Singular Endpoint Research Articles

Friedrichs extensions of a class of discrete Hamiltonian systems with one singular endpoint

AbstractThis paper is concerned with Friedrichs extensions for a class of discrete Hamiltonian systems with one singular endpoint. First, Friedrichs extensions of symmetric Hamiltonian systems are characterized by imposing some constraints on each element of domains of the maximal relations H. Furthermore, it is proved that the Friedrichs extension of each of a class of non‐symmetric systems is also a restriction of the maximal relation H by using a closed sesquilinear form. Then, the corresponding Friedrichs extensions are characterized. In addition, ‐self‐adjoint Friedrichs extensions are studied, and two results are given for elements of , which make the expression of the Friedrichs extension simpler. All results are finally applied to Sturm–Liouville equations with matrix‐valued coefficients.

Friedrichs extensions for Sturm–Liouville operators with complex coefficients and their spectra

In this paper, we study the Friedrichs extensions of Sturm–Liouville operators with complex coefficients according to the classification of B. M. Brown et al. [3]. We characterize the Friedrichs extensions both by boundary conditions at regular endpoint and asymptotic behaviours of elements in the maximal operator domains at singular endpoint. Some of spectral properties are also involved.

Singular fourth-order Sturm–Liouville operators and acoustic black holes

AbstractWe derive conditions for a one-term fourth-order Sturm–Liouville operator on a finite interval with one singular endpoint to have essential spectrum equal to $[0,\infty )$ or $\varnothing $. Of particular usefulness are Kummer–Liouville transformations which have been a valuable tool in the study of second-order equations. Applications to a mechanical beam with a thickness tapering to zero at one of the endpoints are considered. When the thickness $2h$ satisfies $c_1x^{\nu }\leq h(x)\leq c_2x^{\nu }$, we show that the essential spectrum is empty if and only if $\nu < 2$. As a final application, we consider a tapered beam on a Winkler foundation and derive sufficient conditions on the beam thickness and the foundational rigidity to guarantee the essential spectrum is equal to $[0,\infty )$.

Renormalized oscillation theory for singular linear Hamiltonian systems

Working with a general class of linear Hamiltonian systems on intervals with at least one singular endpoint which can be limit-point, limit-circle, or limit-intermediate, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of C2n. In the first part of the analysis we associate our linear Hamiltonian systems with families of well-defined self-adjoint operators, and in the latter part we employ the renormalized oscillation approach to count the number of eigenvalues these operators have on fixed intervals (λ1,λ2) whose closures do not intersect the essential spectrum of the operators. We conclude the analysis with two illustrative examples, indicating how the theory can be implemented in practice. This extends previous work by the authors for regular linear Hamiltonian systems.

On essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpoint

This paper is concerned with essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpoint. For semi-bounded systems, the characterization of each element of the essential numerical range in terms of certain singular sequences is given, the concept of form perturbation small at the singular endpoint is introduced, and the stability of the essential numerical range is obtained under this perturbation, which shows the stability of the infimum or supremum of the essential spectrum. Some sufficient conditions for the invariance of the essential numerical range are given in terms of coefficients of Hamiltonian systems.

On the Spectra of General Ordinary Quasi-Differential Operators and Their <i>L</i><sup>2</sup><sub><i>w</i></sub>-Solutions

In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ+ on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.

Post-endoscopy Esophageal Neoplasia in Barrett’s Esophagus: Consensus Statements From an International Expert Panel

Post-endoscopy Esophageal Neoplasia in Barrett’s Esophagus: Consensus Statements From an International Expert Panel

Essential spectra of singular Hamiltonian differential operators of arbitrary order under a class of perturbations

AbstractThe main object of this paper is to study the essential spectrum of a Hamiltonian system of arbitrary order with one singular endpoint under a class of perturbations. We first present a characterization of the essential spectrum in terms of singular sequences and then give the concept of perturbations small at singular endpoints of Hamiltonian systems. Based on the above characterization, the invariance of essential spectra of Hamiltonian systems under these perturbations is shown. It is noted that these perturbations are given by using the associated pre‐minimal operator , which provides great convenience in the study of essential spectra of Hamiltonian systems since each element of the domain of has compact support. As applications, some sufficient conditions for the invariance of essential spectra of some systems are obtained in terms of coefficients of systems and perturbations terms. Further, essential spectra of Hamiltonian systems with different weight functions are discussed. Here, Hamiltonian systems may be non‐symmetric.

