arXiv subjects
Publications and source records attributed to Boris Mityagin.
We formulate several open questions related to enumerative combinatorics, which arise in the spectral analysis of Hill operators with trigonometric polynomial potentials.
We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator $L= -d^2/dx^2 + v(x), $ $x \in [0,π], $ with $H_{per}^{-1} $-potential and the free operator $L^0=-d^2/dx^2, $ subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $$ \|S_N - S_N^0: L^a \to L^b \| \to 0 \quad \text{if} \;\; 1<a \leq b< \infty, \;\; 1/a - 1/b <1/2, $$ where $S_N$ and $S_N^0 $ are the $N$-th partial sums of the spectral decompositions of $L$ and $L^0.$ Moreover, if $v \in H^{-α} $ with $1/2 < α< 1$ and $\frac{1}{a}=(3/2)-α, $ then we obtain uniform equiconvergence: $\|S_N - S_N^0: L^a \to L^\infty \| \to 0 $ as $N \to \infty. $
For one-dimensional Dirac operators $$ Ly= i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{dy}{dx} + v y, \quad v= \begin{pmatrix} 0 & P \\ Q & 0 \end{pmatrix}, \;\; y=\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, $$ subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in $L^2 ([0,π], \mathbb{C}^2).$ In particular, if the potential matrix $v$ is skew-symmetric (i.e., $\overline{Q} =-P$), or more generally if $\overline{Q} =t P$ for some real $t \neq 0,$ then there exists a Riesz basis that consists of root functions of the operator $L.$
One dimensional Dirac operators $$ L_{bc}(v) y = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,π]$$, considered with $L^2$-potentials $ v(x) = 0 & P(x) Q(x) & 0$ and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc,$ the spectrum of the free operator $ L_{bc}(0) $ is simple while the spectrum of $ L_{bc}(v) $ is eventually simple, and the corresponding normalized root function systems are Riesz bases. For expansions of functions of bounded variation about these Riesz bases, we prove the uniform equiconvergence property and point-wise convergence on the closed interval $[0,π].$ Analogous results are obtained for regular but not strictly regular $bc.$
We study the system of root functions (SRF) of Hill operator $Ly = -y^{\prime \prime} +vy $ with a singular potential $v \in H^{-1}_{per}$ and SRF of 1D Dirac operator $ Ly = i {pmatrix} 1 & 0 0 & -1 {pmatrix} \frac{dy}{dx} + vy $ with matrix $L^2$-potential $v={pmatrix} 0 & P Q & 0 {pmatrix},$ subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in $L^p$-spaces and other rearrangement invariant function spaces.
We analyze the perturbations $T+B$ of a selfadjoint operator $T$ in a Hilbert space $H$ with discrete spectrum $\{t_k \}$, $T ϕ_k = t_k ϕ_k$, as an extension of our constructions in arXiv: 0912.2722 where $T$ was a harmonic oscillator operator. In particular, if $t_{k+1}-t_k \geq c k^{α- 1}, \quad α> 1/2$ and $\| B ϕ_k \| = o(k^{α- 1})$ then the system of root vectors of $T+B$, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in $H$.
One dimensional Dirac operators $$ L_{bc}(v) \, y = i \begin{pmatrix} 1 & 0 0 & -1 \end{pmatrix} \frac{dy}{dx} + v(x) y, \quad y = \begin{pmatrix} y_1 y_2 \end{pmatrix}, \quad x\in[0,π],$$ considered with $L^2$-potentials $ v(x) = \begin{pmatrix} 0 & P(x) Q(x) & 0 \end{pmatrix} $ and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc,$ it is shown that every eigenvalue of the free operator $L^0_{bc}$ is simple and has the form $λ_{k,α}^0 = k + τ_α$ where $ \; α\in \{1,2\}, \; k \in 2 \mathbb{Z} $ and $τ_α=τ_α(bc);$ if $|k|>N(v, bc) $ each of the discs $D_k^α= \{z: \; |z-λ_{k,α}^0| <ρ=ρ(bc) \} , $ $α\in \{1,2\}, $ contains exactly one simple eigenvalue $λ_{k,α} $ of $L_{bc} (v) $ and $(λ_{k,α} -λ_{k,α}^0)_{k\in 2\mathbb{Z}} $ is an $\ell^2 $-sequence. Moreover, it is proven that the root projections $ P_{n,α} = \frac{1}{2πi} \int_{\partial D^α_n} (z-L_{bc} (v))^{-1} dz $ satisfy the Bari--Markus condition $$\sum_{|n| > N} \|P_{n,α} - P_{n,α}^0\|^2 < \infty, \quad n \in 2\mathbb{Z}, $$ where $P_n^0 $ are the root projections of the free operator $L^0_{bc}.$ Hence, for strictly regular $bc,$ there is a Riesz basis consisting of root functions (all but finitely many being eigenfunctions). Similar results are obtained for regular but not strictly regular $bc$ -- then in general there is no Riesz basis consisting of root functions but we prove that the corresponding system of two-dimensional root projections is a Riesz basis of projections.
