Eigenvalues of periodic difference operators on lattice octant

Consider a difference operator $H$ with periodic coefficients on the octant of the lattice. We show that for any integer $N$ and any bounded interval $I$, there exists an operator $H$ having $N$ eigenvalues, counted with multiplicity on this interval, and does not exist other spectra on the interval. Also right and to the left of it are spectra and the corresponding subspaces have an infinite dimension. Moreover, we prove similar results for other domains and any dimension. The proof is based on the inverse spectral theory for periodic Jacobi operators.Comment: 2 figures, 17 pages. arXiv admin note: text overlap with arXiv:1712.0889

Schr\"odinger operator with periodic plus compactly supported potentials on the half-line

We consider the Schr\"odinger operator $H$ with a periodic potential $p$ plus a compactly supported potential $q$ on the half-line. We prove the following results: 1) a forbidden domain for the resonances is specified, 2) asymptotics of the resonance-counting function is determined, 3) in each nondegenerate gap \g_n for $n$ large enough there is exactly an eigenvalue or an antibound state, 4) the asymptotics of eigenvalues and antibound states are determined at high energy, 5) the number of eigenvalues plus antibound states is odd $\ge 1$ in each gap, 6) between any two eigenvalues there is an odd number $\ge 1$ of antibound states, 7) for any potential $q$ and for any sequences (\s_n)_{1}^\iy, \s_n\in \{0,1\} and (\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0, there exists a potential $p$ such that each gap length |\g_n|=\vk_n, n\ge 1 and $H$ has exactly \s_n eigenvalues and 1-\s_n antibound state in each gap \g_n\ne \es for $n$ large enough, 8) if unperturbed operator (at $q=0$) has infinitely many virtual states, then for any sequence (\s)_1^\iy, \s_n\in \{0,1\}, there exists a potential $q$ such that $H$ has \s_n bound states and 1-\s_n antibound states in each gap open \g_n for $n$ large enough

Resonances of third order differential operators

We consider resonances for third order ordinary differential operator with compactly supported coefficients on the real line. Resonance are defined as zeros of a Fredholm determinant on a non-physical sheet of three sheeted Riemann surface. We determine upper bounds of the number of resonances in complex discs at large radius. We express the trace formula in terms of resonances only.Comment: 24 pages, 1 figur

Global estimates of resonances for 1D Dirac operators

We discuss resonances for 1D massless Dirac operators with compactly supported potentials on the line. We estimate the sum of the negative power of all resonances in terms of the norm of the potential and the diameter of its support

Asymptotics of resonances for 1d Stark operators

We consider the Stark operator perturbed by a compactly supported potentials on the real line. We determine forbidden domain for resonances, asymptotics of resonances at high energy and asymptotics of the resonance counting function for large radius.Comment: 12 page

Resonance theory for perturbed Hill operator

We consider the Schr\"odinger operator $Hy=-y"+(p+q)y$ with a periodic potential $p$ plus a compactly supported potential $q$ on the real line. The spectrum of $H$ consists of an absolutely continuous part plus a finite number of simple eigenvalues below the spectrum and in each spectral gap \g_n\ne \es, n\ge1. We prove the following results: 1) the distribution of resonances in the disk with large radius is determined, 2) the asymptotics of eigenvalues and antibound states are determined at high energy gaps, 3) if $H$ has infinitely many open gaps in the continuous spectrum, then for any sequence (\vk)_1^\iy, \vk_n\in \{0,2\}, there exists a compactly supported potential $q$ with $\int_\R qdx=0$ such that $H$ has \vk_n eigenvalues and 2-\vk_n antibound states (resonances) in each gap \g_n for $n$ large enough.Comment: 25 pages. arXiv admin note: repeats content from arXiv:0904.287

Estimates of 1D resonances in terms of potentials

We discuss resonances for Schr\"odinger operators with compactly supported potentials on the line and the half-line. We estimate the sum of the negative power of all resonances and eigenvalues in terms of the norm of the potential and the diameter of its support. The proof is based on harmonic analysis and Carleson measures arguments.Comment: 12 page

Trace formulas for Schr\"odinger operators with complex potentials

We consider 3-dim Schr\"odinger operators with a complex potential. We obtain new trace formulas. In order to prove these results we study analytic properties of a modified Fredholm determinant. In fact we reformulate spectral theory problems as the problems of analytic functions from Hardy spaces in upper half-plane

Conformal spectral theory for the monodromy matrix

For any N\ts N monodromy matrix we define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy. We show that the endpoints of each gap are periodic (anti-periodic) eigenvalues or resonances (real branch points). Moreover, the following results are obtained: 1) we define the quasimomentum as an analytic function on the Riemann surface of the Lyapunov function; various properties and estimates of the quasimomentum are obtained, 2) we construct the conformal mapping with imaginary part given by the Lyapunov exponent and we obtain various properties of this conformal mapping, which are similar to the case of the Hill operator, 3) we determine various new trace formulae for potentials and the Lyapunov exponent, 4) we obtain a priori estimates of gap lengths in terms of the Dirichlet integral. We apply these results to the Schr\"odinger operators and to first order periodic systems on the real line with a matrix valued complex self-adjoint periodic potential

Schr\"odinger operator with a junction of two 1-dimensional periodic potentials

The spectral properties of the Schr\"odinger operator $T_ty= -y''+q_ty$ in $L^2(\R)$ are studied, with a potential $q_t(x)=p_1(x), x<0,$ and $q_t(x)=p(x+t), x>0,$ where $p_1, p$ are periodic potentials and $t\in \R$ is a parameter of dislocation. Under some conditions there exist simultaneously gaps in the continuous spectrum of $T_0$ and eigenvalues in these gaps. The main goal of this paper is to study the discrete spectrum and the resonances of $T_t$. The following results are obtained: i) In any gap of $T_t$ there exist $0,1$ or 2 eigenvalues. Potentials with 0,1 or 2 eigenvalues in the gap are constructed. ii) The dislocation, i.e. the case $p_1=p$ is studied. If $t\to 0$, then in any gap in the spectrum there exist both eigenvalues ($\le 2$) and resonances ($\le 2$) of $T_t$ which belong to a gap on the second sheet and their asymptotics as $t\to 0$ are determined. iii) The eigenvalues of the half-solid, i.e. $p_1={\rm constant}$, are also studied. iv) We prove that for any even 1-periodic potential $p$ and any sequences \{d_n\}_1^{\iy}, where $d_n=1$ or $d_n=0$ there exists a unique even 1-periodic potential $p_1$ with the same gaps and $d_n$ eigenvalues of $T_0$ in the n-th gap for each $n\ge 1.