Gegenbauer-solvable quantum chain model

Journal Article · Mon Nov 15 00:00:00 EST 2010 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.82.052113· OSTI ID:21528570

Znojil, Miloslav ^[1]

---

1. Nuclear Physics Institute ASCR, 250 68 Rez (Czech Republic)

An N-level quantum model is proposed in which the energies are represented by an N-plet of zeros of a suitable classical orthogonal polynomial. The family of Gegenbauer polynomials G(n,a,x) is selected for illustrative purposes. The main obstacle lies in the non-Hermiticity (aka crypto-Hermiticity) of Hamiltonians H{ne}H{sup {dagger}.} We managed to (i) start from elementary secular equation G(N,a,E{sub n})=0, (ii) keep our H, in the nearest-neighbor-interaction spirit, tridiagonal, (iii) render it Hermitian in an ad hoc, nonunique Hilbert space endowed with metric {Theta}{ne}I, (iv) construct eligible metrics in closed forms ordered by increasing nondiagonality, and (v) interpret the model as a smeared N-site lattice.

• OSTI ID: 21528570
• Journal Information: Physical Review. A, Vol. 82, Issue 5; Other Information: DOI: 10.1103/PhysRevA.82.052113; (c) 2010 The American Physical Society; ISSN 1050-2947
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
HAMILTONIANS
HILBERT SPACE
INTERACTIONS
METRICS
POLYNOMIALS
QUANTUM MECHANICS
SECULAR EQUATION
BANACH SPACE
EQUATIONS
FUNCTIONS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
QUANTUM OPERATORS
SPACE