A two-dimensional Schrödinger operator with a constant magnetic field perturbed by a smooth compactly supported potential is considered. The spectrum of this operator consists of eigenvalues which accumulate to the Landau levels. We call the set of eigenvalues near the $n$'th Landau level an $n$'th eigenvalue cluster, and study the distribution of eigenvalues in the $n$'th cluster as $n\to\infty$. A complete asymptotic expansion for the eigenvalue moments in the $n$'th cluster is obtained and some coefficients of this expansion are computed. A trace formula involving the first eigenvalue moments is obtained.
We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform…
The paper is devoted to investigation of the spectrum of a perturbed Laplace-Beltrami operator on manifolds with closed geodesics of the same length and where metric is a perturbation of the metric of the unit sphere. As the result, we give the regularized trace formula for the eigenvalues ??of this operator when the metric of manifold is presented in abstract form.
The Landau Hamiltonian governing the behavior of a quantum particle in dimension 2 in a constant magnetic field is perturbed by a compactly supported magnetic field and a similar electric field. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.
In this paper we study a semiclassical heat trace expansion for perturbations of the harmonic oscillator, by adapting to the semiclassical setting techniques developed by Hitrik and Polterovich in [HP]. We use the expansion to obtain certain inverse spectral results.
The Landau Hamiltonian, describing the behavior of a quantum particle in dimension 2 in a constant magnetic field, is perturbed by a magnetic field with power-like decay at infinity and a similar electric potential. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.
Under a perturbation by a decaying electric potential, the Landau Hamiltonian acquires some discrete eigenvalues between the Landau levels. We study the perturbation by an "expanding" electric potential $V(t^{-1}x)$, $t>0$, and derive a quasi-classical formula for the counting function of the discrete spectrum as $t\to \infty$.
We establish a sharp estimate on the size of the spectral clusters of the Landau Hamiltonian with $L^p$ potentials in two dimensions as the cluster index tends to infinity. In three dimensions, we prove a new limiting absorption principle as well as a unique continuation theorem. The results generalize to higher dimensions.
We consider self-adjoint Dirac operators $\ham{D}=\ham{D}_0 + V(x)$, where $\ham{D}_0$ is the free three-dimensional Dirac operator and $V(x)$ is a smooth compactly supported Hermitian matrix. We define resonances of $\ham{D}$ as poles of the meromorphic continuation of its cut-off resolvent. An upper bound on the number of resonances in disks, an estimate on the scattering determinant and the Lifshits-Krein trace formula then leads to a global Poisson wave trace formula for resonances of $\ham{D}$.
A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly supported potential is considered. For the corresponding eigenvalues $λ_n$, a complete asymptotic expansion for large $n$ is obtained, and the coefficients of this expansion are expressed in terms of the heat invariants. A sequence of trace formulas is obtained, expressing regularised sums of integer powers of eigenvalues $λ_n$ in terms of the heat invariants.
We derive explicit inequalities for sums of eigenvalues of one-dimensional Schrödinger operators on the whole line. In the case of the perturbed harmonic oscillator, these bounds converge to the corresponding trace formula in the limit as the number of eigenvalues covers the whole spectrum.
In this note, we study Lifshitz tails for a 2D Landau Hamiltonian perturbed by a random alloy-type potential constructed with single site potentials decaying at least at a Gaussian speed. We prove that, if the Landau level stays preserved as a band edge for the perturbed Hamiltonian, at the Landau levels, the integrated density of states has a Lifshitz behavior of the type $e^{-\log^2|E-2bq|}$.
We consider metric perturbations of the Landau Hamiltonian. We investigate the asymptotic behaviour of the discrete spectrum of the perturbed operator near the Landau levels, for perturbations with power-like decay, exponential decay or compact support.
We consider the Landau Hamiltonian perturbed by a long-range electric potential $V$. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we obtain an estimate of the rate of the shrinking of these clusters to the Landau levels as the number of the cluster $q$ tends to infinity. Further, we assume that there exists an appropriate $\V$, homogeneous of order $-ρ$ with $ρ\in (0,1)$, such that $V(x) = \V(x) + O(|x|^{-ρ- ε})$, $ε> 0$, as $|x| \to \infty$, and investigate the asymptotic distribution of the eigenvalues within a given clu…
In this note we consider a Landau Hamiltonian perturbed by a random magnetic potential of Anderson type. For a given number of bands, we prove the existence of both strongly localized states at the edges of the spectrum and dynamical delocalization near the center of the bands in the sense that wave packets travel at least at a given minimum speed. We provide explicit examples of magnetic perturbations that split the Landau levels into full intervals of spectrum.
