Quantum star-graph analogues of PT -symmetric square wells: Part II, spectra

For the non-Hermitian equilateral q-pointed star-shaped quantum graphs of our previous paper (Can. J. Phys. 90, 1287 (2012) doi:10.1139/p2012-107 ) we show that because of certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.

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Quantum star-graph analogues of PT-symmetric square wells

Canadian Journal of Physics ◽

10.1139/p2012-107 ◽

2012 ◽

Vol 90 (12) ◽

pp. 1287-1293 ◽

Cited By ~ 8

Author(s):

Miloslav Znojil

Keyword(s):

Boundary Conditions ◽

Bound State ◽

Robin Boundary Conditions ◽

Star Graph ◽

The Real ◽

Rotation Symmetric ◽

Symmetric Quantum ◽

Symmetric Square ◽

Square Well ◽

Robin Boundary

We recall the solvable [Formula: see text]-symmetric quantum square well on an interval of x ∈ (–L, L) := [Formula: see text] (with an α-dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just replace the support interval [Formula: see text] (reinterpreted as an equilateral two-pointed star graph with Kirchhoff matching at the vertex x = 0) with a q-pointed equilateral star graph [Formula: see text] endowed with the simplest complex-rotation-symmetric external α-dependent Robin boundary conditions. The remarkably compact form of the secular determinant is then deduced. Its analysis reveals that (i) at any integer q = 2, 3, …, there exists the same q-independent and infinite subfamily of the real energies, and (ii) at any special q = 2, 6, 10, …, there exists another, additional, q-dependent infinite subfamily of the real energies. In the spirit of the recently proposed dynamical construction of the Hilbert space of a quantum system, the physical bound-state interpretation of these eigenvalues is finally proposed.

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An Inverse Spectral Problem for the Sturm-Liouville Operator on a Three-Star Graph

ISRN Applied Mathematics ◽

10.5402/2012/132842 ◽

2012 ◽

Vol 2012 ◽

pp. 1-23 ◽

Cited By ~ 1

Author(s):

I. Dehghani Tazehkand ◽

A. Jodayree Akbarfam

Keyword(s):

Inverse Problem ◽

Spectral Problem ◽

Liouville Operator ◽

Inverse Spectral Problem ◽

Spectral Characteristics ◽

Star Graph ◽

Dirichlet Problems ◽

Sturm Liouville ◽

Inverse Spectral ◽

Robin Boundary

We study an inverse spectral problem for the Sturm-Liouville operator on a three-star graph with the Dirichlet and Robin boundary conditions in the boundary vertices and matching conditions in the internal vertex. As spectral characteristics,we consider the spectrum of the main problem together with the spectra of two Dirichlet-Dirichlet problems and one Robin-Dirichlet problem on the edges of the graph and investigate their properties and asymptotic behavior. We prove that if these four spectra do not intersect, then the inverse problem of recovering the operator is uniquely solvable.We give an algorithm for the solution of the inverse problem with respect to this quadruple of spectra.

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RT-Symmetric Laplace Operators on Star Graphs: Real Spectrum and Self-Adjointness

Advances in Mathematical Physics ◽

10.1155/2015/649795 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

Maria Astudillo ◽

Pavel Kurasov ◽

Muhammad Usman

Keyword(s):

Quantum Mechanics ◽

Spectral Properties ◽

Block Structure ◽

Circulant Matrices ◽

Real Spectrum ◽

Star Graph ◽

Quantum Graphs ◽

Star Graphs ◽

Rotationally Symmetric ◽

Symmetric Quantum

How ideas ofPT-symmetric quantum mechanics can be applied to quantum graphs is analyzed, in particular to the star graph. The class of rotationally symmetric vertex conditions is analyzed. It is shown that all such conditions can effectively be described by circulant matrices: real in the case of odd number of edges and complex having particular block structure in the even case. Spectral properties of the corresponding operators are discussed.

