The Schroedinger equation on the half line is considered with a real-valued, integrable potential having a finite first moment. It is shown that the potential and the boundary conditions are uniquely determined by the data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This result extends the celebrated two-spectrum uniqueness theorem of Borg and Marchenko to the case where there is also a continuous spectrum
The aim of the paper is to derive spectral estimates on the eigenvalue moments of the magnetic Schroedinger operators defined on the two-dimensional disk with a radially symmetric magnetic field and radially symmetric electric potential.
This paper presents a thorough analysis of 1-dimensional Schroedinger operators whose potential is a linear combination of the Coulomb term 1/r and the centrifugal term 1/r^2. We allow both coupling constants to be complex. Using natural boundary conditions at 0, a two parameter holomorphic family of closed operators is introduced. We call them the Whittaker operators, since in the mathematical literature their eigenvalue equation is called the Whittaker equation. Spectral and scattering theory for Whittaker operators is studied. Whittaker operators appear in quantum mechanics as the radial pa…
We develop direct and inverse scattering theory for one-dimensional Schroedinger operators with steplike potentials which are asymptotically close to different finite-gap periodic potentials on different half-axes. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.
Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schroedinger operator and hence construct a Darboux transformation an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discusss the specialisation to the case of graph with trivial compact part, ie. diagonal matrix potential.
We investigate the eigenvalues of perturbed spherical Schrödinger operators under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show that the square roots of eigenvalues are given by the square roots of the unperturbed eigenvalues up to an decaying error depending on the behavior of $q(x)$ near $x=0$. Furthermore, we provide sets of spectral data which uniquely determine $q(x)$.
A one-dimensional generalized nonlinear Schroedinger equation is considered, and the corresponding inverse scattering problem is analyzed when the potential is compactly supported and depends on the wave function. The unique recovery of the potential is established from an appropriate set of scattering data.
In this paper we consider magnetic Schroedinger operators on the two-dimensional unit disk with a radially symmetric magnetic field which explodes to infinity at the boundary. We prove a bound for the eigenvalue moments and a bound for the number of negative eigenvalues for such operators.
We show that fixed energy scattering measurements for the magnetic Schroedinger operator uniquely determine the magnetic field and electric potential in dimensions $n \geq 3$. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This improves an earlier result of Eskin and Ralston which considered potentials with many derivatives. The proof is close to arguments in inverse boundary problems, and is based on constructing complex geometrical optics solutions to the Schroedinger equation via a pseudodifferential conjugation argument.
We relate the scattering theory of the focusing AKNS system with vanishing boundary conditions to that of the matrix Schroedinger equation. The corresponding Miura transformation which allows this connection, converts the focusing matrix nonlinear Schroedinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method to derive the multisoliton solutions of the nonlocal integrable equation, thus proposing a new method to solve the matrix NLS equation.
In this paper I prove a L^p-L^p' estimate for the solutions of the one-dimensional Schroedinger equation with a potential in L^1_gamma where in the generic case gamma > 3/2 and in the exceptional case (i.e. when there is a half-bound state of zero energy) gamma > 5/2. I use this estimate to construct the scattering operator for the nonlinear Schroedinger equation with a potential. I prove moreover, that the low-energy limit of the scattering operator uniquely determines the potential and the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonl…
This paper presents the spectral analysis of 1-dimensional Schroedinger operator on the half-line whose potential is a linear combination of the Coulomb term 1/r and the centrifugal term 1/r^2. The coupling constants are allowed to be complex, and all possible boundary conditions at 0 are considered. The resulting closed operators are organized in three holomorphic families. These operators are closely related to the Whittaker equation. Solutions of this equation are thoroughly studied in a large appendix to this paper. Various special cases of this equation are analyzed, namely the degenerate…
We study the heat semigroup generated by two-dimensional Schroedinger operators with compactly supported magnetic field. We show that if the field is radial, then the large time behavior of the associated heat kernel is determined by its total flux. We also establish some on-diagonal heat kernel estimates and discuss their applications for solutions to the heat equation. An exact formula for the heat kernel, and for its large time asymptotic, is derived in the case of the Aharonov-Bohm magnetic field.
