Peter Yuditskii

Johannes Kepler University of Linz | JKU · Department of Dynamical Systems and Approximation Theory

In general, point spectrum of an almost periodic Jacobi matrix can depend on the element of the hull. In this paper, we study the hull of the limit-periodic Jacobi matrix corresponding to the equilibrium measure of the Julia set of the polynomial $z^2-\lambda$ with large enough $\lambda$; this is the leading model in inverse spectral theory of ergo...

In spectral theory, j-monotonic families of \(2\times 2\) matrix functions appear as transfer matrices of many one-dimensional operators. We present a general theory of such families, in the perspective of canonical systems in Arov gauge. This system resembles a continuum version of the Schur algorithm, and allows to restore an arbitrary Schur func...

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $$[-1,1]$$ [ - 1 , 1 ] if they are bounded by 1 on a subset of $$[-1,1]$$ [ - 1 , 1 ] of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomi...

We describe a program to construct a counterexample to the Deift conjecture, that is, an almost periodic function whose evolution under the KdV equation is not almost periodic in time. The approach is based on a dichotomy found by Volberg and Yuditskii in their solution of the Kotani problem, which states that there exists an analytic condition tha...

We consider canonical systems and investigate the Szegő class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the Arov gauge, in which we prove that the entropy integral is equal to an integral involving the coefficients of the ca...

We develop a comprehensive theory of reflectionless canonical systems with an arbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy Theorem property. This generalizes, to an infinite gap setting, the constructions of finite gap quasiperiodic (algebro-geometric) solutions of stationary integrable hierarchies. Instead of theta functions o...

In spectral theory, $j$-monotonic families of $2\times 2$ matrix functions appear as transfer matrices of many one-dimensional operators. We present a general theory of such families, in the perspective of canonical systems in Arov gauge. This system resembles a continuum version of the Schur algorithm, and allows to restore an arbitrary Schur func...

Let an algebraic polynomial Pn(ζ) of degree n be such that |Pn(ζ)|⩽1 for ζ∈E⊂T and |E|⩾2π-s. We prove the sharp Remez inequality supζ∈T|Pn(ζ)|⩽Tnsecs4,where Tn is the Chebyshev polynomial of degree n. The equality holds if and only if Pn(eiz)=ei(nz/2+c1)Tnsecs4cosz-c02,c0,c1∈R.This gives the solution of the long-standing problem on the sharp consta...

We establish exact conditions for non triviality of all subspaces of the standard Hardy space in the upper half plane, that consist of the character automorphic functions with respect to the action of a discrete subgroup of \(SL_2({\mathbb {R}})\). Such spaces are the natural objects in the context of the spectral theory of almost periodic differen...

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $[-1,1]$ if they are bounded by $1$ on a subset of $[-1,1]$ of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if t...

We consider canonical systems and investigate the Szeg\H{o} class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the Arov gauge, in which we prove that the entropy integral is equal to an integral involving the coefficients of th...

Reflectionless operators in one dimension are particularly amenable to inverse scattering and are intimately related to integrable systems like KdV and Toda. Recent work has indicated a strong (but not equivalent) relationship between reflectionless operators and almost periodic potentials with absolutely continuous spectrum. This makes the realm o...

We derive Fourier integral associated to the complex Martin function in the Denjoy domain of Widom type with the Direct Cauchy Theorem (DCT). As an application we study reflectionless Weyl-Titchmarsh functions in such domains, related to them canonical systems and transfer matrices. The DCT property appears to be crucial in many aspects of the unde...

We establish exact conditions for non triviality of all subspaces of the standard Hardy space in the upper half plane, that consist of character automorphic functions with respect to the action of a discreet subgroup of $SL_2(\mathbb R)$. Such spaces are natural objects in the context of the spectral theory of almost periodic differential operators...

Let an algebraic polynomial $P_n(\zeta)$ of degree $n$ be such that $|P_n(\zeta)|\le 1$ for $\zeta\in E\subset\mathbb{T}$ and $|E|\ge 2\pi -s$. We prove the sharp Remez inequality $$ \sup_{\zeta\in\mathbb{T}}|P_n(\zeta)|\le \mathfrak{T}_{n}\left(\sec \frac{s} 4\right),$$ where $\mathfrak{T}_{n}$ is the Chebyshev polynomial of degree $n$. The equali...

