On a boundary pair of a dissipative operator
The aim of this brief note is to demonstrate that the boundary pair of a dissipative operator is determined by the unitary boundary pair of its symmetric part.
math.FA↗
arXiv subjects
Rytis Jursenas
Publications and source records attributed to Rytis Jursenas.
On the similarity of boundary triples for dual pairs
The Weyl family of a dual pair $A\subseteq B^c$ of operators in a Krein space determines a minimal boundary triple uniquely up to similarity; if $A=B$, a necessary and sufficient condition in order that the similarity should be unitary is given.
math.FA↗
Locus of non-real eigenvalues of a class of linear relations in a Krein space
It is a classical result that, if a maximal symmetric operator $T$ in a Krein space $\mathcal{H}=\mathcal{H}^-[\oplus]\mathcal{H}^+$ has the property $\mathcal{H}^-\subseteq\mathcal{D}_T$, then the imaginary part of its eigenvalue $λ$ from upper or lower half-plane is bounded by $\lvert \mathrm{Im}\,λ\rvert\leq2\lVert TP^- \rVert$. We prove that in both half-planes $\lvert \mathrm{Im}\,λ\rvert$ never exceeds $t_0\lVert TP^- \rVert$ for some constant $t_0\approx1.84$. The result applies to a closed symmetric relation $T$ and carries on a suitable, most notably dissipative, extension.
math.SP↗
On the similarity of boundary triples of symmetric operators in Krein spaces
It is a classical result that the Weyl function of a simple symmetric operator in a Hilbert space determines a boundary triple uniquely up to unitary equivalence. We generalize this result to a simple symmetric operator in a Pontryagin space, where unitary equivalence is replaced by the similarity realized via a standard unitary operator.
math.FA↗
The peak model for finite rank supersingular perturbations
In its original form the peak model for rank one supersingular perturbations of class $\mathfrak{H}_{-4}$ or higher of a nonnegative self-adjoint operator requires that the Gram matrix of the model should be diagonal. Here we remove the restriction on the Gram matrix. In particular we explain the origin of the Krein $Q$-function associated with the Gram matrix.
math.FA↗
On necessary and sufficient conditions relating the adjoint of a column to a row of linear relations
A row and a column of two linear relations in Hilbert spaces are presented respectively as a sum and an intersection of two linear relations. As an application, necessary and sufficient conditions for the adjoint of a column to be a row are examined. Several outcomes are discussed as well.
math.FA↗
The A-model with mutually equal model parameters can lead to a Hilbert space model
It is known that the A-model for higher order singular perturbations can be considered as a Hilbert space model if the model parameters are mutually distinct, and that it is necessarily a Pontryagin space model if otherwise. In this note we demonstrate that the A-model with mutually equal model parameters can nonetheless lead to a Hilbert space model if the extensions in the model space are instead described by suitable linear relations.
math.FA↗
On some extensions of the A-model
The A-model for finite rank singular perturbations of class $\mathfrak{H}_{-m-2}\smallsetminus\mathfrak{H}_{-m-1}$, $m\in\mathbb{N}$, is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces $(\mathfrak{H}_n)_{n\in\mathbb{Z}}$ admit an orthogonal decomposition $\mathfrak{H}^-_n\oplus\mathfrak{H}^+_n$, with the corresponding projections satisfying $P^\pm_{n+1}\subseteq P^\pm_n$, nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.
math.FA↗
Weyl families of essentially unitary pairs
It is known that the Weyl families corresponding to unitary boundary pairs $(\mathcal{H},Γ)$ belong to the class $\tilde{\mathcal{R}}(\mathcal{H})$ of Nevanlinna families. Here we extend the theorem to the case of essentially unitary boundary pairs by showing that the closures of members of the Weyl families belong to the class $\tilde{\mathcal{R}}(\mathcal{H})$. Thus bounded Weyl functions of essentially unitary boundary pairs are of class $\mathcal{R}[\mathcal{H}]$.
math.FA↗
Irreducible tensor form for the AME coupling
Using the multipole expansion of electromagnetic (EM) field, we present the angular magnetoelectric (AME) coupling in irreducible tensor form. We evaluate the matrix elements when the radiation source is described by electronic transitions in atomic systems. The results indicate that the energy corrections increase for short wavelengths and large charge number.
physics.atom-ph↗
Computation of unitary group for the Rashba spin-orbit coupled operator, with application to point-interactions
We compute an explicit formula for the one-parameter unitary group of the single-particle Rashba spin-orbit coupled operator in dimension three. As an application, we derive the formula for the Green function for the two-particle operator, and then prove that the spin-dependent point-interaction is of class $\mathcal{H}_{-4}$. The latter is thus the example of a supersingular perturbation for which no self-adjoint operator can be constructed.