For 1D Dirac operators Ly= i J y' + v y, where J is a diagonal 2x2 matrix with entrees 1,-1 and v(x) is an off-diagonal matrix with L^2 [0,π]-entrees P(x), Q(x) we characterize the class X of pi-periodic potentials v such that: (i) the smoothness of potentials v is determined only by the rate of decay of related spectral gaps gamma (n) = | λ(n,+) - λ(n,-)|, where λ(..) are the eigenvalues of L=L(v) considered on [0,π] with periodic (for even n) or antiperiodic (for odd n) boundary conditions (bc); (ii) there is a Riesz basis which consists of periodic (or antiperiodic) eigenfunctions and (at most finitely many) associated functions. In particular, X contains symmetric potentials X_{sym} (\overline{Q} =P), skew-symmetric potentials X_{skew-sym} (\overline{Q} =-P), or more generally the families X_t defined for real nonzero t by \overline{Q} =t P. Finite-zone potentials belonging to X_t are dense in X_t. Another example: if P(x)=a exp(2ix)+b exp(-2ix), Q(x)=Aexp(2ix)+Bexp(-2ix) with complex a, b, A, B \neq 0, then the system of root functions of L consists eventually of eigenfunctions. For antiperiodic bc this system is a Riesz basis if |aA|=|bB| (then v \in X), and it is not a basis if |aA| \neq |bB|. For periodic bc the system of root functions is a Riesz basis (and v \in X) always.
We prove the following. For any complex valued $L^p$-function $b(x)$, $2 \leq p < \infty$ or $L^\infty$-function with the norm $\| b | L^{\infty}\| < 1$, the spectrum of a perturbed harmonic oscillator operator $L = -d^2/dx^2 + x^2 + b(x)$ in $L^2(\mathbb{R}^1)$ is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in $L^2(\mathbb{R})$.
We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq π, $$ subject to periodic or antiperiodic boundary conditions, with potentials $v$ which are trigonometric polynomials with nonzero coefficients, of the form (i) $ ae^{-2ix} +be^{2ix}; $ (ii) $ ae^{-2ix} +Be^{4ix}; $ (iii) $ ae^{-2ix} +Ae^{-4ix} + be^{2ix} +Be^{4ix}. $ Then the system of eigenfunctions and (at most finitely many) associated functions is complete but it is not a basis in $L^2 ([0,π], \mathbb{C})$ if $|a| \neq |b| $ in the case (i), if $|A| \neq |B| $ and neither $-b^2/4B$ nor $-a^2/4A$ is an integer square in the case (iii), and it is never a basis in the case (ii) subject to periodic boundary conditions.
