Quasi-Exact Solvability in Local Field Theory. First Steps

The quantum mechanical concept of quasi-exact solvability is based on the idea of partial algebraizability of spectral problem. This concept is not directly extendable to the systems with infinite number of degrees of freedom. For such systems a new concept based on the partial Bethe Ansatz solvability is proposed. In present paper we demonstrate the constructivity of this concept and formulate a simple method for building quasi-exactly solvable field theoretical models on a one-dimensional lattice. The method automatically leads to local models described by hermitian hamiltonians.Comment: LaTeX, 11 page

Non-linear Quantization of Integrable Classical Systems

It is demonstrated that the so-called "unavoidable quantum anomalies" can be avoided in the farmework of a special non-linear quantization scheme. A simple example is discussed in detail.Comment: LaTeX, 14 p

Quasi Exactly Solvable 2×\times2 Matrix Equations

We investigate the conditions under which systems of two differential eigenvalue equations are quasi exactly solvable. These systems reveal a rich set of algebraic structures. Some of them are explicitely described. An exemple of quasi exactly system is studied which provides a direct counterpart of the Lam\'e equation.Comment: 14 pages, Plain Te

Quartic Anharmonic Oscillator and Random Matrix Theory

In this paper the relationship between the problem of constructing the ground state energy for the quantum quartic oscillator and the problem of computing mean eigenvalue of large positively definite random hermitean matrices is established. This relationship enables one to present several more or less closed expressions for the oscillator energy. One of such expressions is given in the form of simple recurrence relations derived by means of the method of orthogonal polynomials which is one of the basic tools in the theory of random matrices.Comment: 12 pages in Late

Shape invariance in prepotential approach to exactly solvable models

100學年度研究獎補助論文[[abstract]]In supersymmetric quantum mechanics, exact-solvability of one-dimensional quantum systems can be classified only with an additional assumption of integrability, the so-called shape invariance condition. In this paper we show that in the prepotential approach we proposed previously, shape invariance is automatically satisfied and needs not be assumed.[[journaltype]]國外[[incitationindex]]SCI[[booktype]]紙本[[countrycodes]]US

Breached pairing in trapped three-color atomic Fermi gases

We introduce an exactly solvable model for trapped three-color atom gases. Applications to a cigar-shaped trapped cold fermions reveal a complex structure of breached pairing phases. We find two competing superfluid phases at weak and intermediate couplings, each one with two color pair condensates, that can be distinguished from density profile measurements.Comment: 4 pages, 5 figure

Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces

We introduce two new families of quasi-exactly solvable (QES) extensions of the oscillator in a $d$-dimensional constant-curvature space. For the first three members of each family, we obtain closed-form expressions of the energies and wavefunctions for some allowed values of the potential parameters using the Bethe ansatz method. We prove that the first member of each family has a hidden sl(2,$\mathbb{R}$) symmetry and is connected with a QES equation of the first or second type, respectively. One-dimensional results are also derived from the $d$-dimensional ones with $d \ge 2$, thereby getting QES extensions of the Mathews-Lakshmanan nonlinear oscillator.Comment: 30 pages, 8 figures, published versio