Related papers: Rigorous Methods for Bohr-Sommerfeld Quantization …
We revisit in this Note the well known Bohr-Sommerfeld quantization rule (BS) for 1-D Pseudo-differential self-adjoint Hamiltonians within the algebraic and microlocal framework of Helffer and Sj\"ostrand; BS holds precisely when the Gram…
In a recent paper by Gomes and Adhikari (J.Phys B30 5987(1997)) a matrix formulation of the Bohr-Sommerfield quantization rule has been applied to the study of bound states in one dimension quantum wells. Here we study these potentials in…
Consider the semiclassical limit, as the Planck constant $\hbar\ri 0$, of bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. We show that, for each eigenvalue of the Schr\"odinger operator,…
We review an "exact semiclassical" resolution method for the general stationary 1D Schr\"odinger equation with a polynomial potential. This method avoids having to compute any Stokes phenomena directly; instead, it basically relies on an…
The Schwinger boson theory provides a natural path for treating quantum spin systems with large quantum fluctuations. In contrast to semi-classical treatments, this theory allows us to describe a continuous transition between magnetically…
In this paper we describe the spectrum of the quantum periodic Benjamin-Ono equation in terms of the multi-phase solutions of the underlying classical system (the periodic multi-solitons). To do so, we show that the semi-classical…
We give Bohr-Sommerfeld quantization rules corresponding to quasi-eigenvalues for a 1-D h-Pseudodifferential operator with real principal symbol and verifying PT symmetry.
A Bohr-Sommerfeld quantization rule is generalized for the case of the deformed commutation relation leading to minimal uncertainties in both coordinate and momentum operators. The correctness of the rule is verified by comparing obtained…
In this paper we study the local and global regularity properties of the Zakharov system on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the…
The Schr\"odinger equation is considered on the half line with a selfadjoint boundary condition when the potential is real valued, integrable, and has a finite first moment. It is proved that the potential and the two boundary conditions…
We describe the eigenvalues and eigenvectors of real-analytic, non-self-adjoint Berezin--Toeplitz operators, up to exponentially small error, on complex one-dimensional compact manifolds, under the hypothesis of regularity of the energy…
We consider the scattering transform for the Schr\"odinger equation with a singular potential and no bound states. Using the Riccati representation for real-valued potentials on the line, we obtain invertibility and Lipschitz continuity of…
The Bohr-Sommerfeld quantization rule lies at the heart of the modern semiclassical theory of a Bloch electron in a magnetic field. This rule is predictive of Landau levels and quantum oscillations for conventional metals, as well as for a…
We review an exact analytical resolution method for general one-dimensional (1D) quantal anharmonic oscillators: stationary Schr\"odinger equations with polynomial potentials. It is an exact form of WKB treatment involving spectral (usual)…
A numerical method to solve the direct scattering problem for the Zakharov-Shabat system associated to the initial value problem for the nonlinear Schroedinger equation is proposed. The method involves the numerical solution of Volterra…
The semiclassical quantization conditions for all partial waves are derived for bound states of two interacting anyons in the presence of a uniform background magnetic field. Singular Aharonov-Bohm-type interactions between the anyons are…
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schr\"odinger eigenvalue equation $H\Psi \equiv (\Delta_2 +V)\Psi=E\Psi$ on any 2D Riemannian…
We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schr\"odinger operators in the presence of two potential wells and with an energy-level crossing. We provide…
We study the eigenvalues of the self-adjoint Zakharov-Shabat operator corresponding to the defocusing nonlinear Schrodinger equation in the inverse scattering method. Real eigenvalues exist when the square of the potential has a simple…
We prove an inequality on the Kantorovich-Rubinstein distance --which can be seen as a particular case of a Wasserstein metric-- between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, but with a…