Few-Body Quantum Problem in the Boundary-Condition Model

Systems of three and four quantum particles in the boundary-condition model are considered. The Faddeev-Yakubovsky approach is applied to construct the Fredholm-type integral equations for these systems in framework of the Potential theory. The boundary-value problems are formulated for the Faddeev-Yakubovsky components of wave functions.Comment: 8 pages, 30 kB, REVTEX; Talk given at the International Workshop "Meson--Barion interactions and Few--Body systems", Dubna, 28-30 April, 199

An alternative proof of the a priori tan⁡Θ\tan\Theta Theorem

Let $A$ be a self-adjoint operator in a separable Hilbert space. Suppose that the spectrum of $A$ is formed of two isolated components $\sigma_0$ and $\sigma_1$ such that the set $\sigma_0$ lies in a finite gap of the set $\sigma_1$. Assume that $V$ is a bounded additive self-adjoint perturbation of $A$, off-diagonal with respect to the partition ${\rm spec}(A)=\sigma_0 \cup \sigma_1$. It is known that if $\|V\|<\sqrt{2}{\rm dist}(\sigma_0,\sigma_1)$, then the spectrum of the perturbed operator $L=A+V$ consists of two disjoint parts $\omega_0$ and $\omega_1$ which originate from the corresponding initial spectral subsets $\sigma_0$ and $\sigma_1$. Moreover, for the difference of the spectral projections $E_A(\sigma_0)$ and $E_{L}(\omega_0)$ of $A$ and $L$ associated with the spectral sets $\sigma_0$ and $\omega_0$, respectively, the following sharp norm bound holds: $$\|E_A(\sigma_0)-E_{L}(\omega_0)\|\leq\sin\left(\arctan\frac{\|V\|}{{\rm dist}(\sigma_0,\sigma_1)}\right).$$ In the present note, we give a new proof of this bound for $\|V\|<{\rm dist}(\sigma_0,\sigma_1)$

Progress in methods to solve the Faddeev and Yakubovsky differential equations

We shortly recall the derivation of the Faddeev-Yakubovsky differential equations and point out their main advantages. Then we give a review of the numerical approaches used to solve the bound-state and scattering problems for the three- and four-body systems based on these equations. A particular attention is payed to the latest developments.Comment: A review article based on a talk given at the 20th European Conference on Few-Body Problems in Physic

On applying the subspace perturbation theory to few-body Hamiltonians

We present a selection of results on variation of the spectral subspace of a Hermitian operator under a Hermitian perturbation and show how these results may work for few-body Hamiltonians.Comment: Based on a talk presented at the 22nd European Conference on Few-Body Problems in Physics (September 9-13, 2013, Cracow, Poland

Unphysical energy sheets and resonances in the Friedrichs-Faddeev model

We consider the Friedrichs-Faddeev model in the case where the kernel of the potential operator is holomorphic in both arguments on a certain domain of $\mathbb{C}$. For this model we, first, study the structure of the $T$- and $S$-matrices on unphysical energy sheet(s). To this end, we derive representations that explicitly express them in terms of these same operators considered exclusively on the physical sheet. Furthermore, we allow the Friedrichs-Faddeev Hamiltonian undergo a complex deformation (or even a complex scaling/rotation if the model is associated with an infinite interval). Isolated non-real eigenvalues of the deformed Hamiltonian are called the deformation resonances. For a class of perturbation potentials with analytic kernels, we prove that the deformation resonances do correspond to the scattering matrix resonances, that is, they represent the poles of the scattering matrix analytically continued to the respective unphysical sheet

Removal of the resolvent-like dependence on the spectral parameter from perturbations

The spectral problem (A + V(z))\psi=z\psi is considered with A, a self-adjoint operator. The perturbation V(z) is assumed to depend on the spectral parameter z as resolvent of another self-adjoint operator A': V(z)=-B(A'-z)^{-1}B^{*}. It is supposed that the operator B has a finite Hilbert-Schmidt norm and spectra of the operators A and A' are separated. Conditions are formulated when the perturbation V(z) may be replaced with a ``potential'' W independent of z and such that the operator H=A+W has the same spectrum and the same eigenfunctions (more precisely, a part of spectrum and a respective part of eigenfunctions system) as the initial spectral problem. The operator H is constructed as a solution of the non-linear operator equation H=A+V(H) with a specially chosen operator-valued function V(H). In the case if the initial spectral problem corresponds to a two-channel variant of the Friedrichs model, a basis property of the eigenfunction system of the operator H is proved. A scattering theory is developed for H in the case where the operator A has continuous spectrum.Comment: LaTeX, 4 pages, no figure

