相关论文: Exercises in exact quantization

Similarity symmetries of the factorization chains for one-dimensional differential and finite-difference Schr\"odinger equations are discussed. Properties of the potentials defined by self-similar reductions of these chains are reviewed. In…

To describe a quantum system whose potential is divergent at one point, one must provide proper connection conditions for the wave functions at the singularity. Generalizing the scheme used for point interactions in one dimension, we…

We discussed exact solutions of the Schroedinger equation for a two-dimensional parabolic confinement potential in a homogeneous external magnetic field. It turns out that the two-electron system is exactly solvable in the sense, that the…

Nonlocal Hamiltonian-type operators, like e.g. fractional and quasirelativistic, seem to be instrumental for a conceptual broadening of current quantum paradigms. However physically relevant properties of related quantum systems have not…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0,$ we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity.…

The numerical version of the Hamilton-Jacobi quantization method, recently proposed, is applied to the one dimensional quartic oscillator. A suitable quantization condition is formulated and various energy levels and wave functions are…

We present a polymer quantization of the -lambda/r^2 potential on the positive real line and compute numerically the bound state eigenenergies in terms of the dimensionless coupling constant lambda. The singularity at the origin is handled…

In this paper, we investigate the Schr\"odinger equation for a class of spherically symmetric potentials in a simple and unified manner using the Lie algebraic approach within the framework of quasi-exact solvability. We illustrate that all…

We prove that the singularities of a potential in the two and three dimensional Schr\"odinger equation are the same as the singularities of the Born approximation (Diffraction Tomography), obtained from backscattering inverse data, with an…

A new type of exact solvability is reported. We study the general central polynomial potentials (with 2q anharmonic terms) which satisfy the Magyari's partial exact solvability conditions (this means that they possess a…

A new pseudoperturbative (artificial in nature) methodical proposal [15] is used to solve for Schrodinger equation with a class of phenomenologically useful and methodically challenging anharmonice oscillator potentials V(q)=\alpha_o q^2 +…

The spectral determinant $D(E)$ of the quartic oscillator is known to satisfy a functional equation. This is mapped onto the $A_3$-related $Y$-system emerging in the treatment of a certain perturbed conformal field theory, allowing us to…

We study a class of Calogero-Sutherland type one dimensional N-body quantum mechanical systems, with potentials given by $$ V( x_1, x_2, \cdots x_N) = \sum_{i <j} {g \over {(x_i - x_j)^2}} - \frac{g^{\prime}}{\sum_{i<j}(x_i - x_j)^2} +…

We survey results that have been obtained for self-adjoint operators, and especially Schr\"odinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus…

In the present work, we apply the exact quantization condition, introduced within the framework of Padgett and Leacock's quantum Hamilton-Jacobi formalism, to angular and radial quantum action variables in the context of the Hartmann and…

A class of singular integral operators, encompassing two physically relevant cases arising in perturbative QCD and in classical fluid dynamics, is presented and analyzed. It is shown that three special values of the parameters allow for an…

This work is a continuation of our previos paper \cite{Zel1}, where for the the Schr\"odinger operator $H=-\Delta+ V(\e)\cdot$ $(V(\e)\ge 0)$, acting in the space $L_2(\R^d)\,(d\ge 3)$, some constructive sufficient conditions for…

We present a simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics, in which the Schr\"{o}dinger equation is a difference equation. It reproduces all the known ones whose…

We improve the results by Gr\'ebert and Paturel in \cite{GP} and prove that a linear Schr\"odinger equation on $R^d$ with harmonic potential $|x|^2$ and small $t$-quasiperiodic potential as $$ {\rm i}u_t - \Delta u+|x|^2u+\varepsilon…

Quantum-mechanical WKB-method is elaborated for the known quantum oscillator problem in curved 3-spaces models Euclid, Riemann, and Lobachevsky E_{3}, H_{3}, S_{3} in the framework of the complex variable function theory. Generalized…