Quasilinearization Method and Summation of the WKB Series

Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. Expansion of the $p$-th QLM iterate in powers of $\hbar$ reproduces the structure of the WKB series generating an infinite number of the WKB terms with the first $2^p$ terms reproduced exactly. The QLM quantization condition leads to exact energies for the P\"{o}schl-Teller, Hulthen, Hylleraas, Morse, Eckart potentials etc. For other, more complicated potentials the first QLM iterate, given by the closed analytic expression, is extremely accurate. The iterates converge very fast. The sixth iterate of the energy for the anharmonic oscillator and for the two-body Coulomb Dirac equation has an accuracy of 20 significant figures

Shape invariance and the exactness of quantum Hamilton-Jacobi formalism

Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two parallel methods to determine the spectra of a quantum mechanical systems without solving the Schr\"odinger equation. It was recently shown that the shape invariance, which is an integrability condition in SUSYQM formalism, can be utilized to develop an iterative algorithm to determine the quantum momentum functions. In this paper, we show that shape invariance also suffices to determine the eigenvalues in Quantum Hamilton-Jacobi Theory.Comment: Accepted for publication in Phys. Lett.

On Exactness Of The Supersymmetric WKB Approximation Scheme

Exactness of the lowest order supersymmetric WKB (SWKB) quantization condition $\int^{x_2}_{x_1} \sqrt{E-\omega^2(x)} dx = n \hbar \pi$, for certain potentials, is examined, using complex integration technique. Comparison of the above scheme with a similar, but {\it exact} quantization condition, $\oint_c p(x,E) dx = 2\pi n \hbar$, originating from the quantum Hamilton-Jacobi formalism reveals that, the locations and the residues of the poles that contribute to these integrals match identically, for both of these cases. As these poles completely determine the eigenvalues in these two cases, the exactness of the SWKB for these potentials is accounted for. Three non-exact cases are also analysed; the origin of this non-exactness is shown to be due the presence of additional singularities in $\sqrt{E-\omega^2(x)}$, like branch cuts in the $x-$plane.Comment: 11 pages, latex, 1 figure available on reques

Fingerprints of Equitable Estoppel and Promissory Estoppel on the Statute of Frauds in Contact Law

This Article evaluates a conundrum and identifies a genuine risk faced by state and federal courts in interpreting and applying the Statute of Frauds to contract law disputes. The Article provides a thorough analytical dissection of the Statute of Frauds as it has been interpreted and applied by the courts in light of the inescapable tension between the Statute’s formalities, mandated by the legislature, and the judiciary’s profound goal of attaining justice and fairness in deciding each contract law dispute in which the Statute is implicated. The Article discusses in depth how the Statute has been construed by state and federal courts in the unique factual context presented by each individual case argued before these courts. It investigates how judicial application of the Statute to particular facts has invoked creativity and ingenuity on the part of the courts that has led to the formulation of two equitable, ameliorating doctrines consisting of equitable estoppel and more recently, equitable estoppel’s evolutionary progeny, promissory estoppel. The Article discusses the potential dilemma of rigid application of the Statute at the expense of fair and just decisions, faced by the courts in applying the Statute, in light of the uniqueness of the factual context of each case; however, this Article criticizes impulses to apply promissory estoppel too readily because of the risk of eviscerating the Statute entirely. The Article’s analytical examination of a plethora of recent state and federal court decisions has concluded that the application of equitable estoppel principles in deciding whether to decline enforcement of a contract, based upon the defense of the Statute of Frauds, is viable and vibrant and is serving the legal community very well, but that there may also be a clear and present danger of over exuberance in unrestrained application of promissory estoppel by state and federal courts to override the application of the Statute and thereby nullify its mandate

Lack of Marketability and Minority Discounts in Valuing Close Corporation Stock: Elusiveness and Judicial Synchrony in Pursuit of Equitable Consensus

This Article discusses the often subtle tasks faced by the courts in construing close corporations law, which is state law. The judiciary in individual states has skillfully managed the invention, continuing development and ongoing evolution of lack of marketability and minority discounts as it strives to honor its constitutional mandate to resolve controversies between minority and majority shareholders in close corporations relating to valuing close corporations stock. These controversies arise in the context of share transactions in such corporations. Close corporations are traditionally not listed on stock exchanges, and the legislatures in some states have, in some instances, helped to facilitate the judiciary\u27s ongoing inventive ingenuity in its continuing efforts to resolve these disputes in a context where there is usually no marketplace for the stockholders in close corporations to readily leave by selling their shares and moving on. This Article analyzes the approaches of the judiciary in individual states as the judiciaries in the states collectively pursue almost in synchrony the elusive judicial goal of a fair and equitable resolution of close corporation valuation problems that arise in a plethora of factual share transaction settings, which demand individually tailored solutions

Ma-Xu quantization rule and exact WKB condition for translationally shape invariant potentials

For translationally shape invariant potentials, the exact quantization rule proposed by Ma and Xu is a direct consequence of exactness of the modified WKB quantization condition proved by Barclay. We propose here a very direct alternative way to calculate the appropriate correction for the whole class of translationally shape invariant potentials

Semiclassical approximation with zero velocity trajectories

We present a new semiclassical method that yields an approximation to the quantum mechanical wavefunction at a fixed, predetermined position. In the approach, a hierarchy of ODEs are solved along a trajectory with zero velocity. The new approximation is local, both literally and from a quantum mechanical point of view, in the sense that neighboring trajectories do not communicate with each other. The approach is readily extended to imaginary time propagation and is particularly useful for the calculation of quantities where only local information is required. We present two applications: the calculation of tunneling probabilities and the calculation of low energy eigenvalues. In both applications we obtain excellent agrement with the exact quantum mechanics, with a single trajectory propagation.Comment: 16 pages, 7 figure