Progressive Reinvention or Destination Lost? Half a Century of Cardiovascular Tissue Engineering.

The concept of tissue engineering evolved long before the phrase was forged, driven by the thromboembolic complications associated with the early total artificial heart programs of the 1960s. Yet more than half a century of dedicated research has not fulfilled the promise of successful broad clinical implementation. A historical account outlines reasons for this scientific impasse. For one, there was a disconnect between distinct eras each characterized by different clinical needs and different advocates. Initiated by the pioneers of cardiac surgery attempting to create neointimas on total artificial hearts, tissue engineering became fashionable when vascular surgeons pursued the endothelialisation of vascular grafts in the late 1970s. A decade later, it were cardiac surgeons again who strived to improve the longevity of tissue heart valves, and lastly, cardiologists entered the fray pursuing myocardial regeneration. Each of these disciplines and eras started with immense enthusiasm but were only remotely aware of the preceding efforts. Over the decades, the growing complexity of cellular and molecular biology as well as polymer sciences have led to surgeons gradually being replaced by scientists as the champions of tissue engineering. Together with a widening chasm between clinical purpose, human pathobiology and laboratory-based solutions, clinical implementation increasingly faded away as the singular endpoint of all strategies. Moreover, a loss of insight into the healing of cardiovascular prostheses in humans resulted in the acceptance of misleading animal models compromising the translation from laboratory to clinical reality. This was most evident in vascular graft healing, where the two main impediments to the in-situ generation of functional tissue in humans remained unheeded–the trans-anastomotic outgrowth stoppage of endothelium and the build-up of an impenetrable surface thrombus. To overcome this dead-lock, research focus needs to shift from a biologically possible tissue regeneration response to one that is feasible at the intended site and in the intended host environment of patients. Equipped with an impressive toolbox of modern biomaterials and deep insight into cues for facilitated healing, reconnecting to the “user needs” of patients would bring one of the most exciting concepts of cardiovascular medicine closer to clinical reality.

Spectral Analysis of Singular Matrix-Valued Sturm–Liouville Operators

In the Hilbert space $$L_{W}^{2}([a,b);E)$$ ( $$-\infty<a<b\le +\infty ,$$ $$\dim E=N<+\infty ,$$ $$W>0$$ ) a space of boundary values of the symmetric singular matrix-valued Sturm–Liouville operator with maximal deficiency indices (2N, 2N) (in limit-circle case at singular end point b) is constructed. With the help of the boundary conditions at a and b, all maximal dissipative, maximal accumulative and self-adjoint extensions of such a symmetric operator are established. In particular, the maximal dissipative operators with separated boundary conditions, called ‘dissipative at a’ and ‘self-adjoint at b’ are investigated. A self-adjoint dilation of the dissipative operator is constructed and then its incoming and outgoing spectral representations are determined. This representation allows us to determine the scattering matrix of the dilation with the help of the Weyl matrix-valued function of a self-adjoint matrix-valued Sturm–Liouville operator. Further a functional model of the dissipative operator is determined and its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl function) is established. Finally, a theorem on completeness of the system of root vectors of the dissipative operator is proved.

