In this paper, we study spectral properties of a family of quasi-periodic Schrodinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curves are extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent, and show that the spectrum is purely singular.
In this paper we study spectral properties of a family of quasi-periodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp in \cite{FEKL1}.
In this paper, we consider one dimensional adiabatic quasi-periodic Schrödinger operators in the regime of strong resonant tunneling. We show the emergence of a level repulsion phenomenon which is seen to be very naturally related to the local spectral type of the operator: the more singular the spectrum, the weaker the repulsion.
We construct almost invariant subspaces and the corresponding effective Hamiltonian for magnetic Bloch bands. We also discuss the question of the dynamics related to the effective Hamiltonian. We assume that the magnetic and electric potentials are slowly varying perturbations of the potential of a constant magnetic field and a periodic lattice potential, respectively.
In this paper, we study spectral properties of the one dimensional periodic Schrodinger operator with an adiabatic quasi-periodic perturbation. We show that in certain energy regions the perturbation leads to resonance effects related to the ones observed in the problem of two resonating quantum wells. These effects affect both the geometry and the nature of the spectrum. In particular, they can lead to the intertwining of sequences of intervals containing absolutely continuous spectrum and intervals containing singular spectrum. Moreover, in regions where all of the spectrum is expected to be…
We consider Schrödinger operators $H=-Δ_{g_\varepsilon} + V$ on a fibre bundle $M\stackrelπ{\to}B$ with compact fibres and a metric $g_\varepsilon$ that blows up directions perpendicular to the fibres by a factor ${\varepsilon^{-1}\gg 1}$. We show that for an eigenvalue $λ$ of the fibre-wise part of $H$, satisfying a local gap condition, and every $N\in \mathbb{N}$ there exists a subspace of $L^2(M)$ that is invariant under $H$ up to errors of order $\varepsilon^{N+1}$. The dynamical and spectral features of $H$ on this subspace can be described by an effective operator on the fibre-wise $λ$-e…
We study a one-dimensional non-stationary Schrödinger equation with a potential slowly depending on time. The corresponding stationary operator depends on time as on a parameter. It has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. The eigenvalues move with time to the edge of the continuous spectrum and, having reached it, disappear one after another. We study the asymptotic behavior of a solution close at some moment to an eigenfunction of the stationary operator, and, in particular, the phenomena occurring when the corresponding ei…
Singular Gordon potentials are defined to be distributions from the space W^{-1}_{2,unif}(R) that are sufficiently fast approximated by periodic ones. We prove that Schrödinger operators with singular Gordon potentials have no point spectrum and show that a rich class of quasiperiodic distributions consists of singular Gordon potentials.
We construct examples of potentials $V(x)$ satisfying $|V(x)| \leq \frac{h(x)}{1+x},$ where the function $h(x)$ is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen "twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of singular continuous spectrum.
We show that the spectral measure of discrete Schrödinger operators $ (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n)$ does not have singular continuous component if the potential $V(n)=O(n^{-1})$.
Boundedness of wave operators for Schrödinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive estimates and commutator bounds.
We show that formal Schrödinger operators with singular potentials from the space W^{-1}_{2,unif}(R) can be naturally defined to give selfadjoint and bounded below operators, which depend continuously in the uniform resolvent sense on the potential in the W^{-1}_{2,unif}(R)-norm. In the case of periodic singular potentials we also establish pure absolute continuity and a band and gap structure of the spectrum thus generalising some classical results for singular potentials of one-dimensional quasicrystal theory.
We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on $[a,\infty)$, $a\in\mathbb{R}$, with a regular finite end point $a$ and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and $2\times 2$ matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrödinger operators on $(a,\infty)$ with a potential strongly singular at the end point $a$.…
We consider a family $\{\mathcal{H}^\varepsilon\}_{\varepsilon>0}$ of $\varepsilon\mathbb{Z}^n$-periodic Schrödinger operators with $δ'$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $m\in\mathbb{N}$ surfaces. We show that in the limit when $\varepsilon\to 0$ and the interactions strengths are appropriately scaled, $\mathcal{H}^\varepsilon$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions s…
We established a generalized version of the Christ-Kiselev's multi-linear operator technique to deal with the spectral theory of Schrödinger operators. As applications, several spectral results of perturbed periodic Schrödinger operators are obtained, including WKB solutions, sharp transitions of preservation of absolutely continuous spectra, criteria of absence of singular spectra and sharp bounds of the Hausdorff dimension of singular spectra.
