Stokes phenomenon for the prolate spheroidal wave equation

New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

“…We suppose that f is even (the odd case is similar). We prove, as in [1], that there exists another independent solution h := f log x+1 x−1 + η where η is an entire function. The action of the monodromy around [−1, 1] on f, η, log x+1…”

“…For sake of simplicity, in this paper and our preceding work [1], we worked with prolate differential equations of order 0. It seems easy to extend the results (and the conjectures) to the prolate and oblate differential equations of arbitrary order m ∈ N + : (x 2 − 1)y + 2x y + τ 2 x 2 − m 2 x 2 − 1 y = μy,…”

New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

“…We suppose that f is even (the odd case is similar). We prove, as in [1], that there exists another independent solution h := f log x+1 x−1 + η where η is an entire function. The action of the monodromy around [−1, 1] on f, η, log x+1…”

“…For sake of simplicity, in this paper and our preceding work [1], we worked with prolate differential equations of order 0. It seems easy to extend the results (and the conjectures) to the prolate and oblate differential equations of arbitrary order m ∈ N + : (x 2 − 1)y + 2x y + τ 2 x 2 − m 2 x 2 − 1 y = μy,…”

An exactly solvable system from quantum optics

“…Concerning the original equation, we have the full characterisation of its local Galois group at infinity. Although in this case it coincides with the full global group, the conclusions of [12] suggest, that for systems with other singular points, the Galois group at infinity alone gives quantisation conditions. Namely, for energy values which belong to the spectrum, the group of the corresponding time-independent Schrödinger equation is solvable.…”

“…Finiteness of this norm implies some conditions on local solutions containing real ±∞, see e.g. quantisation of quartic oscillator in [13], and related problem of Stokes phenomena for prolate spheroidal wave equation considered in [12]. In our considerations, asymptotic solutions at all sectors, not just real infinity, are involved.…”

The Riemann Hypothesis: Past, Present and a Letter Through Time

“…This ultraviolet spectral similarity suggests that one has spectrally captured the contribution of the archimedean place to the mysterious zeta spectrum. The ambiguity in the choice of the Dirac square root D is deeply related to the differential Galois theory of the prolate differential equation studied by J. P. Ramis and his collaborators [45,89].…”

New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

“…We suppose that f is even (the odd case is similar). We prove, as in [1], that there exists another independent solution h := f log x+1 x−1 + η where η is an entire function. The action of the monodromy around [−1, 1] on f, η, log x+1…”

“…For sake of simplicity, in this paper and our preceding work [1], we worked with prolate differential equations of order 0. It seems easy to extend the results (and the conjectures) to the prolate and oblate differential equations of arbitrary order m ∈ N + : (x 2 − 1)y + 2x y + τ 2 x 2 − m 2 x 2 − 1 y = μy,…”

An exactly solvable system from quantum optics

“…Concerning the original equation, we have the full characterisation of its local Galois group at infinity. Although in this case it coincides with the full global group, the conclusions of [12] suggest, that for systems with other singular points, the Galois group at infinity alone gives quantisation conditions. Namely, for energy values which belong to the spectrum, the group of the corresponding time-independent Schrödinger equation is solvable.…”

“…Finiteness of this norm implies some conditions on local solutions containing real ±∞, see e.g. quantisation of quartic oscillator in [13], and related problem of Stokes phenomena for prolate spheroidal wave equation considered in [12]. In our considerations, asymptotic solutions at all sectors, not just real infinity, are involved.…”

The Riemann Hypothesis: Past, Present and a Letter Through Time

“…This ultraviolet spectral similarity suggests that one has spectrally captured the contribution of the archimedean place to the mysterious zeta spectrum. The ambiguity in the choice of the Dirac square root D is deeply related to the differential Galois theory of the prolate differential equation studied by J. P. Ramis and his collaborators [45,89].…”

New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

“…We suppose that f is even (the odd case is similar). We prove, as in [1], that there exists another independent solution h := f log x+1 x−1 + η where η is an entire function. The action of the monodromy around [−1, 1] on f, η, log x+1…”

“…For sake of simplicity, in this paper and our preceding work [1], we worked with prolate differential equations of order 0. It seems easy to extend the results (and the conjectures) to the prolate and oblate differential equations of arbitrary order m ∈ N + : (x 2 − 1)y + 2x y + τ 2 x 2 − m 2 x 2 − 1 y = μy,…”

An exactly solvable system from quantum optics

“…Concerning the original equation, we have the full characterisation of its local Galois group at infinity. Although in this case it coincides with the full global group, the conclusions of [12] suggest, that for systems with other singular points, the Galois group at infinity alone gives quantisation conditions. Namely, for energy values which belong to the spectrum, the group of the corresponding time-independent Schrödinger equation is solvable.…”

“…Finiteness of this norm implies some conditions on local solutions containing real ±∞, see e.g. quantisation of quartic oscillator in [13], and related problem of Stokes phenomena for prolate spheroidal wave equation considered in [12]. In our considerations, asymptotic solutions at all sectors, not just real infinity, are involved.…”

The Riemann Hypothesis: Past, Present and a Letter Through Time

“…This ultraviolet spectral similarity suggests that one has spectrally captured the contribution of the archimedean place to the mysterious zeta spectrum. The ambiguity in the choice of the Dirac square root D is deeply related to the differential Galois theory of the prolate differential equation studied by J. P. Ramis and his collaborators [45,89].…”