Exercises in exact quantization

The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schr\"odinger Hamiltonians [-\d^2/\d q^2 + V(q)]^\pm on the half-line $\{q>0\}$, with a Dirichlet (-) or Neumann (+) condition at q=0. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential $V(q)=q^N$ as $N \to +\infty$vs its (solvable) $N=\infty$ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-N properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants \det(-\d^2/\d q^2 + q^4 + v q^2)^\pm are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic $v \to +\infty$ properties of Airy type. The third study addresses the potentials $V(q)=q^N+v q^{N/2-1}$ of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with v as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to N=2); these results probably reflect the presence of supersymmetric potentials in the family above.Comment: latex txt.tex, 2 files, 34 pages [SPhT-T00/078]; v2: corrections and updates as indicated by footnote

Functional Relations in Stokes Multipliers and Solvable Models related to U_q(A^{(1)}_n)

Recently, Dorey and Tateo have investigated functional relations among Stokes multipliers for a Schr{\"o}dinger equation (second order differential equation) with a polynomial potential term in view of solvable models. Here we extend their studies to a restricted case of n+1-th order linear differential equations.Comment: 20 pages, some explanations improved, To appear in J. Phys.

Apparatus description and data analysis of a radiometric technique for measurements of spectral and total normal emittance

The development of a radiometric technique for determining the spectral and total normal emittance of materials heated to temperatures of 800, 1100, and 1300 K by direct comparison with National Bureau of Standards (NBS) reference specimens is discussed. Emittances are measured over the spectral range of 1 to 15 microns and are statistically compared with NBS reference specimens. Results are included for NBS reference specimens, Rene 41, alundum, zirconia, AISI type 321 stainless steel, nickel 201, and a space-shuttle reusable surface insulation

Universality in the flooding of regular islands by chaotic states

We investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori. Depending on dynamical tunneling rates and the Heisenberg time, regular states disappear and chaotic states flood the regular tori. For a quantitative understanding we introduce a random matrix model. The resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system. We discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis.Comment: 11 pages, 12 figure

Spectral zeta functions of a 1D Schr\"odinger problem

We study the spectral zeta functions associated to the radial Schr\"odinger problem with potential V(x)=x^{2M}+alpha x^{M-1}+(lambda^2-1/4)/x^2. Using the quantum Wronskian equation, we provide results such as closed-form evaluations for some of the second zeta functions i.e. the sum over the inverse eigenvalues squared. Also we discuss how our results can be used to derive relationships and identities involving special functions, using a particular 5F_4 hypergeometric series as an example. Our work is then extended to a class of related PT-symmetric eigenvalue problems. Using the fused quantum Wronskian we give a simple method for calculating the related spectral zeta functions. This method has a number of applications including the use of the ODE/IM correspondence to compute the (vacuum) nonlocal integrals of motion G_n which appear in an associated integrable quantum field theory.Comment: 15 pages, version

Fermi Edge Singularities in the Mesoscopic Regime: II. Photo-absorption Spectra

We study Fermi edge singularities in photo-absorption spectra of generic mesoscopic systems such as quantum dots or nanoparticles. We predict deviations from macroscopic-metallic behavior and propose experimental setups for the observation of these effects. The theory is based on the model of a localized, or rank one, perturbation caused by the (core) hole left behind after the photo-excitation of an electron into the conduction band. The photo-absorption spectra result from the competition between two many-body responses, Anderson's orthogonality catastrophe and the Mahan-Nozieres-DeDominicis contribution. Both mechanisms depend on the system size through the number of particles and, more importantly, fluctuations produced by the coherence characteristic of mesoscopic samples. The latter lead to a modification of the dipole matrix element and trigger one of our key results: a rounded K-edge typically found in metals will turn into a (slightly) peaked edge on average in the mesoscopic regime. We consider in detail the effect of the "bound state" produced by the core hole.Comment: 16 page

Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

The knowledge of quantum phase flow induced under the Weyl's association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton's equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phase-space analytic geometry based on the star-product operation can hardly be visualized. The statement "quantum trajectory belongs to a constraint submanifold" can be changed e.g. to the opposite by a unitary transformation. Some of relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the star-composition law and preserves hamiltonian and constraint star-functions.Comment: 27 pages REVTeX, 6 EPS Figures. New references added. Accepted for publication to JM

Solitary Adrenal Metastasis from Esophageal Adenocarcinoma: A Case Report and Review of the Literature

Introduction. In patients with extra-adrenal malignancy, an adrenal mass necessitates investigating the possibility of metastatic tumor. Curable adrenal metastasis are considered as a rare event. Case report. A 52-year-old male suffering from lower esophageal adenocarcinoma with a solitary left adrenal metastasis is presented herein, who underwent concomitant transhiatal esophagectomy and left adrenalectomy. The patient remains disease-free 18 months later. Discussion. Adrenal metastases mostly occur in patients with lung, kidney, breast, and gastrointestinal carcinomas. Primary esophageal adenocarcinoma gives adrenal metastatic deposits according to autopsy series with an incidence of about 3%–12%. When no other evidence of metastatic disease in cancer patients exists, several authors advocate adrenalectomy with curative intent. Isolated cases of long-term survival after resection of solitary adrenal metastasis from esophageal adenocarcinoma, like in our case, have been reported only as case reports. Conclusion. This study concludes that surgical resection may result in survival benefit in selected patients with solitary adrenal metastasis from esophageal adenocarcinoma

Trace formula for noise corrections to trace formulas

We consider an evolution operator for a discrete Langevin equation with a strongly hyperbolic classical dynamics and Gaussian noise. Using an integral representation of the evolution operator we investigate the high order corrections to the trace of arbitary power of the operator. The asymptotic behaviour is found to be controlled by sub-dominant saddle points previously neglected in the perturbative expansion. We show that a trace formula can be derived to describe the high order noise corrections.Comment: 4 pages, 2 figure

On the Convergence of the WKB Series for the Angular Momentum Operator

In this paper we prove a recent conjecture [Robnik M and Salasnich L 1997 J. Phys. A: Math. Gen. 30 1719] about the convergence of the WKB series for the angular momentum operator. We demonstrate that the WKB algorithm for the angular momentum gives the exact quantization formula if all orders are summed.Comment: latex, 9 pages, no figures, to be published in Journal of Physics A: Math. and Ge