Quantum integrability of quadratic Killing tensors

Abstract

Quantum integrability of classical integrable systems given by quadratic Killing tensors on curved configuration spaces is investigated. It is proven that, using a "minimal" quantization scheme, quantum integrability is insured for a large class of classic examples.

Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

“…is the scalar curvature of the metric tensor of (15). Also in this case we recover the fact stated in [19] that K is not a symmetry of the conformally invariant Laplacian.…”

Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

“…is the scalar curvature of the metric tensor of (15). Also in this case we recover the fact stated in [19] that K is not a symmetry of the conformally invariant Laplacian.…”

Dirac equation in Kerr-NUT-(A)dS spacetimes: Intrinsic characterization of separability in all dimensions

“…These operators are constructed from 2nd-rank Killing tensors and Killing vectors that can all be generated from the PCKY tensor. The common eigenfunction of these operators is characterized by operators' eigenvalues and can be found in a separated form-it is precisely the separated solution obtained by Frolov et al In fact, the demonstrated results provide a textbook example of general theory discussed in [15][16][17].…”

Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures

“…Note that only a sketch of the proof is given in [15], and we recommend [18, §III(A)] of C. Duval and G. Valent, from which a more detailed proof can be extracted. More precisely, combining [18,Equations (3.11) and (3.16)] we obtain the above mentioned result of Carter.…”

Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

“…is the scalar curvature of the metric tensor of (15). Also in this case we recover the fact stated in [19] that K is not a symmetry of the conformally invariant Laplacian.…”

Dirac equation in Kerr-NUT-(A)dS spacetimes: Intrinsic characterization of separability in all dimensions

“…These operators are constructed from 2nd-rank Killing tensors and Killing vectors that can all be generated from the PCKY tensor. The common eigenfunction of these operators is characterized by operators' eigenvalues and can be found in a separated form-it is precisely the separated solution obtained by Frolov et al In fact, the demonstrated results provide a textbook example of general theory discussed in [15][16][17].…”

Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures

“…Note that only a sketch of the proof is given in [15], and we recommend [18, §III(A)] of C. Duval and G. Valent, from which a more detailed proof can be extracted. More precisely, combining [18,Equations (3.11) and (3.16)] we obtain the above mentioned result of Carter.…”

Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

“…is the scalar curvature of the metric tensor of (15). Also in this case we recover the fact stated in [19] that K is not a symmetry of the conformally invariant Laplacian.…”

Dirac equation in Kerr-NUT-(A)dS spacetimes: Intrinsic characterization of separability in all dimensions

“…These operators are constructed from 2nd-rank Killing tensors and Killing vectors that can all be generated from the PCKY tensor. The common eigenfunction of these operators is characterized by operators' eigenvalues and can be found in a separated form-it is precisely the separated solution obtained by Frolov et al In fact, the demonstrated results provide a textbook example of general theory discussed in [15][16][17].…”

Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures

“…Note that only a sketch of the proof is given in [15], and we recommend [18, §III(A)] of C. Duval and G. Valent, from which a more detailed proof can be extracted. More precisely, combining [18,Equations (3.11) and (3.16)] we obtain the above mentioned result of Carter.…”