For the one dimensional Schrödinger operator in the case of Dirichlet boundary condition, we show that $β_{cr}$ is positive and zero for the case of Neumann and Robin boundary condition considering the potential energy of the form $V(x)=-βδ(x-a)$ where, $β\geq 0, \ a > 0.$ We prove that the $β_{cr}$ goes to infinity when the delta potential moves towards the boundary in dimension one with Dirichlet boundary condition. We also show that the $β_{cr}>0$ and $β\in (0,\frac{1}{2})$ considering Dirichlet problem with delta potential on the circle in dimension two.
The spectral series of the Schrödinger operator with a delta-potential on a three-dimensional compact spherically symmetric manifold in the semiclassical limit as $h\to0$ are described.
In this note we sharpen the lower bound from [LLP10] on the spectrum of the 2D Schroedinger operator with a delta-interaction supported on a planar angle. Using the same method we obtain the lower bound on the spectrum of the 2D Schroedinger operator with a delta-interaction supported on crossing straight lines. The latter operators arise in the three-body quantum problem with delta-interactions between particles.
We study the level statistics for two classes of 1-dimensional random Schrödinger operators : (1) for operators whose coupling constants decay as the system size becomes large, and (2) for operators with critically decaying random potential. As a byproduct of (2) with our previous result \cite{KN} imply the coincidence of the limits of circular and Gaussian beta ensembles.
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.
Careful exploration of the idea that equation for radial wave function must be compatible with the full Schrodinger equation shows appearance of the delta-function while reduction of full Schrodinger equation in spherical coordinates. Elimination of this extra term produces a boundary condition for the radial wave function, which is the same both for regular and singular potentials.
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations in their spectral parameter. The relation to t…
We obtain the extra delta-like singularity while reduction of the Laplace operator in spherical coordinates, elimination of which restricts the radial wave functions at the origin. This restriction has the form of boundary condition for the radial wave function.
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.
In this paper we consider the time dependent one-dimensional Schrödinger equation with multiple Dirac delta potentials {of different strengths}. We prove that the classical dispersion property holds under some restrictions on the strengths and on the lengths of the finite intervals. The result is obtained in a more general setting of a Laplace operator on a tree with $δ$-coupling conditions at the vertices. The proof relies on a careful analysis of the properties of the resolvent of the associated Hamiltonian. With respect to the analysis done in \cite{MR2858075} for Kirchhoff conditions, here…
Self-adjoint Schrödinger operators with $δ$ and $δ'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman--Schwinger principle and a variant of Krein's formula are shown. Furthermore, Schatten--von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with $δ$ and $δ'$-potentials, and the Schrödinger operator without a singular interaction are…
We consider the Schrödinger operator in the plane with delta-potential supported by a curve. For the cases of an infinite curve and a finite loop we give estimates on the lower bound of the spectrum expressed explicitly through the strength of the interaction and a parameter which characterizes geometry of the curve. Going further we cut the curve into finite number of pieces and estimate the bottom of the spectrum using the parameters for the pieces. As an application of the elaborated theory we consider a curve with a finite number of cusps and "leaky" quantum graphs.
We consider a single band approximation to the random Schroedinger operator in an external magnetic field. The spectrum of such an operator has been characterized in the case where delta impurities are located on the sites of a lattice. In this paper we generalize these results by letting the delta impurites have random positions as well as strengths; they are located in squares of a lattice with a general bounded distribution. We characterize the entire spectrum of this operator when the magnetic field is sufficiently high. We show that the whole spectrum is pure point, the energy coinciding …
We establish the exact solution of the nonlinear Schrodinger equation with a delta-function impurity, representing a point-like defect which reflects and transmits. We solve the problem both at the classical and the second quantized levels. In the quantum case the Zamolodchikov-Faddeev algebra, familiar from the case without impurities, is substituted by the recently discovered reflection-transmission (RT) algebra, which captures both particle-particle and particle-impurity interactions. The off-shell quantum solution is expressed in terms of the generators of the RT algebra and the exact scat…
We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrödinger operators can approximate non-trivial vertex couplings. The latter include not only the delta-couplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric delta'-couplings and conjecture that the same method can be applied to all couplings invariant with respect to the time reversal.
For Jacobi matrices with a_n = 1+(-1)^n alpha n^{-gamma}, b_n = (-1)^n beta n^{-gamma}, we study bound states and the SzegHo condition. We provide a new proof of Nevai's result that if gamma > 1/2, the Szego condition holds, which works also if one replaces (-1)^n by cos(mu n). We show that if alpha = 0, beta not equal to 0, and gamma < 1/2, the Szego condition fails. We also show that if gamma = 1, alpha and beta are small enough (beta^2 + 8 alpha^2 < 1/24 will do), then the Jacobi matrix has finitely many bound states (for alpha = 0, beta large, it has infinitely many).