We study the scattering problem for the Schrödinger equation on the half-line with Robin boundary condition at the origin. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to a representation for the perturbation determinant and to trace identities of Buslaev-Faddeev type.
We consider the Stark operator perturbed by a compactly supported potential (of a certain class) on the real line. We prove the following results: (a) upper and lower bounds on the number of resonances in complex discs with large radii, (b) the trace formula in terms of resonances only, (c) all resonances determine the potential uniquely.
We consider functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the trace norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_1}\le\const(\|A_1-A_2\|_{\bS_1}+\|B_1-B_2\|_{\bS_1})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in the operator norm.
The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian $A_α=(i \nabla + A)^2 + αδ$ in $L^2(R^2)$ with a $δ$-potential supported on a finite $C^{1,1}$-smooth curve $Σ$ are studied. Here $A = \frac{1}{2} B (-x_2, x_1)^\top$ is the vector potential, $B>0$ is the strength of the homogeneous magnetic field, and $α\in L^\infty(Σ)$ is a position-dependent real coefficient modeling the strength of the singular interaction on the curve $Σ$. After a general discussion of the qualitative spectral properties of $A_α$ and its resolvent, one of the main objectives in the present…
The paper discusses the spectrum of Toeplitz operators in Bargmann spaces. Our Toeplitz operators have real symbols with a variable sign and a compact support. A class of examples is considered where the asymptotics of the eigenvalues of such operators can be computed. These examples show that this asymptotics depends on the geometry of the supports of the positive and negative parts of the symbol. Applications to the perturbed Landau Hamiltonian are given.
We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({\mathbb R^2})$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $Λ_q$, $q \in {\mathbb Z}_+$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${\rm Op}^{\rm w}({\mathcal V})$ with real-valued Weyl symbol ${\mathcal V}$, such that ${\rm Op}^{\rm w}({\mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{\mathcal V} = H_0 + {\rm Op}^{\rm w}({\mathcal V})$. First, we construct symbols ${\mathcal V}…
For a Dirac-type operator D with a spectral boundary condition, the associated heat operator trace has an expansion in powers and log-powers of t. Some of the log-coefficients vanish in the Atiyah-Patodi-Singer product case. We here investigate the effect of perturbations of D, by use of a pseudodifferential parameter-dependent calculus for boundary problems. It is shown that the first k log-terms are stable under perturbations of D vanishing to order k at the boundary (and the nonlocal power coefficients behind them are only locally perturbed). For perturbations of D from the APS product case…
We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(\mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $Λ_q$, $q \in \mathbb{Z}_+$, and the perturbed operator $H_\upsilon = H_0 + \upsilonδ_Γ$, where $Γ\subset \mathbb{R}^2$ is a regular Jordan $C^{1,1}$-curve, and $\upsilon \in L^p(Γ;\mathbb{R})$, $p>1$, has a constant sign. We investigate ${\rm Ker}(H_\upsilon -Λ_q)$, $q \in \mathbb{Z}_+$, and show that generically $$0 \leq {\rm dim \, Ker}(H_\upsilon -Λ_q) - {\rm dim \, Ker}(T_q(\upsilon δ_Γ)) < \infty,$$ where $T_q(\upsilon δ_Γ) = p_q (\upsilon δ_Γ)p_q$, is…
We prove that if $f$ is a Lipschitz function on $\R$, $A$ and $B$ are self-adjoint operators such that ${\rm rank} (A-B)=1$, then $f(A)-f(B)$ belongs to the weak space $\boldsymbol{S}_{1,\be}$, i.e., $s_j(A-B)\le{\rm const} (1+j)^{-1}$. We deduce from this result that if $A-B$ belongs to the trace class $\boldsymbol{S}_1$ and $f$ is Lipschitz, then $f(A)-f(B)\in\boldsymbol{S}_Ω$, i.e., $\sum_{j=0}^ns_j(f(A)-f(B))\le\const\log(2+n)$. We also obtain more general results about the behavior of double operator integrals of the form $Q=\iint(f(x)-f(y))(x-y)^{-1}dE_1(x)TdE_2(y)$, where $E_1$ and $E_2…
We consider the 3D Schrödinger operator $H = H_0 + V$ where $H_0 = (-i\nabla - A)^2$, $A$ is a magnetic potential generating a constant magnetic field of strength $b>0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a meromorphic extension from the upper half-plane to an appropriate complex manifold ${\mathcal M}$, and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed Landau level $2bq$, $q \in {\mathbb N…
We consider the Schrödinger operator on a star shaped graph with $n$ edges joined at a single vertex. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to representations for the perturbation determinant and the spectral shift function, and to an analog of Levinson's formula.