Bosentan‐based, treat‐to‐target therapy in patients with pulmonary arterial hypertension: results from the COMPASS‐3 study

The phase 4 COMPASS-3 study evaluated whether a singular endpoint produces clinically meaningful outcomes in patients with pulmonary arterial hypertension (PAH). The relationship between cardiac magnetic resonance imaging (cMRI)-derived parameters and right heart catheterization (RHC) measurements was also examined. In COMPASS-3 (ClinicalTrials.gov NCT00433329), 100 patients with PAH received bosentan monotherapy for 16 weeks. Patients continued monotherapy if their 6-min walk distance (6MWD) was ≥380 m, or otherwise received add-on sildenafil for an additional 12 weeks. 6MWD, RHC, and cMRI were performed at baseline, week 16, and week 28 (6MWD and cMRI). Baseline median 6MWD was 274 m and 82% of patients had WHO Functional Class III/IV. At week 16, 17% (n = 16) of remaining patients achieved the 6MWD threshold and 78 (83%) did not. In the intention-to-treat population, median 6MWD increased significantly relative to baseline (week 16 = 308 m; week 28 = 327 m; P < 0.001). At week 28, 9/16 (monotherapy) and 15/76 (20%; add-on sildenafil) patients met the target threshold. Baseline cMRI-derived and RHC-derived parameters showed moderate-to-strong correlations (e.g. right to left ventricular end-diastolic ratio [RVEDV:LVEDV] correlated strongly with pulmonary vascular resistance [r = +0.729, P < 0.0001]). cMRI-derived parameters predicted clinical worsening/decline (e.g. week 16 RVEDV:LVDEV [P = 0.0172]). Time to clinical worsening/decline did not differ between patients based on 6MWD threshold achievement. No unexpected safety events were reported. A substantial proportion of patients failed to achieve the goal of 380 m, regardless of treatment. Several cMRI parameters predicted clinical worsening/decline and its non-invasive nature further supports its use in future clinical trials.

Regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint

This paper is concerned with regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint. For any given self-adjoint subspace extension (SSE) of the corresponding minimal subspace, its spectrum can be approximated by eigenvalues of a sequence of induced regular SSEs, generated by the same difference expression on smaller finite intervals. It is shown that every SSE of the minimal subspace has a pure discrete spectrum, and the k-th eigenvalue of any given SSE is exactly the limit of the k-th eigenvalues of the induced regular SSEs; that is, spectral exactness holds, in the limit circle case. Furthermore, error estimates for the approximations of eigenvalues are given in this case. In addition, in the limit point and intermediate cases, spectral inclusive holds.

Dependence of eigenvalues on the boundary conditions of Sturm–Liouville problems with one singular endpoint

This paper is concerned with continuous and differentiable dependence of isolated eigenvalues on the boundary conditions of self-adjoint Sturm–Liouville problems with one singular endpoint. Locally continuous dependence of eigenvalues on the boundary conditions is proved. Especially, in the limit point case, the continuous dependence of isolated eigenvalues is shown by the relationships between Weyl–Titchmarsh m(λ)-function and the spectrum of the singular problems. Then continuous eigenvalue branches through each isolated eigenvalue over the space of boundary conditions are formed. It is rigorously shown that the eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch can be continuously chosen with uniform bound in Lw2 norm. Its proof is different from that in the regular Sturm–Liouville problems. The derivative formulas of the continuous simple eigenvalue branch with respect to all the parameters in the boundary conditions are given, and thus its monotonicity with respect to some parameters is derived.

On Eigenfunction Expansions of First-Order Symmetric Systems and Ordinary Differential Operators of an Odd Order

We study general (not necessarily Hamiltonian) first-order symmetric systems \({J y'-B(t)y=\Delta(t) f(t)}\) on an interval \({\mathcal{I}=[a,b \rangle}\) with the regular endpoint a. It is assumed that the deficiency indices \({n_{\pm}(T_{\rm min})}\) of the minimal relation \({T_{\rm min}}\) satisfy \({n_{+}(T_{\rm min})\leq n_{-}(T_{\rm min})}\). We define special \({\lambda}\)-depending boundary conditions with Nevanlinna-type spectral parameter \({\tau=\tau(\lambda)}\) at the singular endpoint b. With boundary value problem involving such conditions, we associate an exit space self-adjoint extension \({\tilde{T}}\) of \({T_{\rm min}}\) and the \({N \times N}\)-matrix m-function \({m(\cdot)}\) of the size \({N={\rm dim ker} (iJ+I)}\). The role of \({m(\cdot)}\) is similar to that of the Weyl function for a Hamiltonian system. Using the m-function, we obtain the eigenfunction expansion with \({N \times N}\)-matrix spectral function \({\Sigma(\cdot)}\). Moreover, we represent \({\Sigma(\cdot)}\) immediately in terms of a boundary parameter \({\tau}\). We also characterize certain spectral properties of the extension \({\tilde{T}}\). Application of these results to ordinary differential operators of an odd order enables us to complete the results by Everitt and Krishna Kumar on the Titchmarsh–Weyl theory of such operators.

