Isometric Reflectionless Eigenfunction Transforms for Higher-order AOs

Ruijsenaars, S. N. M.

doi:10.2991/jnmp.2005.12.s1.45

Cited by 3 publications

(7 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let ℜz = x 0 be a line on which µ(z) has no poles. Then we can write 16) with the Fourier coefficients c n having exponential decay as |n| → ∞. From (2.13) we now deduce…”

Section: A Class Of A∆os and Their Eigenfunctionsmentioning

confidence: 95%

“…When the coefficients in the A∆O A (2.3) are not only non-constant, but N is also greater than 1, very little seems to be known about solutions to the single eigenvalue problem (2.4). In fact, we are only aware of results for quite special coefficients, which give rise to 'reflectionless' solutions [15,16]. On the other hand, the first order case N = 1 is far more accessible, just as for linear ODEs.…”

Section: A Class Of A∆os and Their Eigenfunctionsmentioning

confidence: 99%

“…To treat the contribution from the second term we choose r such that 16) and set ψ(x) = ρ R (x)φ(x). Then the assumptions of Lemma C.1 are satisfied, so we may use (C.34) with s replaced by −t to deduce that the second term also yields a vanishing contribution for t → −∞.…”

Section: Scattering Theory and Unitarity Of T R And T Lmentioning

confidence: 99%

See 2 more Smart Citations

A unitary joint eigenfunction transform for the AOs exp(ia±d/dz) + exp(2z/a)

Ruijsenaars¹

2005

JNMP

Self Cite

View full text Add to dashboard Cite

To cite this article: S N M Ruijsenaars (2005) AbstractFor positive parameters a + and a − the commuting difference operators exp(ia ± d/dz) + exp(2πz/a ∓ ), acting on meromorphic functions f (z), z = x + iy, are formally self-adjoint on the Hilbert space H = L 2 (R, dx). Volkov showed that they admit joint eigenfunctions. We prove that the joint eigenfunctions for positive eigenvalues exp(2πp/a ∓ ), p ∈ R, give rise to a unitary transform, thus associating commuting self-adjoint operators on H to the analytic difference operators.

show abstract

“…Let ℜz = x 0 be a line on which µ(z) has no poles. Then we can write 16) with the Fourier coefficients c n having exponential decay as |n| → ∞. From (2.13) we now deduce…”

Section: A Class Of A∆os and Their Eigenfunctionsmentioning

confidence: 95%

Section: A Class Of A∆os and Their Eigenfunctionsmentioning

confidence: 99%

See 1 more Smart Citation

A unitary joint eigenfunction transform for the AOs exp(ia±d/dz) + exp(2z/a)

Ruijsenaars¹

2005

JNMP

Self Cite

View full text Add to dashboard Cite

show abstract

“…A large class of one-variable A Os yielding reflectionless unitary eigenfunction transforms has been introduced in [20]. (The reflectionless A Os studied in [6] form a tiny subclass.)…”

Section: Repulsive and Attractive Eigenfunctions For General Couplingmentioning

confidence: 99%

“…Finally, we also need the summation identity [14]. From this we obtain a second identity 20) by using c…”

Section: Downloaded Frommentioning

confidence: 99%

Hilbert space theory for relativistic dynamics with reflection. Special cases

Haworth

Ruijsenaars

2015

J Integrable Syst

View full text Add to dashboard Cite

We present and study a novel class of one-dimensional Hilbert space eigenfunction transforms that diagonalize analytic difference operators encoding the (reduced) two-particle relativistic hyperbolic Calogero-Moser dynamics. The scattering is described by reflection and transmission amplitudes t and r with function-theoretic features that are quite different from nonrelativistic amplitudes. The axiomatic Hilbert space analysis in the appendices is inspired by and applied to the attractive two-particle relativistic Calogero-Moser dynamics for a sequence of special couplings. Together with the scattering function u of the repulsive case, this leads to a triple of amplitudes u, t and r satisfying the Yang-Baxter equations.

show abstract

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case

Ruijsenaars

2015

SIGMA

View full text Add to dashboard Cite

Abstract. The 'relativistic' Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schrödinger equation for an analytic difference operator introduced by van Diejen. We study Hilbert space features of this operator and its 'modular partner', based on an in-depth analysis of the eigenvectors of a Hilbert-Schmidt integral operator whose integral kernel has a previously known relation to the two difference operators. With suitable restrictions on the parameters, we show that the commuting difference operators can be promoted to a modular pair of self-adjoint commuting operators, which share their eigenvectors with the integral operator. Various remarkable spectral symmetries and commutativity properties follow from this correspondence. In particular, with couplings varying over a suitable ball in R 8 , the discrete spectra of the operator pair are invariant under the E 8 Weyl group. The asymptotic behavior of an 8-parameter family of orthonormal polynomials is shown to be shared by the joint eigenvectors.

show abstract

Isometric Reflectionless Eigenfunction Transforms for Higher-order AOs

Cited by 3 publications

References 17 publications

A unitary joint eigenfunction transform for the AOs exp(ia±d/dz) + exp(2z/a)

A unitary joint eigenfunction transform for the AOs exp(ia±d/dz) + exp(2z/a)

Hilbert space theory for relativistic dynamics with reflection. Special cases

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case

Contact Info

Product

Resources

About