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23
Nonselfadjoint differential operators
, 2002
"... We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely ..."
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Cited by 40 (6 self)
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We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator.
Spectra, Pseudospectra, and Localization for Random Bidiagonal Matrices
 Comm. Pure Appl. Math
"... There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the correspon ..."
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Cited by 16 (4 self)
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There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the nonperiodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the in nitedimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of nite bidiagonal matrices, in nite bidiagonal matrices (\stochastic Toeplitz operators"), nite periodic matrices, and doubly in nite bidiagonal matrices (\stochastic Laurent operators").
Wave Packet Pseudomodes of Twisted Toeplitz Matrices
, 2004
"... The pseudospectra of banded, nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered. Such matrices are characterized by a symbol that depends on both position (x) and wave number (k). It is shown that when a certain winding number or twist condition is satisfied, relate ..."
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Cited by 12 (1 self)
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The pseudospectra of banded, nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered. Such matrices are characterized by a symbol that depends on both position (x) and wave number (k). It is shown that when a certain winding number or twist condition is satisfied, related to Hörmander’s commutator condition for partial differential equations, εpseudoeigenvectors of such matrices for exponentially small values of ε exist in the form of localized wave packets. The symbol need not be smooth with respect to x, just differentiable at a point (or less).
Eigenvalue curves of asymmetric tridiagonal random matrices
 ELECTRONIC JOURNAL OF PROBABILITY
, 2000
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The Thouless formula for random nonHermitian Jacobi matrices
 Israel J. Math
, 2005
"... Random nonHermitian Jacobi matrices Jn of increasing dimension n are considered. We prove that the normalized eigenvalue counting measure of Jn converges weakly to a limiting measure µ as n → ∞. We also extend to the nonHermitian case the Thouless formula relating µ and the Lyapunov exponent of th ..."
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Cited by 11 (0 self)
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Random nonHermitian Jacobi matrices Jn of increasing dimension n are considered. We prove that the normalized eigenvalue counting measure of Jn converges weakly to a limiting measure µ as n → ∞. We also extend to the nonHermitian case the Thouless formula relating µ and the Lyapunov exponent of the secondorder difference equation associated with the sequence Jn. The measure µ is shown to be logHölder continuous. 1
Timelike boundary liouville theory
"... The timelike boundary Liouville (TBL) conformal field theory consisting of a negative norm boson with an exponential boundary interaction is considered. TBL and its close cousin, a positive norm boson with a nonhermitian boundary interaction, arise in the description of the c = 1 accumulation point ..."
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Cited by 9 (1 self)
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The timelike boundary Liouville (TBL) conformal field theory consisting of a negative norm boson with an exponential boundary interaction is considered. TBL and its close cousin, a positive norm boson with a nonhermitian boundary interaction, arise in the description of the c = 1 accumulation point of c < 1 minimal models, as the worldsheet description of open string tachyon condensation in string theory and in scaling limits of superconductors with line defects. Bulk correlators are shown to be exactly soluble. In contrast, due to OPE singularities near the boundary interaction, the computation of boundary correlators is a challenging problem which we address but do not fully solve. Analytic continuation from the known correlators of spatial boundary Liouville to TBL encounters an infinite accumulation of poles and zeros. A particular contour prescription is proposed which cancels the poles against the zeros in the boundary correlator d(ω) of two operators of weight ω 2 and yields a finite result. A general relation is proposed between twopoint CFT correlators and stringy Bogolubov coefficients, according to which the magnitude of d(ω) determines the rate of open string pair creation during tachyon
Spectral properties of random nonselfadjoint matrices and operators
, 2001
"... We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also desc ..."
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Cited by 9 (4 self)
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We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the nonselfadjoint Anderson model changes suddenly as one passes to the infinite volume limit.
On the eigenproblems of PTsymmetric oscillators
 J. Math. Phys
, 2001
"... Abstract. We consider the nonHermitian Hamiltonian H = − d2 dx 2 + P(x 2) − (ix) 2n+1 on the real line, where P(x) is a polynomial of degree at most n ≥ 1 with all nonnegative real coefficients (possibly P ≡ 0). It is proved that the eigenvalues λ must be in the sector  argλ  ≤ π d2 2n+3. Also ..."
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Cited by 7 (4 self)
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Abstract. We consider the nonHermitian Hamiltonian H = − d2 dx 2 + P(x 2) − (ix) 2n+1 on the real line, where P(x) is a polynomial of degree at most n ≥ 1 with all nonnegative real coefficients (possibly P ≡ 0). It is proved that the eigenvalues λ must be in the sector  argλ  ≤ π d2 2n+3. Also for the case H = −dx2 − (ix) 3, we establish a zerofree region of the eigenfunction u and its derivative u ′ and we find some other interesting properties of eigenfunctions. We are considering the eigenproblem Preprint. 1.
Spectral theory of pseudoergodic operators
 Commun. Math. Phys
, 2001
"... We define a class of pseudoergodic nonselfadjoint Schrödinger operators acting in spaces l 2 (X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a nonselfadjoint Anderson model acting on l 2 (Z), and find the precise condition for 0 t ..."
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Cited by 7 (3 self)
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We define a class of pseudoergodic nonselfadjoint Schrödinger operators acting in spaces l 2 (X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a nonselfadjoint Anderson model acting on l 2 (Z), and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for such operators.
On the Spectra and Pseudospectra of a Class of NonSelfAdjoint Random Matrices and Operators
, 2011
"... In this paper we develop and apply methods for the spectral analysis of nonselfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudoergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). ..."
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Cited by 7 (4 self)
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In this paper we develop and apply methods for the spectral analysis of nonselfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudoergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). As a major application to illustrate our methods we focus on the “hopping sign model” introduced by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433–6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ±1’s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semiinfinite and biinfinite matrix cases, for example showing that the numerical range and pnorm εpseudospectra (ε> 0, p ∈ [1, ∞]) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum Σ. We also propose a sequence of inclusion sets for Σ which we show is convergent to Σ, with the nth element of the sequence computable by calculating smallest singular values of (large numbers of) n × n matrices. We propose similar convergent approximations for the 2norm εpseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.