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Let H be a Hilbert space. Suppose A is a positive self-adjoint operator on H and φ ∈ H is a cyclic unit vector. For each λ ∈ R , we can define the rank one perturbation of A by A λ = A + λ 〈 φ , ⋅ 〉 φ . To each A λ we can consider the spectral measure of φ, which we denote by μ λ . This generates a family of measures, { μ λ } , and we analyze the packing dimension of this family. Past results have determined that the Hausdorff dimension of this family can be determined if the limit inferior of a ratio involving μ is constant on a Lebesgue typical set. This ratio is sometimes called the pointwise dimension of μ and is related to the upper derivative of μ. Work has been done to make a similar argument for the packing dimension, but with little success. Using the theory of rank one perturbations and Borel transforms, we introduce the concept of Lebesgue exact dimension for μ, which allows us to determine the packing dimension of spectral measures of almost every rank one perturbation μ λ . If the Lebesgue exact dimension for μ is 1 α 2 then the packing dimension of Lebesgue almost every μ λ is 2 − α . As a corollary, we find that this limit condition implies a stronger result: the Hausdorff and packing dimensions are equal for almost every μ λ .

We describe the general form of a Hermitian operator on the injective tensor product of a uniform algebra and a complex Banach space. As an application, we characterize a surjective unital isometry between the injective tensor product of a uniform algebra and a unital factor C ⁎ -algebra. In particular, we give a simple proof that a unital factor C ⁎ -algebra has the Banach–Stone property.

A famous theorem of Coburn says that a nonzero Toeplitz operator on the Hardy space of the unit disk is injective or its adjoint operator is injective. In this paper we study the corresponding problem on the Hardy space of the bidisk. We first show the Coburn type theorem fails generally on the bidisk. But, we show that certain pluriharmonic symbols or product symbols of one variable functions induce Toeplitz operators satisfying the Coburn type theorem.

In this paper, a discontinuous non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition, and with two singular endpoints is studied. The interface conditions are imposed on the discontinuous point. Firstly, we pass the considered problem to a maximal dissipative operator L h by using operator theoretic formulation. The self-adjoint dilation T h of L h in the space H is constructed, furthermore, the incoming and outgoing representations of T h and functional model are also constructed, hence in light of Lax–Phillips theory, we derive the scattering matrix. Using the equivalence between scattering matrix and characteristic function, completeness theorem on the eigenvectors and associated vectors of this dissipative operator is proved.

In this paper we characterize a class of normal (isometric, coisometric, unitary, hyponormal) singular Cauchy integral operators with operator-valued (or matrix-valued) symbols on vector-valued (or C n -valued) L 2 spaces.

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator A in L 2 ( − 1 , 1 ) which has the Legendre polynomials { P n } n = 0 ∞ as eigenfunctions. As a consequence, they explicitly determined the domain D ( A 2 ) of the self-adjoint operator A 2 . However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each end point x = ± 1 in L 2 ( − 1 , 1 ) so D ( A 2 ) should exhibit four boundary conditions. In this paper, we show that this domain can, in fact, be expressed using four separated boundary conditions using the classical GKN (Glazman–Krein–Naimark) theory. In addition, we determine a new characterization of D ( A 2 ) that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple – and natural – and are equivalent to the boundary conditions obtained from the GKN theory.

Let L be a non-negative self-adjoint operator acting on L 2 ( R n ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A p weight on R n × R n , 1 p ∞ . In this article we obtain a weighted q-atomic decomposition with q ≥ p for the weighted Hardy space H L , w 1 ( R n × R n ) associated to L.

We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates. These results extend, and improve, classical perturbation results by Kato and by Gohberg/Krein. Further, we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator; in the latter case, we can even allow for perturbations with relative bound $\ge 1$. The generality of our results is illustrated by several applications, massive and massless Dirac operators, point-coupled periodic systems, and two-channel Hamiltonians with dissipation., Comment: 25 pages, 3 figures

A celebrated theorem of Coburn asserts that, on the setting of the Hardy space, if a Toeplitz operator is nonzero, then either it is one-to-one or its adjoint operator is one-to-one. In this paper, we show that an analogous result holds for Toeplitz operators acting on the Dirichlet space.