Let $Ω\subset \mathbb{R}^d$ be bounded with $C^1$ boundary. In this paper we consider Schrödinger operators $-Δ+ W$ on $Ω$ with $W(x)\approx\mathrm{dist}(x, \partialΩ)^{-2}$ as $\mathrm{dist}(x, \partialΩ)\to 0$. Under weak assumptions on $W$ we derive a two-term asymptotic formula for the sum of the eigenvalues of such operators.
We say that a discrete set $X =\{x_n\}_{n\in\dN_0}$ on the half-line $$0=x_0 < x_1 <x_2 <x_3<... <x_n<... <+\infty$$ is sparse if the distances $Δx_n = x_{n+1} -x_n$ between neighbouring points satisfy the condition $\frac{Δx_{n}}{Δx_{n-1}} \rightarrow +\infty$. In this paper half-line Schrödinger operators with point $δ$- and $δ^\prime$-interactions on a sparse set are considered. Assuming that strengths of point interactions tend to $\infty$ we give simple sufficient conditions for such Schrödinger operators to have non-empty singular continuous spectrum and to have purely singular continuou…
We prove a quantitative unique continuation principle for Schrödinger operators $H=-Δ+V$ on $\mathrm{L}^2(Ω)$, where $Ω$ is an open subset of $\mathbb{R}^d$ and $V$ is a singular potential: $V \in \mathrm{L}^\infty(Ω) + \mathrm{L}^p(Ω)$. As an application, we derive a unique continuation principle for spectral projections of Schrödinger operators with singular potentials.
We study the time evolution of a density matrix in a quantum mechanical system described by an ergodic magnetic Schrödinger operator with singular magnetic and electric potentials, the electric field being introduced adiabatically. We construct a unitary propagator that solves weakly the corresponding time-dependent Schrödinger equation, and solve a Liouville equation in an appropriate Hilbert space.
This paper summarizes the contents of a plenary talk at the Pan African Congress of Mathematics held in Rabat in July 2017. We provide a survey of recent results on spectral properties of Schrödinger operators with singular interactions supported by manifolds of codimension one and of Robin billiards with the focus on the geometrically induced discrete spectrum and its asymptotic expansions in term of the model parameters.
We discuss the spectral properties of singular Schrödinger operators in three dimensions with the interaction supported by an equilateral star, finite or infinite. In the finite case the discrete spectrum is nonempty if the star arms are long enough. Our main result concerns spectral optimization: we show that the principal eigenvalue is uniquely maxi\-mized when the arms are arranged in one of the known five sharp configurations known as solutions of the closely related Thomson problem.
The paper deals with singular Schrödinger operators of the form \begin{gather*} -{\mathrm{d}^2\over \mathrm{d} x^2 } + \sum_{k\in\mathbb{Z} }γ_k δ(\cdot-z_k),\quad γ_k\in\mathbb{R}, \end{gather*} in $\mathsf{L}^2(\ell_-,\ell_+)$, where $(\ell_-,\ell_+)$ is a bounded interval, and $ δ(\cdot-z_k)$ is the Dirac delta-function supported at $z_k\in (\ell_-,\ell_+)$. It will be shown that the interaction strengths $γ_k$ and the points $z_k$ can be chosen in such a way that the essential spectrum and a bounded part of the discrete spectrum of this self-adjoint operator coincide with prescribed sets o…
Norm resolvent approximation for a wide class of point interactions in one dimension is constructed. To analyse the limit behaviour of Schrödinger operators with localized singular rank-two perturbations coupled with δ-like potentials as the support of perturbation shrinks to a point, we show that the set of limit operators is quite rich. Depending on parameters of the perturbation, the limit operators are described by both the connected and separated boundary conditions. In particular an approximation for a four-parametric subfamily of all the connected point interactions is built. We give ex…
For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as εgoes to 0, of a family S_εof one-dimensional Schrödinger operators on the line of the form S_ε:= -D^2 + ε^{-2} V(x/ε). Under certain conditions the family of potentials converges in the sense of distributions to the first derivative of the Dirac delta-function, and then the limit of S_εmight be considered as a "physically motivated" interpretation of the one-dimensional Schrödinger operator with potential δ'.
Simon's subshift conjecture states that for every aperiodic minimal subshift of Verblunsky coefficients, the common essential support of the associated measures has zero Lebesgue measure. We disprove this conjecture in this paper, both in the form stated and in the analogous formulation of it for discrete Schrödinger operators. In addition we prove a weak version of the conjecture in the Schrödinger setting. Namely, under some additional assumptions on the subshift, we show that the density of states measure, a natural measure associated with the operator family and whose topological support i…
We consider the Schrödinger operators $H_V=-Δ_g+V$ with singular potentials $V$ on general $n$-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this pertubation. We prove that the pointwise Weyl law holds for potentials in the Kato class, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. Moreover, we show that the pointwise Weyl law with the standard sharp error term $O(λ^{n-1})$ holds for potentials in $L^n(M)$.