In the framework of the Moutard transformation formalism we find multi-point delta-type potentials of two-dimensional Schrodinger operators and their isospectral deformations on the zero energy level. In particular, these potentials are "reflectionless" in the sense of the Faddeev generalized "scattering" data.
The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schrödinger operators with a $δ$-potential. These operators arise by replacing the discrete Laplacian by a strictly increasing $C^1$-function of the discrete Laplacian. The dependence of the results on this function and the lattice dimension are explicitly derived. It is found that while in the case of the discrete Schrödinger operator these behaviours are the same no matter which end of the continuous spectrum is considered, an asymmetry occur…
Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d>2 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. We show for generic potentials V that the number of resonances of -Delta+V with modulus less than r is approximatively equal to a constant times a^dr^d.
We study a system of N bosons in the plane interacting with delta function potentials. After a coupling constant renormalization we show that the Hamiltonian defines a self-adjoint operator and obtain a lower bound for the energy. The same results hold if one includes a regular inter-particle potential.
The Energy Eigenvalue for the Singular Wave Function of the Three Dimensional Dirac Delta Schrodinger Potential via Distributionally Generalized Quantum Mechanics
Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d odd larger than 2. Here V is a bounded real- or complex-valued function vanishing outside the closed ball of center 0 and radius a. If V belongs to the class of potentials introduced by Christiansen, we show that when r goes to infinity, the resonances of -Delta+V, scaled down by the factor r, are asymptotically distributed, with respect to an explicit probability distribution on the closed lower unit half-disc of the complex plane. The rate of convergence is also considered for subclasses of potentials.
The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich - Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of th…
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.
Exact solutions of two-particle relativistic equations of quantum field theory describing the scattering $s$-states and the bound $s$-states are found in the cases of delta-shell potential and superposition of delta-shell potentials. Some properties of obtained relativistic wave functions, scattering amplitudes and quantization conditions are investigated. The resonance character of scattering processes is demonstrated by the behavior of amplitudes. It is shown that the non-relativistic limits of these relativistic values coincide with respective non-relativistic ones obtained from the Schrödi…
The Exact Point Spectrum and Eigenvector of the Unique Continuous L$^2(\mathbb{R}^2)$ Bound State Solution to the Dirac Delta Schrodinger Potential in Two Dimensions
We use relative zeta functions technique of W. Muller \cite{Mul} to extend the classical decomposition of the zeta regularized partition function of a finite temperature quantum field theory on a ultrastatic space-time with compact spatial section to the case of non compact spatial section. As an application, we study the case of Schrödinger operators with delta like potential, as described by Albeverio & alt. in \cite{AGHH}.
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger's equation. This paper, in contrast, investigates the integral form of Schrödinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger's diffe…
In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and the initial condition u_0 satisfies: u_0 in L^1(R^n), integral_{R^n} u_0(x) dx = 0, and |e^{t Delta}u_0|_{L^1(R^n)} <= Ct^{-beta/2} for fixed beta in (0,1), all t>0, and a constant C, we show that the L^1-norm of the solution to the convection-diffusion equation decays with the rate t^{-beta/2} as t --> infinity. Moreover, we prove that, for small initial co…
The Schrödinger equation with attractive delta potential has been previously studied in the supersymmetric quantum mechanical approach by a number of authors, but they all used only the particular superpotential solution. Here, we introduce a one-parameter family of strictly isospectral attractive delta function potentials, which is based on the general superpotential (general Riccati) solution, we study the problem in some detail and suggest possible applications
The paper discusses nonlinear singular perturbations of delta type of the fractional Schrödinger equation $\imath\partial_tψ=\left(-\triangle\right)^sψ$, with $s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local and global well posedness (in a strong sense), conservations laws and existence of blow-up solutions and standing waves.
The evolution problem for a quantum particle confined in a 1D box and interacting with one fixed point through a time dependent point interaction is considered. Under suitable assumptions of regularity for the time profile of the Hamiltonian, we prove the existence of strict solutions to the corresponding Schrödinger equation. The result is used to discuss the stability and the steady-state local controllability of the wavefunction when the strenght of the interaction is used as a control parameter.
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'.
This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form $q=u'+u^2$ for some $u\in L^2(R)$. We consider potentials for which there exist `left' and `right' Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev--Marchenko potentials in $L^1(R,(1+|x|)dx)$ generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding…