Let ( X , ρ ) be a locally compact metric space endowed with a doubling measure μ, and L be a non-negative self-adjoint operator on L 2 ( X , d μ ) . Assume that the semigroup P t = e − t L generated by L consists of integral operators with (heat) kernel p t ( x , y ) enjoying Gaussian upper bound but having no information on the regularity in the variables x and y. In this paper, we introduce Besov and Triebel–Lizorkin spaces associated with L , and present an atomic decomposition of these function spaces.

For autonomous linear integral equations, we establish an explicit asymptotic representation formula of solutions, developing the spectral analysis for the generator of the solution semigroup as well as the one for the formal adjoint operator associated with the generator.

This paper studies resolvent convergence and spectral approximations of sequences of self-adjoint subspaces (relations) in complex Hilbert spaces. Concepts of strong resolvent convergence, norm resolvent convergence, spectral inclusion, and spectral exactness are introduced. Fundamental properties of resolvents of subspaces are studied. By applying these properties, several equivalent and sufficient conditions for convergence of sequences of self-adjoint subspaces in the strong and norm resolvent senses are given. It is shown that a sequence of self-adjoint subspaces is spectrally inclusive under the strong resolvent convergence and spectrally exact under the norm resolvent convergence. A sufficient condition is given for spectral exactness of a sequence of self-adjoint subspaces in an open interval lacking essential spectral points. In addition, criteria are established for spectral inclusion and spectral exactness of a sequence of self-adjoint subspaces that are defined on proper closed subspaces.

Everitt and Markus characterized the domains of self-adjoint operator realizations of very general even and odd order symmetric ordinary differential equations in terms of Lagrangian subspaces of symplectic spaces. Recently, for the even order case with real coefficients, Wang, Sun and Zettl constructed limit-circle (LC) solutions and Hao, Wang, Sun and Zettl characterized the self-adjoint domains in terms of LC solutions. These LC solutions are higher order analogues of the celebrated Titchmarsh–Weyl limit-circle solutions in the second-order case. This LC characterization has been used to obtain information about the discrete, continuous, and essential spectra of these operators. In this paper we investigate the connection between these two very different kinds of characterizations and thus add the methods of symplectic geometry to the techniques available for the investigation of the spectrum of self-adjoint operators.

We consider a class of self-adjoint extensions using the boundary triple technique. Assuming that the associated Weyl function has the special form $M(z)=\big(m(z)\Id-T\big) n(z)^{-1}$ with a bounded self-adjoint operator $T$ and scalar functions $m,n$ we show that there exists a class of boundary conditions such that the spectral problem for the associated self-adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for $T$. As a motivating example we consider differential operators on equilateral metric graphs, and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete laplacians., Comment: 19 pages

In this paper, we propose a class of stochastic heat equations with first order fractional noises. We define a first order noise through the adjoint operator of the first order operator, where the operation of the stochastic integral can be avoided. In this framework, the existence and uniqueness of the solution of the equation will be established. Further, we give the regularity of the solution. Finally, we model the term structure of forward rate with the solutions and give the conditions under which the market is arbitrary-free.

In this paper, non-self-adjoint Sturm–Liouville operators in Weyl’s limit-circle case are studied. We first determine all the non-self-adjoint boundary conditions yielding dissipative operators for each allowed Sturm–Liouville differential expression. Then, using the characteristic determinant, the completeness of the system of eigenfunctions and associated functions for these dissipative operators is proved.

Let X be a metric space with doubling measure, and L be a non-negative self-adjoint operator such that the corresponding heat kernels satisfy Gaussian bounds. This article is concerned with some weighted norm inequalities for area integrals associated with L . As an application, we obtain sharp estimates for the operator norm of the area integrals on L p ( X ) as p becomes large.

A theorem that is of aid in computing the domain of the adjoint operator is provided. It may serve e.g. as a criterion for selfadjointness of a symmetric operator, for normality of a formally normal operator or for H -selfadjointness of an H -symmetric operator. Differential operators and operators given by an infinite matrix are considered as examples.

Asymptotic and oscillatory behaviours near x = 0 of all solutions y = y ( x ) of self-adjoint linear differential equation ( P p q ) : ( p y ′ ) ′ + q y = 0 on ( 0 , T ] , will be studied, where p = p ( x ) and q = q ( x ) satisfy the so-called Hartman–Wintner type condition. We show that the oscillatory behaviour near x = 0 of ( P p q ) is characterised by the nonintegrability of q / p on ( 0 , T ) . Moreover, under this condition, we show that the rectifiable (resp. unrectifiable) oscillations near x = 0 of ( P p q ) are characterised by the integrability (resp. nonintegrability) of q / p 3 4 on ( 0 , T ) . Next, some invariant properties of rectifiable oscillations in respect to the Liouville transformation are proved. Also, Sturmʼs comparison type theorem for the rectifiable oscillations is stated. Furthermore, previous results are used to establish such kind of oscillations for damped linear second-order differential equation y ″ + g ( x ) y ′ + f ( x ) y = 0 , and especially, the Bessel type damped linear differential equations are considered. Finally, some open questions are posed for the further study on this subject.