Выдвигается гипотеза, что асимптотика многочленов Чебышeва в области на комплексной плоскости может быть найдена в терминах воспроизводящих ядер подходящего гильбертова пространства аналитических функций в этой области. Гипотеза основана на двух классических результатах П. Р. Гарабедяна и Г. Видома. Для подтверждения этой гипотезы изучается асимпто...

We establish precise spectral criteria for potential functions $V$ of reflectionless Schr\"odinger operators $L_V = -\partial_x^2 + V$ to admit solutions to the Korteweg de-Vries (KdV) hierarchy with $V$ as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Rie...

We prove Szeg\H{o}-Widom asymptotics for the Chebyshev polynomials of a compact subset of $\mathbb{R}$ which is regular for potential theory and obeys the Parreau-Widom and DCT conditions.

We present a conjecture that the asymptotics for Chebyshev polynomials in a complex domain can be given in terms of the reproducing kernels of a suitable Hilbert space of analytic functions in this domain. It is based on two classical results due to Garabedian and Widom. To support this conjecture we study the asymptotics for Ahlfors extremal polyn...

One of the first theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class, which destroys completely the absolutely continuous spectrum of A (von Neumann). However, if A is the discrete free 1-D Schr\"odinger operator and B is a Jacobi matrix the a.c. spectrum remain...

We give a free parametric representation for the coefficient sequences of Jacobi matrices whose spectral measures satisfy the Killip–Simon condition with respect to two (arbitrary) disjoint intervals. This parametrization is given by means of the Jacobi flow on SMP matrices, which we introduce here.

One of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class with arbitrary small operator norm, which destroys completely the absolutely continuos (a.c.) spectrum of the initial operator A (von Neumann). However, if A is the...

Functions of bounded characteristic in simply connected domains have a classical factorization to Blaschke, outer and singular inner parts. The latter has a singular measure on the boundary assigned to it. The exponential speed of change of a function when approaching a point of a boundary (mean type) corresponds to a point mass at this point. In t...

The Kotani-Last conjecture states that every ergodic operator in one space dimension with non-empty absolutely continuous spectrum must have almost periodic coefficients. This statement makes sense in a variety of settings; for example, discrete Schr\"odinger operators, Jacobi matrices, CMV matrices, and continuum Schr\"odinger operators. In the ma...

The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices.

We generalize the Korkin-Zolotarev theorem to the case of entire functions having the smallest L^1 norm on a system of intervals E. If C@?E is a domain of Widom type with the Direct Cauchy Theorem, we give an explicit formula for the minimal deviation. Important relations between the problem and the theory of canonical systems with reflectionless r...

In 1969 Harold Widom published his seminal paper, which gave a complete description of orthogonal and Chebyshev polynomials on a system of smooth Jordan curves. When there were Jordan arcs present the theory of orthogonal polynomials turned out to be just the same, but for Chebyshev polynomials Widom's approach proved only an upper estimate, which...

This is the second part of the paper arXiv:1309.0959v2 on the theory of SMP (Strong Moment Problem) matrices and their relation to the Killip-Simon problem on two disjoint intervals. In this part we define and study the Jacobi flow on SMP matrices.

Jacobi matrices probably are the most classical object in spectral theory, while CMV matrices are a comparably fresh one, although they are related to a very classical topic, namely to orhtogonal polynomials on the unit circle (in the same way as Jacobi matrices are related to orhogonal polynomials on the real axis). We will discuss the third membe...

It is a small theory of non almost periodic ergodic families of Jacobi matrices with pure (however) absolutely continuous spectrum. And the reason why this effect may happen: under our "axioms" we found an analytic condition on the resolvent set that is responsible for (exactly equivalent to) this effect.

We generalized the Korkin-Zolotarev theorem to the case of entire functions having the smallest $L^1$ norm on a system of intervals $E$. If $\bbC\setminus E$ is a domain of Widom type with the Direct Cauchy Theorem we give an explicit formula for the minimal deviation. Important relations between the problem and the theory of canonical systems with...

The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices.

We discuss a class of regions and conformal mappings which are useful in several problems of approximation theory, harmonic analysis and spectral theory.

We describe polynomials of the best uniform approximation to sgn(x) on the union of two intervals in terms of special conformal mappings. This permits us to find the exact asymptotic behavior of the error of this approximation.

We discuss several questions which remained open in our joint work with M. Sodin "Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character--automorphic functions". In particular, we show that there exists a non-homogeneous set $E$ such that the Direct Cauchy Theorem (DCT) holds...

We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first c...