math-ph↗
Direct integral description of angulon
We propose a representation of angulon in which the angulon operator is decomposable relative to the field of Hilbert spaces over the probability measure space, and the probability measure corresponds to the total-number operator of phonons. In this representation we are able to find the system of $N+1$ equations whose solutions form the eigenspace of the angulon operator, where $1\leq N<\infty$ is the number of phonon excitations. Using this result we estimate the infimum of the spectrum. In the special case $N=1$, the lowest energy approximates to the value which is already known in the literature. Our findings indicate that two-phonon excitations ($N=2$) contribute notably to the energy of a molecule in superfluid $^4$He.
math-ph↗
A model for reducing the angulon operator
We propose a mathematical model for the recently introduced angulon. In our formulation, the angulon operator is decomposable relative to the field of Hilbert spaces over the probability measure space. That is, we transfer the population of phonons from the inner structure of the many-body Hamiltonian to the definition of the measure. We do not compute the measure itself. However, we demonstrate that the approach allows us to perform angular reduction thereby considerably simplifying the spectral analysis.
math-ph↗
Spectrum of a family of spin-orbit coupled Hamiltonians with singular perturbation
The present study is the first such attempt to examine rigorously and comprehensively the spectral properties of a three-dimensional ultracold atom when both the spin-orbit interaction and the Zeeman field are taken into account. The model operator is the Rashba spin-orbit coupled operator in dimension three. The self-adjoint extensions are constructed using the theory of singular perturbations, where regularized rank two perturbations describe spin-dependent contact interactions. The spectrum of self-adjoint extensions is investigated in detail laying emphasis on the effects due to spin-orbit coupling. When the spin-orbit-coupling strength is small enough, the asymptotics of eigenvalues is obtained. The conditions for the existence of eigenvalues above the threshold are discussed in particular.
math.SP↗
Series expansion for the Fourier transform of a rational function in three dimensions
In Rashba-Dresselhaus spin-orbit coupled systems, the calculation of Green's function requires the knowledge of the inverse Fourier transform of rational function $P(p)/Q(p)$, where $P(p)$ takes the values $1$ and $p^{2}$, and where \[ Q(p)=(p^{2}-ζ)^{2}- α^{2}(p_{1}^{2}+p_{2}^{2})-β^{2} \] with suitable parameters $α$, $β\geq0$, $ζ\in\mathbb{C}$. While a two-dimensional problem, with $p=(p_{1},p_{2})$, has been recently solved [J. Brüning et al, J. Phys. A: Math. Theor. 40 (2007)], its three-dimensional analogue, with $p=(p_{1},p_{2},p_{3})$, remains open. In this paper, a hypergeometric series expansion for the triple integral is provided. Convergence of the series dependent on the parameters is studied in detail.
math-ph↗
A note on the spectrum of a two-particle Rashba Hamiltonian
In a series of recent papers it was shown that, when the attractive s-wave interaction is dominant, the spin-orbit coupled fermions form a bound state. Attributed to a convenient momentum representation, it became a common condition of agreement to express the bound state as a function of the center-of-mass momentum Q. In this letter we prove that the bound state of Rashba fermions does not depend on the chosen representation. That is, all the states characterized by nonzero Q fail to obey the translation symmetry.
cond-mat.quant-gas↗
On the definite integral of two confluent hypergeometric functions related to the Kampé de Fériet double series
The Kampé de Fériet double series $F_{1:1;1}^{1:1;1}$ is studied through the solution to the associated first-order nonhomogeneous differential equation. It is shown that the integral of $t^{β+l}M(\cdot;β;λt)M(\cdot;β;-λt)$ over $t\in[0,T]$, $T\geq0$, $l=0,1,\ldots$, $\Reβ+l>-1$, is a linear combination of functions $F_{1:1;1}^{1:1;1}$. The integral is a generalization of a class of so-called Coulomb integrals involving regular Coulomb wave functions.
math.CA↗
Bound states of the spin-orbit coupled ultracold atom in a one-dimensional short-range potential
We solve the bound state problem for the Hamiltonian with the spin-orbit and the Raman coupling included. The Hamiltonian is perturbed by a one-dimensional short-range potential V which describes the impurity scattering. In addition to the bound states obtained by considering weak solutions through the Fourier transform or by solving the eigenvalue equation on a suitable domain directly, it is shown that ordinary point-interaction representations of V lead to spin-orbit induced extra states.
math-ph↗