We define and examine certain matrix-valued multiplicative functionals with local Kato potential terms and use probabilistic techniques to prove that the semigroups of the corresponding partial differential operators with matrix-valued coefficients are spatially continuous and have a jointly continuous integral kernel. These partial differential operators include Yang-Mills type Hamiltonians and Pauli type Hamiltonians, with "electrical" potentials that are elements of the matrix-valued local Kato class., 26 pages; will appear in a slightly different form in the Journal of Mathematical Analysis and Applications

In the present paper we investigate the set ΣJ of all J-self-adjoint extensions of an operator S which is symmetric in a Hilbert space H with deficiency indices 〈2,2〉 and which commutes with a non-trivial fundamental symmetry J of a Krein space (H,[⋅,⋅]),SJ=JS. Our aim is to describe different types of J-self-adjoint extensions of S, which, in general, are non-self-adjoint operators in the Hilbert space H. One of our main results is the equivalence between the presence of J-self-adjoint extensions of S with empty resolvent set and the commutation of S with a Clifford algebra Cl2(J,R), where R is an additional fundamental symmetry with JR=−RJ. This enables one to parameterize in terms of Cl2(J,R) the set of all J-self-adjoint extensions of S with stable C-symmetry. Here an extension has stable C-symmetry if it commutes with a fundamental symmetry and, in turn, this fundamental symmetry commutes with S. Such a situation occurs naturally in many applications, here we discuss the case of indefinite Sturm–Liouville operators and the case of a one-dimensional Dirac operator with point interaction.

This paper uses commutator equations to study the absolute continuity of spectral measures associated with certain subclasses of unbounded self-adjoint Jacobi matrix operators determined by properties of the diagonal and subdiagonal sequences. If the diagonal sequence is the zero sequence, properties of the difference sequence of the subdiagonal determine the choice of a bounded operator for the commutator equation. The structure of the resulting commutator leads to results on absolute continuity.

Motivated by energy space representation of Dirac operators, in the sense of K. Friedrichs, we recently introduced the notion of closely embedded Krein spaces. These spaces are associated to unbounded selfadjoint operators that play the role of kernel operators, in the sense of L Schwartz, and they are special representations of induced Krein spaces. In this article we present a canonical representation of closely embedded Krein spaces in terms of a generalization of the notion of operator range and obtain a characterization of uniqueness. When applied to Dirac operators, the results differ according to a mass or a massless particle in a dramatic way: in the case of a particle with a nontrivial mass we obtain a dual of a Sobolev type space and we have uniqueness, while in the case of a massless particle we obtain a dual of a homogenous Sobolev type space and we lose uniqueness. (C) 2010 Elsevier Inc. All rights reserved.

We study non-self-adjoint Hamiltonian systems on Sturmian time scales, defining Weyl–Sims sets, which replace the classical Weyl circles, and a matrix-valued M-function on suitable cone-shaped domains in the complex plane. Furthermore, we characterize realizations of the corresponding dynamic operator and its adjoint, and construct their resolvents. Even-order scalar equations and the Orr–Sommerfeld equation on time scales are given as examples illustrating the theory, which are new even for difference equations. These results unify previous discrete and continuous theories to dynamic equations on Sturmian time scales.

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on l 2 is skew. We show its domain is a proper subset of the domain of its adjoint S ∗ , and − S ∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L 2 [ 0 , 1 ] . We compare the domain of T with the domain of its adjoint T ∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps–Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.

Let H be a separable Hilbert space and B sa ( H ) the set of all bounded linear self-adjoint operators. We say that A , B ∈ B sa ( H ) quasi-commute if there exists a nonzero ξ ∈ C such that A B = ξ B A . Bijective maps on B sa ( H ) which preserve quasi-commutativity in both directions are classified.

