For a class of non-selfadjoint h-pseudodifferential operators in dimension 2, we determine all eigenvalues in an h-independent domain in the complex plane and show that they are given by a Bohr-Sommerfeld quantization condition. No complete integrability is assumed, and as a geometrical step in our proof, we get a KAM-type theorem (without small divisors) in the complex domain.
In this paper, we give a description of the spectrum of a class of non-selfadjoint perturbations of selfadjoint operators in dimension one and we show that it is given by Bohr-Sommerfeld quantization conditions. To achieve this, we make use of previous work by Michael Hitrik, Anders Melin and Johannes Sj{ö}strand. We also give an application of our result in the case of PT-symmetric pseudo-differential operators.
In this article, we state the Bohr-Sommerfeld conditions around a singular value of hyperbolic type of the principal symbol of a self-adjoint semiclassical Toeplitz operator on a compact connected Kähler surface. These conditions allow the description of the spectrum of the operator in a fixed size neighbourhood of the singularity. We provide numerical computations for three examples, each associated to a different topology.
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.
Consider the semiclassical limit, as the Planck constant $\hbar\ri 0$, of bound states of a quantum particle in a one-dimensional potential well. We justify the semiclassical asymptotics of eigenfunctions and recover the Bohr-Sommerfeld quantization condition.
In this paper, we describe Bohr-Sommerfeld rules for semi-classical completely integrable systems with 2 degrees of freedom with non degenerate singularities (Morse-Bott singularities) under the assumption that the energy level of the first Hamiltonian is non singular. The more singular case of {\it focus-focus} singularities is studied in [Vu Ngoc San, CPAM 2000] and [Vu Ngoc San, PhD 1998] The case of 1 degree of freedom has been studied in [Colin de Verdiere-Parisse, CMP 1999] Our theory is applied to some famous examples: the geodesics of the ellipsoid, the $1:2$-resonance, and Schroedinge…
ETH Zurich, Institute for Mathematical Research（苏黎世联邦理工学院数学研究所）；Department of Mathematical Sciences, Ritsumeikan University（立命馆大学数学科学系）；Department of Engineering, University of Borås（博尔斯大学工程系）
In this article, we state the Bohr-Sommerfeld conditions around a global minimum of the principal symbol of a self-adjoint semiclassical Toeplitz operator on a compact connected Kähler surface, using an argument of normal form which is obtained thanks to Fourier integral operators. These conditions give an asymptotic expansion of the eigenvalues of the operator in a neighbourhood of fixed size of the singularity. We also recover the usual Bohr-Sommerfeld conditions away from the critical point. We end by investigating an example on the two-dimensional torus.
We present a detailed study, in the semi-classical regime $h \to 0$, of microlocal properties of systems of two commuting h-PDO s $P_1(h)$, $P_2(h)$ such that the joint principal symbol $p=(p_1,p_2)$ has a special kind of singularity called a "focus-focus" singularity. Typical examples include the quantum spherical pendulum or the quantum Champagne bottle. In the spirit of Colin de Verdière and Parisse, we show that such systems have a universal behavior described by singular quantization conditions of Bohr-Sommerfeld type. These conditions are used to give a precise description of the joint s…
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 Schrödinger operator, the Bohr-Sommerfeld quantization condition is satisfied at least for one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of a potential barrier. We show that, at least from one side, the barrier fixes the phase of wave functions in the same way as a potential barrier of infinite width. On the other hand, it …
We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to h-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.
The stationary 1D Schrödinger equation with a polynomial potential $V(q)$ of degree N is reduced to a system of exact quantization conditions of Bohr-Sommerfeld form. They arise from bilinear (Wronskian) functional relations pairing spectral determinants of (N+2) generically distinct operators, all the transforms of one quantum Hamiltonian under a cyclic group of complex scalings. The determinants' zeros define (N+2) semi-infinite chains of points in the complex spectral plane, and they encode the original quantum problem. Each chain can now be described by an exact quantization condition whic…
We study the eigenvalues of the non-self adjoint problem $-y^{\prime\prime}+V(x)y=E y$ on the half-line $0\leq x<+\infty$ under the Robin boundary condition at $x=0$, where $V$ is a monic polynomial of degree $\geq 3$. We obtain a Bohr-Sommerfeld-like asymptotic formula for $E_n$ that depends on the boundary conditions. Consequently, we solve certain inverse spectral problems, recovering the potential $V$ and boundary condition from the first $(m+2)$ terms of the asymptotic formula.
In this note we compare two recent results about the distribution of eigenvalues for semi-classical pseudodifferential operators in two dimensions. For classes of analytic operators A. Melin and the author obtained a complex Bohr-Sommerfeld rule, showing that the eigenvalues are situated on a distorted lattice. On the other hand, with M. Hager we showed in any dimension that Weyl asymptotics holds with probability close to 1 for small random perturbations of the operator. In both cases the eigenvalues distribute to leading order according two smooth densities and we show here that the two dens…