Operator interpretation of resonances arising in spectral problems for 2 x 2 operator matrices

We consider operator matrices {\bf H}= (A_0 B_{01} \\ B_{10} A_{1}) with self-adjoint entries A_i, i=0,1, and bounded B_{01}=B_{10}^*, acting in the orthogonal sum {\cal H}={\cal H}_0\oplus{\cal H}_1 of Hilbert spaces {\cal H}_0 and {\cal H}_1. We are especially interested in the case where the spectrum of, say, A_1 is partly or totally embedded into the continuous spectrum of A_0 and the transfer function M_1(z)=A_1-z+V_1(z), where V_1(z)=B_{10}(z-A_0)^{-1}B_{01}, admits analytic continuation (as an operator-valued function) through the cuts along branches of the continuous spectrum of the entry A_0 into the unphysical sheet(s) of the spectral parameter plane. The values of z in the unphysical sheets where M_1^{-1}(z) and consequently the resolvent (H-z)^{-1} have poles are usually called resonances. A main goal of the present work is to find non-selfadjoint operators whose spectra include the resonances as well as to study the completeness and basis properties of the resonance eigenvectors of M_1(z) in {\cal H}_1. To this end we first construct an operator-valued function V_1(Y) on the space of operators in {\cal H}_1 possessing the property: V_1(Y)\psi_1=V_1(z)\psi_1 for any eigenvector \psi_1 of Y corresponding to an eigenvalue z and then study the equation H_1=A_1+V_1(H_1). We prove the solvability of this equation even in the case where the spectra of A_0 and A_1 overlap. Using the fact that the root vectors of the solutions H_1 are at the same time such vectors for M_1(z), we prove completeness and even basis properties for the root vectors (including those for the resonances).Comment: LaTeX 2e (REVTeX style), 62 pages. A slightly extended version. In particular a figure was added. A number of corrections was made. Submitted to Math. Nach

Solvability of the operator Riccati equation in the Feshbach case

We consider a bounded block operator matrix of the form $$L=\left(\begin{array}{cc} A & B \\ C & D \end{array} \right),$$ where the main-diagonal entries $A$ and $D$ are self-adjoint operators on Hilbert spaces $H_{_A}$ and $H_{_D}$, respectively; the coupling $B$ maps $H_{_D}$ to $H_{_A}$ and $C$ is an operator from $H_{_A}$ to $H_{_D}$. It is assumed that the spectrum $\sigma_{_D}$ of $D$ is absolutely continuous and uniform, being presented by a single band $[\alpha,\beta]\subset\mathbb{R}$, $\alpha<\beta$, and the spectrum $\sigma_{_A}$ of $A$ is embedded into $\sigma_{_D}$, that is, $\sigma_{_A}\subset(\alpha,\beta)$. We formulate conditions under which there are bounded solutions to the operator Riccati equations associated with the complexly deformed block operator matrix $L$; in such a case the deformed operator matrix $L$ admits a block diagonalization. The same conditions also ensure the Markus-Matsaev-type factorization of the Schur complement $M_{_A}(z)=A-z-B(D-z)^{-1}C$ analytically continued onto the unphysical sheet(s) of the complex $z$ plane adjacent to the band $[\alpha,\beta]$. We prove that the operator roots of the continued Schur complement $M_{_A}$ are explicitly expressed through the respective solutions to the deformed Riccati equations.Comment: 18 pages, 2 figure

On invariant graph subspaces of a J-self-adjoint operator in the Feshbach case

We consider a J-self-adjoint 2x2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry is embedded into the absolutely continuous spectrum of the other main-diagonal entry. We work with the analytic continuation of one of the Schur complements of L to the unphysical sheets of the spectral parameter plane. We present the conditions under which the continued Schur complement has operator roots, in the sense of Markus-Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding J-orthogonal invariant subspaces of L. The presentation ends with an explicitly solvable example

Bounds on variation of the spectrum and spectral subspaces of a few-body Hamiltonian

We overview the recent results on the shift of the spectrum and norm bounds for variation of spectral subspaces of a Hermitian operator under an additive Hermitian perturbation. Along with the known results, we present a new subspace variation bound for the generic off-diagonal subspace perturbation problem. We also demonstrate how some of the abstract results may work for few-body Hamiltonians.Comment: arXiv admin note: text overlap with arXiv:1311.660