We consider radial tree extensions of one-dimensional quasi-periodic Schroedinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The sufficiency criterion for that is the existence of Bloch-Floquet states for the one dimensional operator corresponding to the radial problem.
We show, in general, how to transform the nonautonomous nonlinear Schroedinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear science. Derivation of the corresponding equivalent nonisospectral Lax pair is outlined. A few simple integrable systems are discussed.
In this work, we study inverse resonance problems for the Schrödinger operator on the real line with the potential supported in $[0,1]$. In general, all eigenvalues and resonances can not uniquely determine the potential. (i) It is shown that if the potential is known a priori on $[0,1/2]$, then the unique recovery of the potential on the whole interval from all eigenvalues and resonances is valid. (ii) If the potential is known a priori on $[0,a]$, then for the case $a>1/2$, infinitely many eigenvalues and resonances can be missing for the unique determination of the potential, and for the ca…
We give a short description of the proof of asymptotic-completeness for NLS-type equations, including time dependent potential terms, with radial data in three dimensions. We also show how the method applies for the two-body Quantum Scattering case.
The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a second moment, it is shown that the corresponding scattering matrix is differentiable at zero energy. An explicit formula is provided for the derivative of the scattering matrix at zero energy. The previously established results when the potential has only the first moment are improved when the second moment exists, by presenting the small-energy asymptotics for the related Jost …
The paper is devoted to operators given formally by the expression \begin{equation*} -\partial_x^2+\big(α-\frac14\big)x^{-2}. \end{equation*} This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real $α$, or closed operator for complex $α$, we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on $L^2({\mathbb R}_+)$, which we denote $H_{m,κ}$ and $H_0^ν$, with $m^2=α$, $-1<\Re(m)<1$, and where $κ,ν\in{\mathbb C}\cup\{\infty\}$ specify the boundary condition at $0$. …
The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a first moment, it is shown that the corresponding scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the corresponding Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.
We introduce Laplace transformations of 2D semi-discrete hyperbolic Schroedinger operators and show their relation to a semi-discrete 2D Toda lattice. We develop the algebro-geometric spectral theory of 2D semi-discrete hyperbolic Schroedinger operators and solve the direct spectral problem for 2D discrete ones (the inverse problem for discrete operators was already solved by Krichever). Using the spectral theory we investigate spectral properties of the Laplace transformations of these operators. This makes it possible to find solutions of the semi-discrete and discrete 2D Toda lattices in te…
The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin. When the matrix potential is integrable, the high-energy asymptotics are established for the related Jost matrix, the inverse of the Jost matrix, and the scattering matrix. Under the additional assumption that the matrix potential has a first moment, Levinson's theorem is derived, relating the number of bound states to the change in the argument of the determinant of the scattering matrix.
We describe inverse scattering for the matrix Schroedinger operator with general selfadjoint boundary conditions at the origin using the Marchenko equation. Our approach allows the recovery of the potential as well as the boundary conditions. It is easily specialised to inverse scattering on star-shaped graphs with boundary conditions at the node.
For non-relativistic Schroedinger equations the lowering of their degree by substitution Psi(r) \to F(r) =Psi'(r) /Psi(r) is known to facilitate our understanding and use of their (incomplete, so called quasi-exact) solvability. We show that and how the radial Dirac relativistic equation may quasi-exactly be solved in similar spirit.