We prove upper and lower bounds on the eigenvalues and discuss their asymptotic behaviour (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lyapounov–Schmidt reduction. The results are applied to a class of semilinear elliptic operators in bounded domains of R N and in particular to Sturm–Liouville operators.

We characterize functions u from the real line into a Hilbert space that are the orbits of a unitary group { U ( t ) } t ∈ R ; that is, u ( t ) = U ( t ) u ( 0 ) , for all real t. One of the characterizations is that u be the Fourier transform of a certain type of vector-valued measure Z; we then use our characterizations to construct Z from u.

Let B ( H ) be the C * -algebra of all bounded linear operators acting on a complex Hilbert space H . In this note, we shall show that if S is an invertible normal operator in B ( H ) the following estimation holds ‖ S ⊗ S −1 + S −1 ⊗ S ‖ λ ⩽ ‖ S ‖ ‖ S −1 ‖ + 1 ‖ S ‖ ‖ S −1 ‖ where ‖ . ‖ λ is the injective norm on the tensor product B ( H ) ⊗ B ( H ) . This last inequality becomes an equality when S is invertible self-adjoint. On the other hand, we shall characterize the set of all invertible normal operators S in B ( H ) satisfying the relation ‖ S ⊗ S −1 + S −1 ⊗ S ‖ λ = ‖ S ‖ ‖ S −1 ‖ + 1 ‖ S ‖ ‖ S −1 ‖ and also we shall give some characterizations of some subclasses of normal operators in B ( H ) by inequalities or equalities.

The nonlinear ill-posed Cauchy problem d u d t = A u ( t ) + h ( t , u ( t ) ) , u ( 0 ) = χ , where A is a positive self-adjoint operator on a Hilbert space H , χ ∈ H , and h : [ 0 , T ) × H → H is a uniformly Lipschitz function, is studied in order to establish continuous dependence results for solutions to approximate well-posed problems. The authors show here that solutions of the problem, if they exist, depend continuously on solutions to corresponding approximate well-posed problems, if certain stabilizing conditions are imposed. The approximate problem is given by d v d t = f ( A ) v ( t ) + h ( t , v ( t ) ) , v ( 0 ) = χ , for suitable functions f. The main result is that ‖ u ( t ) − v ( t ) ‖ ⩽ C β 1 − t T M t T , where C and M are computable constants independent of β and 0 β 1 . This work extends to the nonlinear case earlier results by the authors and by Ames and Hughes.

This paper deals with the existence and uniqueness for the periodic boundary value problem of the semilinear evolution equation in a Hilbert space H{u′(t)+Au(t)=f(t,u(t)),00 and f:[0,ω]×H→H satisfy Caratheodory condition. We present some spectral conditions for the nonlinearity f(t,u) to guarantee the existence and uniqueness. These spectral conditions are the generalization for nonresonance condition of the self-adjoint elliptic boundary value problems.

The ill-posed parabolic equation backward in time { u t + A u = 0 , 0 t T , ‖ u ( T ) − f ‖ ⩽ ϵ subject to the constraint ‖ u ( 0 ) ‖ ⩽ E with the positive self-adjoint unbounded operator A and E > ϵ > 0 being given is regularized by the well-posed non-local boundary value problem { u t + A u = 0 , 0 t T , α u ( 0 ) + u ( T ) = f , α > 0 . The error estimates of Holder type of the regularized solutions are obtained. These estimates improve the related results by Mel'nikova, Denche and Bessila.

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.

Suppose that $\Gamma$ is a continuous and self-adjoint Hankel operator on $L^2(0, \infty)$ and that $Lf=-(d/dx(a(x)df/dx))+b(x)f(x)$ with $a(0)=0$. If $a$ and $b$ are both quadratic, hyperbolic or trigonometric functions, and $\phi$ satisfies a suitable form of Gauss's hypergeometric equation, or the confluent hypergeometric equation, then $L\Gamma =\Gamma L$. The paper catalogues the commuting pairs $\Gamma$ and $L$, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half plane., Comment: 18 pages

In this note, the estimate of norms of commutators of self-adjoint operators is established.