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Legal Update: North Dakota Supreme Court: Third Edition of Guides is “Most Current”

Guides Newsletter ◽

10.1001/amaguidesnewsletters.1999.janfeb04 ◽

1999 ◽

Vol 4 (1) ◽

pp. 6-7

Author(s):

James J. Mangraviti

Keyword(s):

Internal Rotation ◽

External Rotation ◽

Measurement Techniques ◽

Fourth Edition ◽

Hip Motion ◽

The Difference ◽

The Right ◽

Hip Flexion ◽

Hip Rotation ◽

Hip Extension

Abstract The accurate measurement of hip motion is critical when one rates impairments of this joint, makes an initial diagnosis, assesses progression over time, and evaluates treatment outcome. The hip permits all motions typical of a ball-and-socket joint. The hip sacrifices some motion but gains stability and strength. Figures 52 to 54 in AMA Guides to the Evaluation of Permanent Impairment (AMA Guides), Fourth Edition, illustrate techniques for measuring hip flexion, loss of extension, abduction, adduction, and external and internal rotation. Figure 53 in the AMA Guides, Fourth Edition, illustrates neutral, abducted, and adducted positions of the hip and proper alignment of the goniometer arms, and Figure 52 illustrates use of a goniometer to measure flexion of the right hip. In terms of impairment rating, hip extension (at least any beyond neutral) is irrelevant, and the AMA Guides contains no figures describing its measurement. Figure 54, Measuring Internal and External Hip Rotation, demonstrates proper positioning and measurement techniques for rotary movements of this joint. The difference between measured and actual hip rotation probably is minimal and is irrelevant for impairment rating. The normal internal rotation varies from 30° to 40°, and the external rotation ranges from 40° to 60°.

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Clinical Update: Measuring Shoulder Joint Motion

Guides Newsletter ◽

10.1001/amaguidesnewsletters.1998.sepoct02 ◽

1998 ◽

Vol 3 (5) ◽

pp. 4-5

Author(s):

Christopher R. Brigham

Keyword(s):

External Rotation ◽

Joint Motion ◽

Fourth Edition ◽

Single Plane ◽

Reconstructive Procedures ◽

Results Of Treatment ◽

Shoulder Disorders ◽

Shoulder Motion ◽

Motion Measurements ◽

Upper Extremity Impairment

Abstract Accurate measurement of shoulder motion is critical in assessing impairment following shoulder disorders. To this end, measuring and recording joint motion are important steps in diagnosing, determining the severity and progression of a disorder, assessing the results of treatment, and evaluating impairment. Shoulder movement usually is composite rather than in a single plane, so isolating single movements is challenging. Universal goniometers with long arms are used to measure shoulder motion, and testing must be performed and recorded consistently. Passive motion may be carried out cautiously by the examiner; two measurements of the same patient by the same examiner should lie within 10° of each other. Shoulder extension and flexion are illustrated. Maximal flexion of the shoulder also includes slight external rotation and abduction, and controlling or eliminating these components during evaluation is challenging. Abduction and adduction are illustrated. Deficits in external rotation may occur in patients who have undergone reconstructive procedures with an anterior approach; deficits in internal rotation may result from issues with shoulder instability. The authors recommend recording the shoulder's range of motion measurements according to the Upper Extremity Impairment Evaluation Record in the AMA Guides to the Evaluation of Permanent Impairment, Fourth Edition.

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Isolated Lateral Meniscal Tear in the Dog

Veterinary and Comparative Orthopaedics and Traumatology ◽

10.1055/s-0038-1633280 ◽

1988 ◽

Vol 01 (03/04) ◽

pp. 152-154

Author(s):

S. Johnson ◽

D. Hulse

Keyword(s):

External Rotation ◽

Femoral Condyle ◽

Meniscal Tear ◽

Lateral Meniscus ◽

Surgical Excision ◽

Human Beings ◽

Stifle Joint ◽

Microscopic Pathology ◽

Bucket Handle Tear ◽

Bucket Handle

degenerative changes of the involved stifle joint associated with a “bucket handle” tear of the caudal body of the lateral meniscus. Surgical excision of the torn section of meniscus was beneficial in the first patient but this patient had persistant difficulty with the leg after exercise. Gross and microscopic pathology of the involved stifle in the second patient showed the meniscal lesion to be associated with severe cartilage fibrillation of the overlying lateral femoral condyle. As in human beings, the mechanism of injury may have been placement of the foot during vigorous external rotation of the femur with the stifle flexed. Extension of the limb from this position could have resulted in an isolated tear of the lateral meniscus.

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