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Plamen Djakov

Publications and source records attributed to Plamen Djakov.

At least 19 recordsLinked to original sources
arXiv2014

Riesz basis property of Hill operators with potentials in weighted spaces

Consider the Hill operator $L(v) = - d^2/dx^2 + v(x) $ on $[0,π]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2 $ there are one Dirichlet eigenvalue $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+ $ (counted with multiplicity). We describe classes of complex potentials $v(x)= \sum_{2\mathbb{Z}} V(k) e^{ikx}$ in weighted spaces (defined in terms of the Fourier coefficients of $v$) such that the periodic (or antiperiodic) root function system of $L(v) $ contains a Riesz basis if and only if $$V(-2n) \asymp V(2n) \quad \text{as} \;\; n \in 2\mathbb{N}\;\; (\text{or} \; n \in 1+ 2\mathbb{N}), \;\; n \to \infty.$$ For such potentials we prove that $λ_n^+ - λ_n^- \sim \pm 2\sqrt{V(-2n)V(2n)} $ and $$μ_n - \frac{1}{2}(λ_n^+ + λ_n^-) \sim -\frac{1}{2} (V(-2n) + V(2n)).$$

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arXiv2013

Asymptotics of spectral gaps of 1D Dirac operator with two exponential terms potential

The one-dimensional Dirac operator \begin{equation*} L = i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{d}{dx} +\begin{pmatrix} 0 & P(x) \\ Q(x) & 0 \end{pmatrix}, \quad P,Q \in L^2 ([0,π]), \end{equation*} considered on $[0,π]$ with periodic and antiperiodic boundary conditions, has discrete spectra. For large enough $|n|,\, n \in \mathbb{Z}, $ there are two (counted with multiplicity) eigenvalues $λ_n^-,λ_n^+ $ (periodic if $n$ is even, or antiperiodic if $n$ is odd) such that $|λ_n^\pm - n |<1/2.$ We study the asymptotics of spectral gaps $γ_n =λ_n^+ - λ_n^-$ in the case $$P(x)=a e^{-2ix} + A e^{2ix}, \quad Q(x)=b e^{-2ix} + B e^{2ix},$$ where $a, A, b, B$ are nonzero complex numbers. We show, for large enough $m,$ that $γ_{\pm 2m}=0 $ and \begin{align*} γ_{2m+1} = \pm 2 \frac{\sqrt{(Ab)^m (aB)^{m+1}}}{4^{2m} (m!)^2 } \left[ 1 + O \left( \frac{\log^2 m}{m^2}\right) \right], \end{align*} \begin{align*} γ_{-(2m+1)} = \pm 2\frac{\sqrt{(Ab)^{m+1} (aB)^m}}{4^{2m} (m!)^2} \left[ 1 + O \left( \frac{\log^2 m}{m^2}\right) \right]. \end{align*}

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arXiv2013

Asymptotic formulas for spectral gaps and deviations of Hill and 1D Dirac operators

Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,π].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large enough $|n|$ close to $n^2 $ in the Hill case, or close to $n, \; n\in \mathbb{Z}$ in the Dirac case, there are one Dirichlet eigenvalue $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+ $ (counted with multiplicity). We give estimates for the asymptotics of the spectral gaps $γ_n = λ_n^+ - λ_n^-$ and deviations $ δ_n =μ_n - λ_n^+$ in terms of the Fourier coefficients of the potentials. Moreover, for special potentials that are trigonometric polynomials we provide precise asymptotics of $γ_n$ and $δ_n.$

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arXiv2012

Divergence of spectral decompositions of Hill operators with two exponential term potentials

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 ($bc$) with potentials of the form $$ v(x) = a e^{-2irx} + b e^{2isx}, \quad a, b \neq 0, r,s \in \mathbb{N}, r\neq s. $$ It is shown that the system of root functions does not contain a basis in $L^2 ([0,π], \mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic and $r, s$ are odd or $r=1$ and $s \geq 3. $

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arXiv2012

Bohr property of bases in the space of entire functions and its generalizations

We prove that if $(φ_n)_{n=0}^\infty, \; φ_0 \equiv 1, $ is a basis in the space of entire functions of $d$ complex variables, $d\geq 1,$ then for every compact $K\subset \mathbb{C}^d$ there is a compact $K_1 \supset K$ such that for every entire function $f= \sum_{n=0}^\infty f_n φ_n$ we have $\sum_{n=0}^\infty |f_n|\, \sup_{K}|φ_n| \leq \sup_{K_1} |f|.$ A similar assertion holds for bases in the space of global analytic functions on a Stein manifold with the Liouville Property.

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arXiv2012

Improved asymptotics of the spectral gap for the Mathieu operator

The Mathieu operator {equation*} L(y)=-y"+2a \cos{(2x)}y, \quad a\in \mathbb{C},\;a\neq 0, {equation*} considered with periodic or anti-periodic boundary conditions has, close to $n^2$ for large enough $n$, two periodic (if $n$ is even) or anti-periodic (if $n$ is odd) eigenvalues $λ_n^-$, $λ_n^+$. For fixed $a$, we show that {equation*} λ_n^+ - λ_n^-= \pm \frac{8(a/4)^n}{[(n-1)!]^2} [1 - \frac{a^2}{4n^3}+ O (\frac{1}{n^4})], \quad n\rightarrow\infty. {equation*} This result extends the asymptotic formula of Harrell-Avron-Simon, by providing more asymptotic terms.

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arXiv2011

Equiconvergence of spectral decompositions of Hill-Schrödinger operators

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. $

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arXiv2011

Riesz bases consisting of root functions of 1D Dirac operators

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 $L.$

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arXiv2011

Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

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 $bc.$

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arXiv2011

Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

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.

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arXiv2010

Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

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.

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arXiv2010

1D Dirac operators with special periodic potentials

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.

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arXiv2009

Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

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.

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arXiv2009

Spectral gaps of Schrödinger operators with periodic singular potentials

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 $v \in L^2_{loc}(\mathbb{R}).$

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arXiv2009

Bari-Markus property for Riesz projections of 1D periodic Dirac operators

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 $L^2.$

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arXiv2008

Bari-Markus property for Riesz projections of Hill operators with singular potentials

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.

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arXiv2008

Deviations of Riesz projections of Hill operators with singular potentials

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. $

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