We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in ℂ 2 by polynomials of lower degree with complex coefficients. We reduce the problem to a one-dimensional weighted minimization problem on an interval. In a sense, the corresponding extremal polynomials are uniform counterparts of the classic...

We develop a scattering theory for CMV matrices, similar to the Faddeev--Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for the uniqueness, which are connected with the Helson--Szeg\H o and the Strong Szeg\H o theorems....

We give a complete solution of the scattering problem for Jacobi matrices from a class which was recently introduced by E. Ryckman. We characterize the scattering data for this class and illustrate the inverse scattering on some simple examples.

The direct and inverse Geronimus relations between Verblunsky parameters of measures on the unit circle and Jacobi parameters of their Szego{double acute} transforms have been used to prove that Guseinov's class of Jacobi parameters, is in a canonical correspondence with the followingclass of Verblunsky parameters αn → 0. © L. Golinskii, A. Kheifet...

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of |x|p on two symmetric intervals. Note that, in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the b...

We develop a modern extended scattering theory for CMV matrices with asymptotically constant Verblunsky coefficients. We show that the traditional (Faddeev–Marchenko) condition is too restrictive to define the class of CMV matrices for which there exists a unique scattering representation. The main results are: (1) the class of twosided CMV matrice...

B. Simon proved the existence of the wave operators for the CMV matrices with Szego class Verblunsky coefficients, and therefore the existence of the scattering function. Generally, there is no hope to restore a CMV matrix when we start from the scattering function, in particular, because it does not contain any information about the (possible) sin...

Let f be an entire function of the exponential type, such that the indicator diagram is in [−iσ,iσ], σ>0. Then the upper density of f is bounded by cσ, where c≈1.508879 is the unique solution of the equationlog(c2+1+c)=1+c−2. This bound is optimal. To cite this article: A. Eremenko, P. Yuditskii, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

We find the exact upper estimate for the upper density of zeros of entire functions of exponential type whose indicator diagram is contained in a given interval.

In 2002 C. Berg, Y. Chen, and M. Ismail found a nice relation between the determinancy of the Hamburger moment problem and asymptotic behavior of the smallest eigenvalues of the corresponding Hankel matrices. We investigate whether an analog of this statement holds for the Nevanlinna--Pick interpolation problem.

We consider Nehari's problem in the case of non-uniqueness of solution. The solution set is then parametrized by the unit ball of $H^{\infty}$ by means of so-called {\em regular generators} -- bounded holomorphic functions $\phi$. The definition of {\em regularity} is given below, but let us mention now that 1) the following assumption on modulus o...

We construct mesures supported on a compact subset E of the real line having zero principal value of their Cauchy integral a.e. on E with respect to Lebesgue measure and having singular components. E is sufficiently regular (Widom property is satisfied) but not homogeneous as for homogeneous spectrum such construction is impossible. This impossibil...

Our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity...

Adamjan-Arov (Lax--Phillips) model space is considered as a scattering representation space for a CMV matrix in context of an extended Marchenko--Faddeev scattering theory. That is, there exists a basis in which the multiplication by independent variable is a CMV matrix. This basis as well as Verblunski coefficients are computed explicitly in terms...

The algebraic structure of V.P. Potapov's Fundamental Matrix Inequality (FMI) is discussed and its interpolation meaning is analyzed. Functional model spaces are involved. A general Abstract Interpolation Problem is formulated which seems to cover all the classical and recent problems in the field and the solution set of this problem is described u...

We study the best uniform approximation by polynomials of fixed degree of the function sgn(x) on the union of two intervals symmetric with respect to the origin. We obtain precise asymptotics, with explicit constants, for the error of the best approximation as the degrees tend to infinity. Our approach is based on a new representation of the extrem...

We are going to prove a Lipschitz property of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron–Frobenius–Ruelle operators associated to hyperbolic polynomial dynamics. This Lipschitz estimate will not depend on the dimension of the Jacobi matrix. It is obtained using some sufficient cond...

Minor modifications are given to prove the Main Theorem under the Blaschke (instead of Carleson) condition as well as a small historical comment. Because of the reference [1] on our paper [4] a certain historical comment is needed. In [4] we generalized H. Widom’s Theorem [5] based on an absolutely new idea, dealing with one dimensional perturbatio...

We study the correspondence between almost periodic difference operators and algebraic curves (spectral surfaces). An especial role plays the parametrization of the spectral curves in terms of, so called, branching divisors. The multiplication operator by the covering map with respect to the natural basis in the Hardy space on the surface is the $2...