We describe two-dimensional potential Schrodinger and Dirac operators which are finite-gap at one energy level and have singular spectral curves. It appears that the singularities can be rather complicated. Such Dirac operators appear as the spectral curves of tori immersed into the three-space.
Using an extension of the Hörmander product of distributions, we obtain an intrinsic formulation of one-dimensional Schrödinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz} distributions and does not require (as some previous approaches) the use of more general distributions or generalized functions. We determine, in the new formulation, the action and domain of the Schrödinger operators with arbitrary singular boundary potentials. We also consider the inverse problem, and obtain a general procedure for constructing the singular (…
It is proven that the absolutely continuous spectrum of matrix Schrödinger operators coincides (with the multiplicity taken into account) with the spectrum of the unperturbed operator if the (matrix) potential is square integrable. The same result is also proven for some classes of slower decaying potentials if they are smooth.
We consider Schrödinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a finite curve $Γ$. We present a proper definition of the operators and study their properties, in particular, we show that the discrete spectrum can be empty if $Γ$ is short enough. If it is not the case, we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length $2ε$. We derive an asymptotic expansion with the leading term which a multiple of $ε\lnε$.
By careful exploration of separation of variables into the Laplacian in spherical coordinates, we obtain the extra delta-like singularity, elimination of which restricts the radial wave function at the origin. This constraint has the form of boundary condition for the radial Schrodinger equation.
Physical realizations of non-standard singular self-adjoint extensions for one-dimensional Schrödinger operator in terms of the mass-jump are considered. It is shown that corresponding boundary conditions can be realized for the Hamiltonian with the position-dependent effective mass in two qualitatively different profiles of the effective mass inhomogeneity: the mass-jump and the mass-bump. The existence of quantized magnetic flux in a case of the mass-jump is proven by explicit demonstration of the Zeeman-like splitting for states with the opposite projections of angular momentum.
We consider Schrödinger operators with ergodic potential $V_ω(n)=f(T^n(ω))$, $n \in \Z$, $ω\in Ω$, where $T:Ω\to Ω$ is a non-periodic homeomorphism. We show that for generic $f \in C(Ω)$, the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani Theory.
We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space $\mathbb{R}^3$, in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighbourhood…
We consider singular self-adjoint extensions for one-dimensional Schrödinger operator for two-component wave function within the framework of the distribution theory for discontinuous test functions \cite{funcan_deltadistr_kurasov_jmathan1996}. We show that among $\mathds{C}^{4}$-parameter set of boundary conditions with state mixing there is only $\mathds{R}^2$-parameter subset compatible with the spin interpretation of the two-component structure of the wave function. For the spin interpretation of such wave function they can be identified as the point-like spin-momentum (Rashba) interaction…
In this paper, I consider one-dimensional periodic Schr{ö}dinger operators perturbed by a slowly decaying potential. In the adiabatic limit, I give an asymptotic expansion of the eigenvalues in the gaps of the periodic operator. When one slides the perturbation along the periodic potential, these eigenvalues oscillate. I compute the exponentially small amplitude of the oscillations.
Consider a five-point discretization of a two-dimensional finite-gap for a fixed energy Schrödinger operator. We construct the Green's function of the operator. In appears as the explicit formula in terms of the integral by the specific contour on the spectral curve of the differential which is constructed by the wave function and its double. The formula is parametrized by the point on the spectral curve and for almost every parameter value it gives the Green's function with known growth at infinity.
We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families o…
We investigate a class of generalized Schrödinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a smooth curve $Γ$. We find a strong-coupling asymptotic expansion of the discrete spectrum in case when $Γ$ is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schrödinger operator with a potential determined by the curvature of $Γ$. In the same way we obtain an asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if $Γ$ is not a strai…
We consider singular self-adjoint extensions for the Schrödinger operator of spin-$1/2$ particle in one dimension. The corresponding boundary conditions at a singular point are obtained. There are boundary conditions with the spin-flip mechanism, i.e. for these point-like interactions the spin operator does not commute with the Hamiltonian. One of these extensions is the analog of zero-range $δ$-potential. The other one is the analog of so called $δ^{(1)}$-interaction. We show that in physical terms such contact interactions can be identified as the point-like analogues of Rashba Hamiltonian (…