Quantum non-linear SCHROEDINGER equation is equivalent to Lieb-Liniger model. It has non-trivial conservation laws. Recently these conservation laws were used for evaluation of the three-body recombination rate for interacting gas of quantum bosons. These conservations laws were known already in 1989. Submitted text is retyping of the preprint of Centre for Mathematical Analysis of Australian National University CMA-R33-89. It was discussed in Leningrad Branch of the V.A. Steklov Mathematical Institute at that time. A copy of the original preprint can be found in the section Quantum Inverse Sc…
In this review paper we carry on our investigations on Schroedinger operators with inverse square potentials on the half-line. Depending on several parameters, such operators possess either a finite number of complex eigenvalues, or an infinite one, but also some spectral singularities embedded in the continuous spectrum (exceptional situations). The spectral and the scattering theory for these operators is recalled, and new results for the exceptional cases are provided. Some index theorems in scattering theory are also developed, and explanations why these results can not be extended to the …
New inverse and half-inverse problems: {\it sliding problems} are introduced. In this way several physically important equations are recovered from the quantum defect. In particular, sliding problems are solved for radial Schrödinger equation, radial Dirac system and multidimensional Schrödinger equation. Systems with Coulomb-type potentials are considered as well.
This work studies the direct and inverse fixed energy scattering problem for two-dimensional Schroedinger equation with rather general nonlinear index of refraction. In particular, using the Born approximation we prove that all singularities of the unknown compactly supported potential from $L^2$-space can be obtained uniquely by the scattering data with fixed positive energy. The proof is based on the new estimates for the Faddeev-Green's function in $L^\infty$-space.
Based on the Caputo fractional derivative the classical, non relativistic Hamiltonian is quantized leading to a fractional Schroedinger type wave equation. The free particle solutions are localized in space. Solutions for the infinite well potential and the radial symmetric ground state solution are presented. It is shown, that the behaviour of these functions may be reproduced with a ordinary Schroeodinger equation with an additional potential, which is of the form V ~ x for $α<1$, corresponding to the confinement potential, introduced phenomenologically to the standard models for non relativ…
For Schroedinger operators with long-range magnetic vector potentials and short-range electric scalar potentials in an exterior domain $Ω$ in ${\bf R}^n$ with $n \geq 2$, we show that there is a one-to-one correspondence between the gauge equivalent classes of Hamiltonians and those of S-matrices if $Ω$ is exterior to a bounded convex obstacle.
The subject of this work are random Schroedinger operators on regular rooted tree graphs $\T$ with stochastically homogeneous disorder. The operators are of the form $H_λ(ω) = T + U + λV(ω)$ acting in $\ell^2(\T)$, with $T $ the adjacency matrix, $U$ a radially periodic potential, and $V(ω)$ a random potential. This includes the only class of homogeneously random operators for which it was proven that the spectrum of $H_λ(ω)$ exhibits an absolutely continuous (ac) component; a results established by A. Klein for weak disorder, in case U=0 and $V(ω)$ given by iid random variables on $\T$. Our m…
Enss (1983) proved a propagation estimate for the usual free Schroedinger operator that turned out later to be very useful for inverse scattering in the work of Enss--Weder (1995). Since then, this method has been called the Enss--Weder time-dependent method. We study the same type of propagation estimate for the fractional powers of the negative Laplacian and, as with the Enss--Weder method, we apply our estimate to inverse scattering. We find that the high-velocity limit of the scattering operator uniquely determines the short-range interactions.
In the quantum frame, for 3-dimensional space, in the two body problem case, we approach the Schrödinger equation (SE) taking in account the potential: Vq(r)=Dr^2+(A/r)+(B/r^2) called by us quasi-harmonic potential with the centrifugal type term, with D,A,B >0 and D<<1. We use Laplace transform method (LTM) and we find for the first time an analytic solution of the Vq-potential problem. Namely, using directly and inverse Laplace transformations, we obtain the complete forms of the energy eigenvalues and wave functions. Furthermore, for this potential Vq, we make considerations about critical o…
This paper is the continuation of our work with Victor Guillemin; Victor and I proved that the Taylor expansion of the potential at a generic non degenerate critical point is determined by the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value. Here, I show that, under some genericity assumptions, the potential of the 1D Schroedinger operator is determined by its semi-classical spectrum. Moreover, there is an explicit reconstruction. This paper is strongly related to a paper of David Gurarie (J. Math. Phys. 36:1934--1944 (1995)).
