy (2001), Handy and Wang (2001)) is applied to the $H_\alpha \equiv P^2 + iX^3 + i\alpha X$ 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)).

Generating converging eigenenergy bounds for the discrete states of the -ix3non-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 $|\Psi(x)|^2$. 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.

Continuous wavelet transform analysis of one-dimensional quantum bound states from first principles.

Over the last decade, Handy and Bessis have developed a moment-problem-based, multiscale quantization theory, the eigenvalue moment method (EMM), which has proven effective in solving singular, strongly coupled, multidimensional Schr{umlt o}dinger Hamiltonians. We extend the scope of EMM by demonstrating its essential role in the generation of wavelet transforms for one-dimensional quantum systems. Combining this with the function-wavelet reconstruction formulas currently available, we are able to recover the wave function systematically, from first principles, through a multiscale process proceeding from large spatial scales to smaller ones. This accomplishment also addresses another outstanding problem, that of reconstructing a function from its moments. For the class of problems considered, the combined EMM-wavelet analysis yields a definitive solution. {copyright} {ital 1996 The American Physical Society.}

Ground-state energy bounds of singular potentials

The eigenvalue moment method is an effective eigenenergy bounding theory. Its application can be further enhanced by incorporating important asymptotic features of the bound state wavefunction, Ψ( r), at r = 0, and ∞. We demonstrate this by examining potentials of the form λ r 2α ( α = 2 and 3).

Variational formulation of a moment problem quantization method

The eigenvalue moment method (EMM) has proven to be an effective technique for generating converging lower and upper bounds to the bosonic ground state energy of singular, strongly coupled, quantum systems. Application of EMM theory requires an appropriate linearization of the highly nonlinear Hankel–Hadamard (HH) moment determinant constraints for the (n+1)×(n+1) Hankel matrices ℳn[u]≡M̂n0+∑i=1msM̂niu i), dependent on the missing moment variables {u(i)}≡u. We propose an alternate variational formulation utilizing the functions Det(ℳn+1[u])/Det(ℳn[u]), which we prove to be locally convex over the missing moment subset satisfying the HH positivity conditions Det(ℳν[u])≳0, for ν≤n. Additional features of this variational formulation facilitate its application to important problems such as the octic, sextic, and quartic anharmonic oscillators.

Application of the eigenvalue moment method to important one-dimensional quantum systems

The authors study the effectiveness of the eigenvalue moment method as applied to various one-dimensional quantum problems which have appeared in the literature. Of particular interest are the radial potential problems (including angular momentum effects) -Ze2/(r+ beta ) and r2+ lambda r2/(1+gr2) studied by De Meyer and Vanden Berghe (1990) (and more recently by Fernandez (1991)), and Witwit (1991) respectively. They also examine the potentials 1/2x2+or-gx4/(1+ alpha gx2), exhaustively studied by Auberson and Boissiere (1983). Finally, they examine the one-dimensional regulated Bohr atom with potential -Z/( mod x mod + alpha ), well studied by Loudon (1959), and Haines and Roberts (1969).

An eigenvalue moment method analysis of theV(x) = [ω2/2]x2+ αx3+ βx4anharmonic oscillator

The anharmonic oscillator potential V(x) = [ω2/2]x 2 + αx 3 + βx 4 has been studied in terms of a hypervirial-Padé analysis (Lai, C. S., and Madan, M. P., 1985, Molec. Phys., 54, 669) and improved perturbative expansions (Fernandez, F. M., Meson, A. M., and Castro, E. A., 1985, Molec. Phys., 58, 365). We compare these results with those of a recently developed technique, the eigenvalue moment method, which is fundamentally simple and capable of yielding converging bounds to the low lying eigenenergies. It is concluded that hypervirial-Padé approaches can be inadequate in ‘high precision’ analysis of eigenenergy spectra.

Application of the eigenvalue moment method to the quartic anharmonic double-well oscillator.

The eingenvalue moment method (EMM) is a precise technique for generating converging lower and upper bounds to the low-lying eigenenergies of singular quantum Hamiltonians. We apply it to the quartic anharmonic double-well oscillator problem, P 2 -Z 2 x 2 +x 4 , recently studied by Saavedra and Buendia [Phys. Rev. A 42, 5073 (1990)]. In addition, we introduce important algorithmic modifications to the conventional EMM formulation, leading to more efficient applications

Euclidean time formulation of the eigenvalue moment method: a moment problem-convexity analysis of Barnsley's theorem

Over the last few years, the eigenvalue moment method (EMM) has been shown to be very effective in generating converging lower and upper bounds to the discrete low-lying spectrum of singular multidimensional Hamiltonians. In this work the authors adapt the EMM approach to the Euclidean time-dependent Schrodinger equation. The result is a new EMM theory which significantly overlaps with other eigenenergy bounding theories and which leads to a more rigorous algorithmic formulation than previously available.