Hill Determinant Method Research Articles

Accurate quasinormal modes of the five-dimensional Schwarzschild-Tangherlini black holes

The objective of this paper is to construct the accurate (say, to 11 decimal places) frequencies of the quasinormal modes of the five-dimensional Schwarzschild-Tangherlini black hole using three major techniques: the Hill determinant method, the continued fractions method, and the WKB-Pad\'e method and to discuss the limitations of each. It is shown that for the massless scalar, gravitational tensor, gravitational vector, and electromagnetic vector perturbations considered in this paper, the Hill determinant method and the method of continued fractions (both with the convergence acceleration) always give identical results, whereas the WKB-Pad\'e method gives the results that are amazingly accurate in most cases. Notable exception are the gravitational vector perturbations ($j=2$ and $\ensuremath{\ell}=2$), for which the WKB-Pad\'e approach apparently does not work. Here we have an interesting situation in which the WKB-based methods (WKB-Pad\'e and WKB--Borel--Le Roy) give the complex frequency that differs from the result obtained within the framework of the continued fraction method and the Hill determinant method. For the fundamental mode, deviation of the real part of frequency from the exact value is 0.5% whereas the deviation of the imaginary part is 2.7%. For $\ensuremath{\ell}\ensuremath{\ge}3$ the accuracy of the WKB results is similar again to the accuracy obtained for other perturbations. The case of the gravitational scalar perturbations is briefly discussed.

The Hill determinant method revisited

A Hill method is shown to give accurate energies and expectation values for several double and triple well sextic potential problems, thus contradicting previous criticisms of the method.

Quasimodes instability analysis of uncertain asymmetric rotor system based on 3D solid element model

Uncertainties are considered in the equation of motion of an asymmetric rotor system. Based on Hill's determinant method, quasimodes stability analysis with uncertain parameters is used to get stochastic boundaries of unstable regions. Firstly, A 3D finite element rotor model was built in rotating frame with four parameterized coefficients, which is assumed as random parameters representing the uncertainties existing in the rotor system. Then the influences of uncertain coefficients on the distribution of the unstable region boundaries are analyzed. The results show that uncertain parameters have various influences on the size, boundary and number of unstable regions. At last, the statistic results of the minimum and maximum spin speeds of unstable regions were got by Monte Carlo simulation. The used method is suitable for real engineering rotor system, because arbitrary configuration of rotors can be modeled by 3D finite element.

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.

Eigenvalues from power-series expansions: an alternative approach

An appropriate rational approximation to the eigenfunction of the Schrödinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of Hankel determinants. The convergence rate of this approach is greater than that for a well-established method based on power-series expansions weighted by a Gaussian factor with an adjustable parameter (the so-called Hill-determinant method).

Subsequence summation and the m function

A complex variable modification of the Hill determinant method is shown to give accurate values for the Weyl-Titchmarsh m 0( λ) function used in the spectral theory of the Schrödinger equation. A method of geometric sampling of partial sums is tested and shown to produce well converged results from some extremely slowly converging spectral sums for the m 0( λ) function associated with the potential V( x) = x 2.

Accurate energy-levels of the sextic-double-well oscillatorV(X)=X 6−3X 4 using modified Hill determinant approach

Using scaled harmonic oscillator wave functions, accurate energy levels of the sextic-double-well oscillatorV(X)=X 6−3X 4 are calculated through modified Hill determinant method. Present groundstate energy remains the same as reported by Tater and Turbiner,J. Phys. A26, 697 (1993).

Eigenvalues of Inverse-power Potentials in N-Dimensional Space

The eigenvalues of the inverse-power potentials are obtained in N-dimensional space by the infinite Hill determinant method. It is shown that exact solutions of this potential exist when there are some restrictions on the parameters by using the formalism of supersymmetric quantum mechanics.

On the improvement of convergence of Hill determinants

We propose a simple approach to analytically determine a parameter of a convergence factor exp(-gx2/4) in the Hill determinant method. It provides much more rapid convergence of the method than the other analytical approaches currently used for the g determination.