Basis problems for self-adjoint matrix valued functions are studied. We suggest a new and nonstandard method to solve basis problems both in finite and infinite dimensional spaces. Although many results in this paper are given for operator functions in infinite dimensional Hilbert spaces, but to demonstrate practicability of this method and to present a full solution of basis problems, in this paper we often restrict ourselves to matrix valued functions which generate Rayleigh systems on the n-dimensional complex space Cn. The suggested method is an improvement of an approach given recently in our paper [M. Hasanov, A class of nonlinear equations in Hilbert space and its applications to completeness problems, J. Math. Anal. Appl. 328 (2007) 1487–1494], which is based on the extension of the resolvent of a self-adjoint operator function to isolated eigenvalues and the properties of quadratic forms of the extended resolvent. This approach is especially useful for nonanalytic and nonsmooth operator functions when a suitable factorization formula fails to exist.

The following generalized eigenproblem is analyzed: Find u ∈ H 0 1 ( Ω ) , u ≠ 0 , and λ ∈ R such that 〈 ∇ u , ∇ v 〉 D = λ 〈 ∇ u , ∇ v 〉 Ω for all v ∈ H 0 1 ( Ω ) , where Ω ⊂ R n is a bounded domain, D is a subdomain with closure contained in Ω, and 〈 ⋅ , ⋅ 〉 Ω is the inner product 〈 ∇ u , ∇ v 〉 Ω = ∫ Ω ∇ u ⋅ ∇ v d x . It is proved that any f ∈ H 0 1 ( Ω ) can be expanded in terms of orthogonal eigenfunctions for the generalized eigenproblem. During the analysis, we present a new inner product on H 1 / 2 ( ∂ D ) with the following properties: (a) the norm associated with the inner product is equivalent to the usual norm on H 1 / 2 ( ∂ D ) , and (b) the double layer potential operator is self adjoint with respect to the new inner product and compact as a mapping from H 1 / 2 ( ∂ D ) into itself. The analysis identifies four classes of eigenfunctions for the generalized eigenproblem: 1. The function Π which is 1 on D and harmonic on Ω ∖ D ; the eigenvalue is 0. 2. Functions in H 0 1 ( Ω ) with support in Ω ∖ D ; the eigenvalue is 0. 3. Functions in H 0 1 ( Ω ) with support in D; the eigenvalue is 1. 4. Excluding Π, the harmonic extension of the eigenfunctions of a double layer potential on ∂D. The eigenvalues are contained in the open interval ( 0 , 1 ) . The only possible accumulation point is λ = 1 / 2 . A positive lower bound for the smallest positive eigenvalue is obtained. These results can be used to evaluate the change in the electric potential due to a lightning discharge.

The Ekeland Variational Principle is used to prove the nonemptiness of the spectrum for positively homogeneous gradient mappings in Hilbert space. Under additional information on the measure of noncompactness of the operator, this leads to the existence of a maximal (or minimal) compact eigenvalue for the operator itself.

It was shown by P. Jonas and H. Langer that a selfadjoint definitizable operator A in a Krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator B is selfadjoint and the resolvent set ρ(B) is nonempty. It is the aim of this note to prove a more general variant of this perturbation result where the assumption on ρ(B) is dropped. As an application a class of singular ordinary differential operators with indefinite weight functions is studied.

A $p$-adic Schr\"{o}dinger-type operator $D^{\alpha}+V_Y$ is studied. $D^{\alpha}$ ($\alpha>0$) is the operator of fractional differentiation and $V_Y=\sum_{i,j=1}^nb_{ij}\delta_{x_i}$ $(b_{ij}\in\mathbb{C})$ is a singular potential containing the Dirac delta functions $\delta_{x}$ concentrated on points $\{x_1,...,x_n\}$ of the field of $p$-adic numbers $\mathbb{Q}_p$. It is shown that such a problem is well-posed for $\alpha>1/2$ and the singular perturbation $V_Y$ is form-bounded for $\alpha>1$. In the latter case, the spectral analysis of $\eta$-self-adjoint operator realizations of $D^{\alpha}+V_Y$ in $L_2(\mathbb{Q}_p)$ is carried out.

The theory of locally convex cones as a branch of functional analysis was presented by K. Keimel and W. Roth in [K. Keimel, W. Roth, Ordered Cones and Approximation, Lecture Notes in Math., vol. 1517, Springer-Verlag, Heidelberg, 1992]. We study some more results about dual cones and adjoint operators on locally convex cones. Moreover we introduce the concept of the uniformly precompact sets and discuss their relations with σ-bounded sets. Some results obtained about inductive limit, projective limit, metrizability and quotients of locally convex cones.

An inverse spectral problem is studied for a non-selfadjoint Sturm–Liouville operator on a finite interval with an arbitrary behavior of the spectrum. The spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function in the selfadjoint case. The connection with other types of spectral characteristics is investigated and a uniqueness theorem is proved. A constructive procedure for solving the inverse problem is given.