We consider the Aharonov-Bohm effect for the Schroedinger operator and the related inverse problem in an exterior domain the 2-dim. Euclidean space with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain and the Hamiltonian up to Gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle is convex.
We show that there is a family Schroedinger operators with scaled potentials which approximates the $δ'$-interaction Hamiltonian in the norm-resolvent sense. This approximation, based on a formal scheme proposed by Cheon and Shigehara, has nontrivial convergence properties which are in several respects opposite to those of the Klauder phenomenon.
For a class of singular Sturm-Liouville equations on the unit interval with explicit singularity $a(a + 1)/x^2, a \in \mathbb{N}$, we consider an inverse spectral problem. Our goal is the global parametrization of potentials by spectral data noted by $λ^a$, and some norming constants noted by $κ^a$. For $a = 0$ and $a=1$, $λ^a\times κ^a$ was already known to be a global coordinate system on $\lr$. With the help of transformation operators, we extend this result to any non-negative integer $a$ and give a description of isospectral sets.
We consider the inverse scattering problem at fixed and sufficiently large energy for the nonrelativistic and relativistic Newton equation in $\R^n$, $n \ge 2$, with a smooth and short range electromagnetic field $(V,B)$. Using results of [Firsov, 1953] or [Keller-Kay-Shmoys, 1956] we obtain a uniqueness result when $B$ is assumed to be zero in a neighborhood of infinity and $V$ is assumed to be spherically symmetric in a neighborhood of infinity.
For a competitive system of k coupled nonlinear Schroedinger equations we prove the existence, when the competition parameter is large, of positive radial solutions on R^N. We show that, when the competition parameter goes to infinity, the profile of each component separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing-sign solutions of the scalar nonlinear Schroedinger equation. Within an Hartree-Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the…
In present article the self-contained derivation of eigenvalue inverse problem results is given by using a discrete approximation of the Schroedinger operator on a bounded interval as a finite three-diagonal symmetric Jacobi matrix. This derivation is more correct in comparison with previous works which used only single-diagonal matrix. It is demonstrated that inverse problem procedure is nothing else than well known Gram-Schmidt orthonormalization in Euclidean space for special vectors numbered by the space coordinate index. All the results of usual inverse problem with continuous coordinate …
It is shown that the radial Schroedinger equation for a power law potential and a particular angular momentum may be transformed using a change of variable into another Schroedinger equation for a different power law potential and a different angular momentum. It is shown that this leads to a mapping of the spectra of the two related power law potentials. It is shown that a similar correspondence between the classical orbits in the two related power law potentials exists. The well known correspondence of the Coulomb and oscillator spectra is a special case of a more general correspondence betw…
We study an inverse scattering problem at fixed energy for radial magnetic Schr{ö}dinger operators on R^2 \ B(0, r\_0), where r\_0 is a positive and arbitrarily small radius. We assume that the magnetic potential A satisfies a gauge condition and we consider the class C of smooth, radial and compactly supported electric potentials and magnetic fields denoted by V and B respectively. If (V, B) and (\tilde{V} , \tilde{B}) are two couples belonging to C, we then show that if the corresponding phase shifts $δ$\_l and \tilde{$δ$}\_l (i.e. the scattering data at fixed energy) coincide for all l $\in…
Inspired by F. Wilczek's QCD Lite, quantum Yang-Mills-Weyl Dynamics (YMWD) describes quantum interaction between gauge bosons (associated with a simple compact gauge Lie group $\mathbb{G}$) and larks (massless chiral fields colored by an irreducible unitary representation of $\mathbb{G}$). Schroedinger representation of this quantum Yang-Mills-Weyl theory is based on a sesqui-holomorphic operator calculus of infinite-dimensional operators with variational derivatives. The spectrum of the quantum YMWD, with initial data in the central euclidean ball of a radius $0<R<+\infty$, is self-similar in…
The Schroedinger equation is considered on the line when the potential is real valued, compactly supported, and square integrable. The nonuniqueness is analyzed in the recovery of such a potential from the data consisting of the ratio of a corresponding reflection coefficient to the transmission coefficient. It is shown that there are a discrete number of potentials corresponding to the data and that their L^2-norms are related to each other in a simple manner. All those potentials are identified, and it is shown how an additional estimate on the L^2-norm in the data can uniquely identify the …