Accurate eigenvalues and eigenfunctions of simple quantum-mechanical systems by means of the Riccati-Hill method

We develop a power-series approach for the calculation of eigenvalues and eigenfunctions of separable quantum-mechanical problems. It improves on the widely used Hill-determinant method through a more convenient determination of the exponential factor of the eigenfunctions from a Riccati-like equation. The present Riccati-Hill approach yields accurate eigenvalues and eigenfunctions for an anharharmonic oscillator which has recently proved intractable by means of the standard method of the Hill determinant.

Improved Hill determinant method for the solution of quantum anharmonic oscillators.

Improved Hill determinant method for the solution of quantum anharmonic oscillators.

Accurate eigenvalues and eigenfunctions of simple quantum-mechanical systems by means of the Riccati-Hill method

We develop a power-series approach for the calculation of eigenvalues and eigenfunctions of separable quantum-mechanical problems. It improves on the widely used Hill-determinant method through a more convenient determination of the exponential factor of the eigenfunctions from a Riccati-like equation. The present Riccati-Hill approach yields accurate eigenvalues and eigenfunctions for an anharharmonic oscillator which has recently proved intractable by means of the standard method of the Hill determinant.

Rational potential using a modified Hill determinant method.

The use of vector recursion relations for calculating successive approximants to banded secular determinants permits a considerable enlargement of the class of potentials whose energy eigenvalues can be evaluated to high precision by the method of Hill determinants. To demonstrate this, we study the rational potential V(x)=${\mathit{x}}^{2}$+\ensuremath{\lambda}${\mathit{x}}^{2}$/(1+${\mathit{gx}}^{2}$) in one dimension for g>0. Using operator methods and Hill determinants in conjunction with vector recursion relations, we display the convenience with which the energy eigenvalues of the problem can be calculated to great accuracy for positive as well as negative values of \ensuremath{\lambda}. The results for \ensuremath{\lambda}0 have not previously been obtained. The energy eigenvalue spectrum of this potential is also shown to possess discontinuities when the coupling parameters \ensuremath{\lambda} and g take suitable limiting values, pointing to the need for exercising care in the construction of perturbative solutions to the problem.

Failure of the Hill determinant method for the sextic anharmonic oscillator

An extended analysis of eigenvalues and wavefunctions resulting from the Hill determinant method is carried out for the one-dimensional sextic anharmonic oscillator. It is shown that, in spite of a seemingly natural ansatz of the method, irrelevant wavefunctions may appear, leading to incorrect eigenvalues. The domain of applicability of the method is limited and its practical use demands a priori investigation before concrete calculations of spectral characteristics. Effective variational calculations based on adequate trial functions are performed and comparison of the results of both approaches is presented.

The anharmonic oscillator and the range of validity of its Hill-determinant construction

For all the interactions V( x)= ax+ bx 2+ cx 3+ dx 4 we prove that (and under which-surprising-condition) the so-called Hill-determinant method leads to the correct bound states.

Improved Hill determinant method: General approach to the solution of quantum anharmonic oscillators.

Improved Hill determinant method: General approach to the solution of quantum anharmonic oscillators.

Polynomial solutions of the planar Coulomb diamagnetic problem.

It is argued that a set of manifestly normalizable solutions of the two-dimensional Coulomb diamagnetic problem generated by the fine-tuning of the external magnetic field is unphysical. An implication of this for the Hill determinant method is pointed out.

The perturbative method of Hill determinants

A new form of perturbation theory is proposed. It is based on the so-called Hill-determinant method, and its main idea lies in taking the related particular, exactly known bound states as zero-order approximants. In detail, a few related technicalities are illustrated on the sextic anharmonic oscillator.

Quantum mechanical sextic anharmonic oscillators: normalisability of wavefunctions and some exact eigenvalues

It is shown that the usually tacitly assumed normalisability of the wavefunctions of the anharmonic oscillators with potential V(y)= 1 2 (ay 2+by 4+cy 6) (y=x for the 1-d problem, r for the 3-d problem) is generally not valid. A criterion for strict invalidity of the truncated Hill-determinant method is given. Some exact eigenvalues are obtained.