Power asymptotics of spectral functions of boundary value problems for generalized second-order differential equations with boundary conditions at a singular endpoint

Let I = (-∞, b), where b ⩽ +∞, and let M(x), x ∈ I, be a nondecreasing function on I such that M(x) > 0 for x ∈ I. In the middle of the past century, it was proved that, in the case where M(x) is Lebesgue integrable on the interval (-∞, c), c ∈ I, the boundary value problem \(- \tfrac{d} {{dM(x)}}y^ + (x) = \lambda y(x)\), x ∈ I, limx→−∞y(x) = 1 is uniquely solvable for any complex λ and has at least one spectral function τ(λ) (“+” denotes right derivative).

On essential spectra of singular linear Hamiltonian systems

The paper is concerned with essential spectra of singular linear Hamiltonian systems of arbitrary order with arbitrary equal defect indices. Several sufficient conditions for the essential spectral points are given in terms of the number of linearly independent square integrable solutions of the corresponding Hamiltonian system, and a sufficient and necessary condition for the essential spectral points is obtained for Hamiltonian systems of even-order with one singular endpoint. An advantage of these results is that one can determine the essential spectral points of Hamiltonian systems by the information of solutions obtained by numerous tools available in the fundamental theory of differential equations. In addition, two illustrative examples are provided to show how to get some information about the essential spectrum by our results.

On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems

We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)−B(t)y(t) = Δ(t) f(t) on an interval \({\mathcal{I}=[a,b) }\) with the regular endpoint a. It is assumed that the deficiency indices n ±(T min) of the minimal relation T min associated with this system in \({L^2_\Delta(\mathcal{I})}\) satisfy \({n_-(T_{\rm min})\leq n_+(T_{\rm min})}\). We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size \({N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}\). Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform \({V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}\) where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension \({\tilde{T}}\) of T min induced by the boundary problem is unitarily equivalent to the multiplication operator in L 2(Σ). Hence the multiplicity of spectrum of \({\tilde{T}}\) does not exceed N Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.

Invariance of deficiency indices under perturbation for discrete Hamiltonian systems

This paper deals with discrete Hamiltonian systems with one singular endpoint. Using Hermitian linear relation generalized by linear Hamiltonian system, the invariance of the minimal and maximal deficiency indices under bounded perturbation for discrete Hamiltonian systems is built. This parallels the well-known results for linear Hamiltonian differential systems obtained by F.V. Atkinson.

Singular critical endpoint and phase diagrams of the anisotropic 3-state Potts model on square lattice

Abstract We have studied the ground state and thermodynamic properties of the anisotropic 3-state Potts model on square lattice by means of the tensor network-based numerical method. The phase diagrams of this model in the ground state and at finite temperature are identified. The singular behavior at the critical endpoint along the phase boundary is carefully investigated. It is discovered that the sharp peaks appear in the second-order derivative of the field as well as the first-order derivative of the magnetization with respect to temperature on the phase boundary. Our numerical results confirm the prediction of Fisher et al. about the critical endpoint.

Mark Krein's method of directing functionals and singular potentials

AbstractIt is shown that M. Krein's method of directing functionals can be used to prove the existence of a scalar spectral measure for certain Sturm–Liouville equations with two singular endpoints. The essential assumption is the existence of a solution of the equation that is square integrable at one singular endpoint and depends analytically on the eigenvalue parameter.