The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal polynomials for which the Koosis theorem seems to be the most natural tool. Namely, we consider the case when a Szeg\...

First, we give a simple proof of a remarkable result due to Videnskii and Shirokov: let B be a Blaschke product with n zeros; then there exists an outer function φ, φ(0) = 1, such that ‖(Bφ)′‖ ⩽ Cn, where C is an absolute constant. Then we apply this result to a certain problem of finding the asymptotics of orthogonal polynomials.

For $a\in (0,1)$ let $L^k_m(a)$ be the error of the best approximation of the function $\sgn(x)$ on the two symmetric intervals $[-1,-a]\cup[a,1]$ by rational functions with the only possible poles of degree $2k-1$ at the origin and of $2m-1$ at infinity. Then the following limit exists \begin{equation} \lim_{m\to \infty}L^k_m(a)(\frac{1+a}{1-a})^{...

We prove that one-dimensional reflectionless Schrödinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class includes all earlier examples of reflectionless almost periodic Schrödinger operators. In addition, we construct...

Abstract We give an explicit parametrization of a set of almost periodic CMV matrices whose,spectrum (is equal to the absolute continuous spectrum and) is a homogenous set E lying on the unit circle, for instance a Cantor set of positive Lebesgue measure. First to every operator of this set we associate a function from a certain subclass of the Sch...

We prove that one-dimensional reflectionless Schr\"odinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class includes all earlier examples of reflectionless almost periodic Schr\"odinger operators. In addition, we constr...

For all hyperbolic polynomials we proved in [J. Funct. Anal. 246, No. 1, 1–30 (2007; Zbl 1125.47023), see the following review] a Lipschitz estimate of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron–Frobenius–Ruelle operators associated to hyperbolic polynomial dynamics (with real Juli...

An original approach to the inverse scattering for Jacobi matrices was suggested in a recent paper by Volberg-Yuditskii. The authors considered quite sophisticated spectral sets (including Cantor sets of positive Lebesgue measure), however they did not take into account the mass point spectrum. This paper follows similar lines for the continuous se...

In recent works we considered an asymptotic problem for orthogonal polynomials when a Szegö measure on the unit circumference is perturbed by an arbitrary Blaschke sequence of point masses outside the unit disk. In the current work we consider a similar problem in the scattering setting. The goal of this work is to consider a new asymptotic problem...

First we give here a simple proof of a remarkable result of Videnskii and Shirokov: let $B$ be a Blaschke product with $n$ zeros, then there exists an outer function $\phi, \phi(0)=1$, such that $\|(B\phi)'\| \leq C n$, where $C$ is an absolute constant. Then we apply this result to a certain problem of finding the asymptotic of orthogonal polynomi...

We consider the Jacobi matrix generated by a balanced measure of hyperbolic polynomial map. The conjecture of Bellissard says that this matrix should have an extremely strong periodicity property. We show how this conjecture is related to a certain noncommutative version of Bowen--Ruelle theory, and how the two weight Hilbert transform naturally ap...

We present a point of view on results of the paper of Geronimo and Johnson [Comm. Math, Phys. 193 (1998)] that allow infinitely dimensional generalization up to the case when spectrum is supported on a Cantor set of positive Lebesgue measure.

We prove a partial result concerning the long-standing problem on limit periodicity of the Jacobi matrix associated with the balanced measure on the Julia set of an expending polynomial. Besides this, connections of the problem with the Faybusovich--Gekhtman flow and many other objects (the Hilbert transform, the Schwarz derivative, the Ruelle and...

This work is in a stream initiated by a paper of Killip and Simon (2003). Using methods of functional analysis and the classical Szegö theorem we prove sum rule identities in a very general form. Then, we apply the result to obtain new asymptotics for orthonormal polynomials.

We study the correspondence between almost periodic difference operators and algebraic curves (spectral surfaces). An especial role plays the parametrization of the spectral curves in terms of, so-called, branching divisors. The multiplication operator by the covering map with respect to the natural basis in the Hardy space on the surface is the 2d...

A lambda-Hankel operator X is a bounded operator on Hilbert space satisfying the operator equation S*X - XS = lambdaX, where S is the (unilateral) forward shift and S* is its adjoint. We prove that there are non-compact lambda-Hankel operators for lambda a complex number of modulus less than 2, by first exhibiting a way to obtain bounded solutions...