In the Euclidean space of any dimension d, we consider the heat semigroup generated by the magnetic Schroedinger operator from which an inverse-square potential is subtracted in order to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behaviour of the heat semigroup is determined by the eigenvalue problem for a magnetic Schroedinger operator on the (d-1)-dimensional sphere whose vector potential reflects the behaviour of the magnetic field at the space infinity. From the spectral problem on th…
We provide a new natural interpretation of the Lax representation for an integrable system; that is, the spectral problem is the linearized form of a Miura transformation between the original system and a modified version of it. On the basis of this interpretation, we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by Miura transformations. Thus, this method can be used to generate new integrable systems from known systems through inverse Miura maps; it can be applied to both continuous and discrete systems in 1+1 dimensi…
We transform an inverse scattering problem to be an interior transmission problem. We find an inverse uniqueness on the scatterer with a knowledge of a fixed interior transmission eigenvalue. By examining the solution in a series of spherical harmonics at far fields, we can decide the perturbation uniquely for the radially symmetric perturbations.
We consider the nonlinear Schroedinger equation on the one dimensional torus, with a defocousing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measure with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant for times of order $β^{2+ς}$, $β$ being the inverse of the temperature and $ς$ a positive number (we prove $ς= 1/10$). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory.
For Schroedinger equations with both time-independent and time-dependent Kato potentials, we give a simple proof of the maximal speed bound. The latter states that the probability to find the quantum system outside the ball of radius proportional to the time lapsed decays as an inverse power of time. We give an explicit expression for the constant of proportionality in terms of the maximal energy available to the initial condition. For the time-independent part of the interaction, we require neither decay at infinity nor smoothness.
We study inverse scattering problems at a fixed energy for radial Schrödinger operators on $\R^n$, $n \geq 2$. First, we consider the class $\mathcal{A}$ of potentials $q(r)$ which can be extended analytically in $\Re z \geq 0$ such that $\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-ρ}$, $ρ\textgreater{} \frac{3}{2}$. If $q$ and $\tilde{q}$ are two such potentials and if the corresponding phase shifts $δ\_l$ and $\tildeδ\_l$ are super-exponentially close, then $q=\tilde{q}$. Secondly, we study the class of potentials $q(r)$ which can be split into $q(r)=q\_1(r) + q\_2(r)$ such that $q\_1(r)$ has…
A new type of supersymmetric transformations of the coupled-channel radial Schroedinger equation is introduced, which do not conserve the vanishing behavior of solutions at the origin. Contrary to usual transformations, these ``non-conservative'' transformations allow, in the presence of thresholds, the construction of potentials with coupled scattering matrices from uncoupled potentials. As an example, an exactly-solvable potential matrix is obtained which provides a very simple model of Feshbach-resonance phenomenon.
The asymptotic iteration method (AIM) is applied to obtain highly accurate eigenvalues of the radial Schroedinger equation with the singular potential V(r)=r^2+λ/r^α(α,λ> 0) in arbitrary dimensions. Certain fundamental conditions for the application of AIM, such as a suitable asymptotic form for the wave function, and the termination condition for the iteration process, are discussed. Several suggestions are introduced to improve the rate of convergence and to stabilize the computation. AIM offers a simple, accurate, and efficient method for the treatment of singular potentials such as V(r) va…
The Self-Adjoint Extension in the Schrodinger equation for potentials behaved as an attractive inverse square at the origin is critically reviewed. Original results are also presented. It is shown that the additional non-regular solutions must be retained for definite interval of parameters, which requires a necessity of performing a Self-Adjoint Extension (SAE) procedure of radial Hamiltonian.The Pragmatic approach is used and some of its consequences are considered for wide class of transitive potentials. Our consideration is based on the established earlier by us a boundary condition for th…