Based on the direct methods of the perturbation theory, sufficient conditions for the finiteness of the discrete spectrum of linear pencils of the form L ( λ ) = B − λ A , where A and B are bounded self-adjoint operators, are established. An estimate formula for the discrete spectrum is also presented. As applications, we study the spectrum of the characteristic equation of radiation energy transfer.

In this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equations Δ [ p ( t ) Δ u ( t − 1 ) ] + q ( t ) u ( t ) = f ( t , u ( t ) ) , t ∈ Z , without any periodicity assumptions on p ( t ) , q ( t ) and f , providing that f ( t , x ) grows superlinearly both at origin and at infinity or is an odd function with respect to x ∈ R , and satisfies some additional assumptions.

We prove a Critical Point Theorem for C 1 functionals on the unit sphere of a separable Hilbert space H which improves a previous result of ours. This is applied in nonlinear eigenvalue theory to study the effect of suitably restricted homogeneous perturbations upon the discrete spectrum of a bounded self-adjoint operator in H.

We give a new and unified proof of the fact that for any eigenvalue of a self-adjoint Sturm–Liouville problem with limit-circle end points, its analytic and geometric multiplicities are equal. This proof is geometric in nature and can be generalized to the case of high-order differential equations.

We revisit the old problem of finding the stability and instability intervals of a second-order elliptic equation on the real line with periodic coefficients (Hill's equation). It is well known that the stability intervals correspond to the spectrum of the Bloch family of operators defined on a single period. Here we propose a characterization of the instability intervals. We introduce a new family of non-self-adjoint operators, formally equivalent to the Bloch ones but with an imaginary Bloch parameter, that we call exponential. We prove that they admit a countable infinite number of eigenvalues which, when they are real, completely characterize the intervals of instability of Hill's equation.

In this paper, geometric properties of the self-adjoint Sturm–Liouville problems are investigated. It is proved that for linear self-adjoint Sturm–Liouville problems, the eigenfunctions correspond exactly to the projections of the curvature lines on the energy functional surface with an appropriate metric and that the eigenvalues correspond exactly to the principal curvatures (at the origin) of the same energy functional surface.

The Reggeon field theory is governed by a non-self adjoint operator constructed as a polynomial in A, A * , the standard Bose annihilation and creation operators. In zero transverse dimension, this Hamiltonian acting in Bargmann space is defined by H λ ′ , μ = λ ′ A * 2 A 2 + μ A * A + i λ A * ( A * + A ) A , where i 2 = − 1 , λ ′ , μ and λ are real numbers and the operators A , A * satisfy the commutation relation [ A , A * ] = I . As the quantum mechanical system described by H λ ′ , μ has a velocity-dependent potential containing powers of momentum up to the fourth, the problem of existence of Hamiltonian path integral for the evolution operator e − t H λ ′ , μ of this theory is of interest on its own. In particular, can we express e − t H λ ′ , μ as a limit of “integral” operators? In this article one considerably reduces the difficulty by studying the Trotter product formula of H λ ′ , μ to reach two objectives: • The first objective is to prove a very specific error estimate for the error in a Trotter product formula in trace-norm for H viewed as the sum of the operators λ ′ A * 2 A 2 and μ A * A + i λ A * ( A * + A ) A . • The second objective of this work is to give a approximation of the semigroup generated by H λ ′ , μ when H λ ′ , μ is split in the sum of λ ′ A * 2 A 2 + μ A * A and i λ A * ( A * + A ) A . We note that this case is entirely different. In fact, the usual Trotter product formula is not defined, because the interaction operator A * ( A * + A ) A is not the infinitesimal generator of a semigroup on Bargmann space. For λ ′ > 0 and ɛ > 0 , we choose an approximation operator θ ɛ = [ I − ɛ i λ A * ( A * + A ) A ] e − ɛ ( λ ′ A * 2 A 2 + μ A * A ) and we give a connection between θ ɛ and e − ɛ H λ ′ , μ . This choice allows us to give in [A. Intissar, Note on the path integral formulation of Reggeon field theory, preprint] a “generalized Trotter product formula” for T μ = μ A * A + i λ A * ( A + A * ) A , i.e., for limit case as λ ′ = 0 and answers to the above question.

In this paper, we study a final value problem for first order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.