Let $E$ be a homogeneous compact set, for instance a Cantor set of positive length. Further let $\sigma$ be a positive measure with $\text{supp}(\sigma)=E$. Under the condition that the absolutely continuous part of $\sigma$ satisfies a Szeg\"o--type condition we give an asymptotic representation, on and off the support, for the polynomials orthono...

We give a simple example of non-uniqueness in the inverse scattering for Jacobi matrices: roughly speaking $S$-matrix is analytic. Then, multiplying a reflection coefficient by an inner function, we repair this matrix in such a way that it does uniquely determine a Jacobi matrix of Szeg\"o class; on the other hand the transmission coefficient remai...

A λ-Hankel operator X is a bounded operator on Hilbert space satisfying the operator equation S*X - XS = λX, where S is the (unilateral) forward shift and S* is its adjoint. We prove that there are non-compact λ-Hankel operators for λ a complex number of modulus less than 2, by first exhibiting a way to obtain bounded solutions to the above equatio...

We give a new proof of a special case of de Branges' theorem on the inverse monodromy problem: when an associated Riemann surface is of Widom type with Direct Cauchy Theorem. The proof is based on our previous result (with M.Sodin) on infinite dimensional Jacobi inversion and on Levin's uniqueness theorem for conformal maps onto comb-like domains....

When solving the inverse scattering problem for a discrete Sturm–Liouville operator with a rapidly decreasing potential, one gets reflection coefficients s ± and invertible operators \(\), where \(\) is the Hankel operator related to the symbol s ±. The Marchenko–Faddeev theorem [8] (in the continuous case, for the discrete case see [4, 6]), guaran...

Let be a positive measure whose support is an interval E plus a denumerable set of mass points which accumulate at the boundary points of E only. Under the assumptions that the mass points satisfy Blaschke's condi- tion and that the absolutely continuous part of satises Szego's condition, asymptotics for the orthonormal polynomials on and o the sup...

For groups of Widom type we prove that the direct integral of the spaces of character automorphic forms of weight 1 with the Haar measure on the group of characters is the space of automorphic forms of weight 1 with respect to commutator of the given group. It gives us an explanation why the whole set of ergodic Jacobi matrices with a fixed homogen...

All three subjects reflected in the title are closely intertwined in the paper. LetJ E be a class of Jacobi matrices acting inl 2(ℤ) with a homogeneous spectrumE (see Definition 3.2) and with diagonal elements of the resolventR(m, m; z) having pure imaginary boundary values a.e. onE. For this class, we extend fundamental results pertaining to the...

A non-uniqueness criterion for the character-automorphic Nehari problem is given. Certain subclass of solutions, connected with “the entropy functional” of the problem, is described. The description yields a character-automorphic counterpart of the Adamyan-Arov-Krein theorem.

Translated from: Operators in Function Spaces and Problems in Function Theory, Kiev 1987, 83-96 (1987; Zbl 0703.41005). The algebraic structure of V. P. Potapov’s Fundamental Matrix Inequality (FMI) is discussed and its interpolation meaning is analyzed. Functional model spaces are involved. A general abstract interpolation problem is formulated wh...

In this paper we generalize some results of M.Abrahamse [1]. M.Abrahamse has given a solvability criterion for the Nevanlinna – Pick problem in multiply connected domains. Almost simultaneously J.Ball [2] has established the operator lifting theorem for domains of this kind. Now these topics are intensively investigated by S.Fedorov and V.Vinnikov....

Being based on the infinite dimensional Jacobi inversion found earlier, we establish the direct generalization of the well-known properties of finite-band Sturm-Liouville operators in the case of operators with a homogeneous and, generally speaking, Cantor-type spectrum, and with pseudocontinuable Weyl functions. In our investigations the group of...

Without Abstract

LetR be a rational function with nonempty set of normality that consists of basins of attraction only and let ( LQ g )( z ) = åR( w ) = z Q( w )g( w )\left( {L_Q g} \right)\left( z \right) = \sum\limits_{R\left( w \right) = z} {Q\left( w \right)g\left( w \right)} be a Ruelle operator with a rational weightQ which acts in a space of locally analy...

There exist several approaches to the generalization and unification of various problems of the Nevanlinna-Pick type [1–4, 5, 9, 10, 12, 14, 28, 29, 31, 35, 38, 40].

Let R be an expanding rational function with a real bounded Julia set, and let {Mathematical expression} be a Ruelle operator acting in a space of functions analytic in a neighbourhood of the Julia set. We obtain explicit expressions for the resolvent function {Mathematical expression} and, in particular, for the Fredholm determinant D(λ)=det(I-λL)...