On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.     

Global extstence of small solutions to semiliner schrödinger equations

We present global existence theorem for semilinear schrödinger equations. In general, Schrödinger-type equations do not admit the classical energy estimates. To avoid this difficulty, we use S. Doi's method for linear Schrödinger-type equations. Combining his method and L^p-L^q estimates, we prove the global existence of solutions with small initial data.     

The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data

We set-up and solve the Cauchy problem for Schrödinger-type differential operators with generalized functions as coefficients, in particular, allowing for distributional coefficients in the principal part. Equations involving such kind of operators appeared in models of deep earth seismology. We prove existence and uniqueness of Colombeau generalized solutions and analyze the relations with classical and distributional solutions. Furthermore, we provide a construction of generalized initial values that may serve as square roots of arbitrary probability measures.     

Schrödinger uncertainty relation,Wigner–Yanase–Dyson skew information and metric adjusted correlation measure

In this paper, we give Schrödinger-type uncertainty relation using the Wigner–Yanase–Dyson skew information. In addition, we give Schrödinger-type uncertainty relation by use of a two-parameter extended correlation measure. We finally show a further generalization of Schrödinger-type uncertainty relation by use of the metric adjusted correlation measure. These results generalize our previous result in [Phys. Rev. A 82 (2010) 034101].     

Explicit second-order accurate schemes for the nonlinear Schrödinger equations

We consider three-level explicit schemes for solving the nonlinear variable coefficient Schrödinger-type equation. Using spectral and energy methods we establish the stability and convergence of these schemes. The existence of discrete conservation laws is investigated. General results are applied for the DuFort-Frankel and leap-frog diffenrence schemes.     

Local Solvability Of First Order Differential Operators Near A Critical Point,Operators With Quadratic Symbols And The Heisenberg Group

We study questions of solvability for operators of the form p(x,D)+b, where p(x,ξ) is a real quadratic form and b?C. As one consequence, we obtain a necessary and sufficient condition for the local solvability of operators of the form L= near the critical point x=0, and prove the existence of tempered fundamental solutions whenever L is locally solvable.Our analysis of these operators is largely based on recent results about the solvabilitiy of left–invariant second order differential operators on the Heisenberg group and a transference principle for the Schrödinger representation.     

Gain of regularity for semilinear Schrödinger equations

We discuss local existence and gain of regularity for semilinear Schr?dinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi's transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schr?dinger-type equations. In particular, the sharp G?rding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof. Received: 14 December 1998     

On Combinatorics of Schrödinger Perturbations

We give a tight upper bound for Schrödinger-type perturbations of integral kernels.     

Bilinear Pseudo-differential Operators with Symbols in $$BS^{m}_{1,1}$$ on Triebel–Lizorkin Spaces

In this paper, bilinear pseudo-differential operators with symbols in the bilinear Hörmander symbol class \(BS^{m}_{1,1}\) on Triebel–Lizorkin spaces are discussed. As a result, we can obtain the Kato–Ponce inequality in local Hardy spaces.     

Bounds on the density of states for Schrödinger operators

We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a “density of states outer-measure” that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.     

Extrapolation from A_{\infty}^{\rho,\infty}, vector-valued inequalities and applications in the Schrödinger settings

In this paper, we generalize the A _∞ extrapolation theorem (Cruz-Uribe–Martell–Pérez, Extrapolation from A _∞ weights and applications, J. Funct. Anal. 213 (2004), 412–439) and the A _p extrapolation theorem of Rubio de Francia to Schrödinger settings. In addition, we also establish weighted vector-valued inequalities for Schrödinger-type maximal operators by using weights belonging to $A_{p}^{\rho,\infty }$ which includes A _p . As applications, we establish weighted vector-valued inequalities for some Schrödinger-type operators.     

Solutions of two-dimensional Schrödinger-type equations in a magnetic field

We use the method of dressing by a linear operator of general form to construct new solutions of Schrödinger-type two-dimensional equations in a magnetic field. In the case of a nonunit metric, we integrate the class of solutions that admit a variable separation before dressing. In particular, we show that the ratio of the coefficients of the differential operators in the unit metric case satisfies the Hopf equation. We establish a relation between the solutions of the two-dimensional eikonal equation with the unit right-hand side and solutions of the Hopf equation.     

On a Schrödinger–Kirchhoff-type equation involving the p(x)-Laplacian

In this paper we deal with a Schrödinger-type equation involving a nonlocal Kirchhoff-type coefficient and depending on two real parameters. Working within the framework of variable exponent spaces and using the variational approach, we obtain several results of existence and multiplicity of solutions, depending on the range of the parameters.     

On the absence of global periodic solutions of a Schrödinger-type nonlinear evolution equation

Theoretical and Mathematical Physics - We study the problem of the absence of global periodic solutions of a nonlinear Schrödinger-type evolution equation with a damped linear term. We prove...     

Boundedness of Schrödinger Type Propagators on Modulation Spaces

It is known that Fourier integral operators arising when solving Schrödinger-type operators are bounded on the modulation spaces ? ^p,q , for 1≤p= q≤∞, provided their symbols belong to the Sjöstrand class M ^∞,1. However, they generally fail to be bounded on ? ^p,q for p≠q. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on ? ^p,q for p≠q, and between ? ^p,q →? ^q,p , 1≤q<p≤∞. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.     

Discrete Crum’s Theorems and Lattice KdV-Type Equations

Theoretical and Mathematical Physics - We develop Darboux transformations (DTs) and their associated Crum’s formulas for two Schrödinger-type difference equations that are themselves...     

Some new results on weak compactness

We consider the scattering problem in Lsu2$\text{Rsu3}$ for Schrödinger operators with momentum-dependent interactions, i.e., for the pair of Hamiltonians H₀ = ? Δ and H = H₀ + T, where T is a pseudodifferential operator. The existence of the wave operators is proved by estimating the integrals appearing in the CookHack convergence criterion, and their asymptotic completeness is established under more restrictive conditions on the symbol of T by using a local trace criterion.     

On a complex fundamental solution of the Schrödinger equation

A second-order Schrödinger differential operator of parabolic type is considered, for which an explicit form of a fundamental solution is derived. Such operators arise in many problems of physics, and the fundamental solution plays the role of the Feynman propagation function.     

Modified Wave Operator for a System of Nonlinear Schrödinger Equations in 2d

We consider a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. We prove the existence of modified wave operators or wave operators under some mass conditions.