Solution of the Bosonic and Algebraic Hamiltonians by using AIM

We apply the notion of asymptotic iteration method (AIM) to determine eigenvalues of the bosonic Hamiltonians that include a wide class of quantum optical models. We consider solutions of the Hamiltonians, which are even polynomials of the fourth order with the respect to Boson operators. We also demonstrate applicability of the method for obtaining eigenvalues of the simple Lie algebraic structures. Eigenvalues of the multi-boson Hamiltonians have been obtained by transforming in the form of the single boson Hamiltonian in the framework of AIM

Is there any relationship between autism and pineal gland volume?

Purpose: Abnormalities in melatonin physiology and circadian rhythm are detected in patients with autism. Melatonin is produced predominantly in the pineal gland and the amount of melatonin released is proportional to the pineal gland volume. This study aimed to examine whether the pineal gland volume in children with autism is different from that in healthy children. Material and methods: Brain magnetic resonance images (MRI) of 120 paediatric patients with autism and 82 control paediatric subjects were examined; pineal parenchymal volume (PPV), pineal cyst rate (PCR), and total pineal gland volume (TPGV) were measured using a multimodality viewer (MMV), but only the TPGVs were measured using a tumour tracking (TT) method. Measurements were taken by 2 separate radiologists. Results: In patients with autism, the PPV and TPGV according to MMV, and the TPGV according to TT were significantly lower, and the PCR was significantly higher. Moreover, the ratio of PPV to TPGV was significantly lower in the autism patient group. In both groups, the TPGVs were significantly lower in the autism patient group than the controls among all age groups. Conclusions: Our study was the first to examine TPGVs in detail in paediatric patients with autism using 2 different methods. Low PPV-TPGV and high PCR have been observed in autism. This study also provides comparable reference values for pineal gland size in healthy children or autistic children aged 2-17 years. These results show promising potential for further research to understand the relationship between autism pathogenesis and the pineal gland

N-fold Supersymmetry in Quantum Systems with Position-dependent Mass

We formulate the framework of N-fold supersymmetry in one-body quantum mechanical systems with position-dependent mass (PDM). We show that some of the significant properties in the constant-mass case such as the equivalence to weak quasi-solvability also hold in the PDM case. We develop a systematic algorithm for constructing an N-fold supersymmetric PDM system. We apply it to obtain type A N-fold supersymmetry in the case of PDM, which is characterized by the so-called type A monomial space. The complete classification and general form of effective potentials for type A N-fold supersymmetry in the PDM case are given.Comment: 18 pages, no figures; Refs. updated, typos correcte

A progressive diagonalization scheme for the Rabi Hamiltonian

A diagonalization scheme for the Rabi Hamiltonian, which describes a qubit interacting with a single-mode radiation field via a dipole interaction, is proposed. It is shown that the Rabi Hamiltonian can be solved almost exactly using a progressive scheme that involves a finite set of one variable polynomial equations. The scheme is especially efficient for lower part of the spectrum. Some low-lying energy levels of the model with several sets of parameters are calculated and compared to those provided by the recently proposed generalized rotating-wave approximation and full matrix diagonalization.Comment: 8pages, 1 figure, LaTeX. Accepted for publication in J. Phys. B: At. Mol. Opt. Phy

Quaternionic Root Systems and Subgroups of the Aut(F4)Aut(F_{4})

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4),SP(2)= SO(5),SO(8),SO(9),F_{4} and E_{8} in terms of the discrete elements of the division algebras. The roots themselves display the group structures besides the octonionic roots of E_{8} which form a closed octonion algebra. The automorphism group Aut(F_{4}) of the Dynkin diagram of F_{4} of order 2304, the largest crystallographic group in 4-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F_{4}.The Weyl groups of many Lie algebras, such as, G_{2},SO(7),SO(8),SO(9),SU(3)XSU(3) and SP(3)X SU(2) have been constructed as the subgroups of Aut(F_{4}). We have also classified the other non-parabolic subgroups of Aut(F_{4}) which are not Weyl groups. Two subgroups of orders192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group $G_{2}$ of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e_{1},e_{2},e_{3} is used to show that SO(7) and SO(9) can be embedded triply symmetric way in SO(8) and F_{4} respectively

Spectrum generating algebras for position-dependent mass oscillator Schrodinger equations

The interest of quadratic algebras for position-dependent mass Schr\"odinger equations is highlighted by constructing spectrum generating algebras for a class of d-dimensional radial harmonic oscillators with $d \ge 2$ and a specific mass choice depending on some positive parameter $\alpha$. Via some minor changes, the one-dimensional oscillator on the line with the same kind of mass is included in this class. The existence of a single unitary irreducible representation belonging to the positive-discrete series type for $d \ge 2$ and of two of them for d=1 is proved. The transition to the constant-mass limit $\alpha \to 0$ is studied and deformed su(1,1) generators are constructed. These operators are finally used to generate all the bound-state wavefunctions by an algebraic procedure.Comment: 21 pages, no figure, 2 misprints corrected; published versio

A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour

The quantum version of a non-linear oscillator, previouly analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form $m={(1+\lambda x^2)}^{-1}$ and with a \la-dependent nonpolynomial rational potential. This \la-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for \la\to 0 all the characteristics of the linear oscillator are recovered. Firstly, the \la-dependent Schr\"odinger equation is exactly solved as a Sturm-Liouville problem and the \la-dependent eigenenergies and eigenfunctions are obtained for both \la>0 and \la<0. The \la-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as \la-deformations of the standard Hermite polynomials. In the second part, the \la-dependent Schr\"odinger equation is solved by using the Schr\"odinger factorization method, the theory of intertwined Hamiltonians and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a \la-dependent Rodrigues formula, a generating function and \la-dependent recursion relations between polynomials of different orders.Comment: 29 pages, 4 figure

Pseudo-Hermitian versus Hermitian position-dependent-mass Hamiltonians in a perturbative framework

We formulate a systematic algorithm for constructing a whole class of Hermitian position-dependent-mass Hamiltonians which, to lowest order of perturbation theory, allow a description in terms of PT-symmetric Hamiltonians. The method is applied to the Hermitian analogue of the PT-symmetric cubic anharmonic oscillator. A new example is provided by a Hamiltonian (approximately) equivalent to a PT-symmetric extension of the one-parameter trigonometric Poschl-Teller potential.Comment: 13 pages, no figure, modified presentation, 6 additional references, published versio

Cantrell pentalojisi

Cantrel pentalojisi nadir görülen bir sendromdur. Gelişen teknoloji ile birlikte son yıllarda erken haftalarda tanı konabilmektedir. Tanı konulan hastalarda gebeliğin sonlandırılması önerilmekte ve bir çok hastada terminasyon uygulanmaktadır. Ancak nadiren de olsa aile terminasyon seçeneğini kabul etmediğinden dolayı terme ulaşan fetuslar görülmektedir. Bizim vakamız da terme kadar ulaşan ve doğum esnasında kaybedilen bir Cantrel pentalojisi vakasıdı