By using quasi--derivatives we develop a Fourier method for studying the spectral gaps of one dimensional Schrödinger operators with periodic singular potentials $v.$ Our results reveal a close relationship between smoothness of potentials and spectral gap asymptotics under a priori assumption $v \in H^{-1}_{loc} (\mathbb{R}).$ They extend and strengthen similar results proved in the classical case $v \in L^2_{loc}(\mathbb{R}).$
The Dirac operators $$ Ly = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,π],$$ with $L^2$-potentials $$ v(x) = 0 & P(x) Q(x) & 0, \quad P,Q \in L^2 ([0,π]), $$ considered on $[0,π]$ with periodic, antiperiodic or Dirichlet boundary conditions $(bc)$, have discrete spectra, and the Riesz projections $$ S_N = \frac{1}{2πi} \int_{|z|= N-{1/2}} (z-L_{bc})^{-1} dz, \quad P_n = \frac{1}{2πi} \int_{|z-n|= {1/4}} (z-L_{bc})^{-1} dz $$ are well--defined for $|n| \geq N$ if $N $ is sufficiently large. It is proved that $$\sum_{|n| > N} \|P_n - P_n^0\|^2 < \infty, $$ where $P_n^0, n \in \mathbb{Z},$ are the Riesz projections of the free operator. Then, by the Bari--Markus criterion, the spectral Riesz decompositions $$ f = S_N f + \sum_{|n| >N} P_n f, \quad \forall f \in L^2; $$ converge unconditionally in $L^2.$
A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov \cite{KC1,KC2} led to an integral operator $K$ -- see (1), (2). They guessed that the equation $E_0(a,T)=1$ where $E_0$ is the largest eigenvalue of the operator $K$ has a solution $T(a)=1-τ(a)$ with $τ(a) \sim a^{2/5}$ when $a$ goes to 0. $τ(a)$ imitates the shift of critical (instability) temperature. We give a rigorous analysis of an anisotropic integral operator $K$ and prove the asymptotic ($*$) -- see Theorem 8 and Proposition 10. Additive Uncertainty Principle (of Landau-Pollack-Slepian [SP], \cite{LP1,LP2}) plays important role in this analysis.
The Hill operators $L y = - y^{\prime \prime} + v(x) y, x \in [0,π],$ with $H^{-1}$ periodic potentials, considered with periodic, antiperiodic or Dirichlet boundary conditions, have discrete spectrum, and therefore, for sufficiently large $N,$ the Riesz projections $$ P_n = \frac{1}{2πi} \int_{C_n} (z-L)^{-1} dz, \quad C_n=\{z: |z-n^2|= n\} $$ are well defined. It is proved that $$\sum_{n>N} \|P_n - P_n^0\|^2_{HS} < \infty, $$ where $P_n^0$ are the Riesz projection of the free operator and $\|\cdot\|_{HS}$ is the Hilbert--Schmidt norm.
It is shown that the deviations $P_n -P_n^0$ of Riesz projections $$ P_n = \frac{1}{2πi} \int_{C_n} (z-L)^{-1} dz, \quad C_n=\{|z-n^2|= n\}, $$ of Hill operators $L y = - y^{\prime \prime} + v(x) y, x \in [0,π],$ with zero and $H^{-1}$ periodic potentials go to zero as $n \to \infty $ even if we consider $P_n -P_n^0$ as operators from $L^1$ to $L^\infty. $ This implies that all $L^p$-norms are uniformly equivalent on the Riesz subspaces $Ran P_n. $
By using quasi--derivatives, we develop a Fourier method for studying the spectral properties of one dimensional Schrödinger operators with periodic singular potentials.
Let $γ_n $ denote the length of the $n$-th zone of instability of the Hill operator $Ly= -y^{\prime \prime} - [4tα\cos2x + 2 α^2 \cos 4x ] y,$ where $α\neq 0, $ and either both $α, t $ are real, or both are pure imaginary numbers. For even $n$ we prove: if $t, n $ are fixed, then, for $ α\to 0, $ $$ γ_n = | \frac{8α^n}{2^n [(n-1)!]^2} \prod_{k=1}^{n/2} (t^2 - (2k-1)^2) | (1 + O(α)), $$ and if $ α, t $ are fixed, then, for $ n \to \infty, $ $$ γ_n = \frac{8 |α/2|^n}{[2 \cdot 4 ... (n-2)]^2} | \cos (\fracπ{2} t) | [ 1 + O (\frac{\log n}{n}) ]. $$ Similar formulae (see Theorems \ref{thm2} and \ref{thm4}) hold for odd $n.$ The asymptotics for $α\to 0 $ imply interesting identities for squares of integers.
For tri-diagonal matrices arising in the simplified Jaynes--Cummings model, we give an asymptotics of the eigenvalues, prove a trace formula and show that the Spectral Riemann Surface is irreducible.