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quant-ph9708011.txt |@word represent:5 consider:1 localizes:3 global:1 focus:1 illustrated:1 latter:2 consists:1 disk:1 χa:1 relationships:1 increase:1 preserves:3 depend:3 calculate:1 aug:1 technique:1 tr:1 γt:16 environment:11 finally:4 does:5 squeezing:12 preserved:2 under:9 stochastic:23 anharmonic:2 norm:7 alber:3 case:17 roc:1 performs:1 evolves:5 very:7 account:1 results:6 sons:2 every:3 lecture:1 affect:1 simplify:1 adaptive:1 condition:8 dαt:2 cmn:1 indicate:1 presented:1 opt:5 knight:4 notes:1 solution:1 large:1 vector:14 small:1 generalize:3 optics:2 programme:1 upper:1 approximately:1 compensated:1 consistent:1 strunz:2 approximation:5 rate:35 equal:1 theo:1 further:1 dynamics:1 choices:1 linear:2 index:2 what:1 nonlinear:2 appear:1 advantageous:1 giving:3 section:4 interactions:1 shares:1 above:2 capital:1 new:9 increasing:1 method:1 publications:1 satisfy:2 evolving:1 αin:4 whose:1 human:1 κa:3 here:2 understanding:2 separation:1 irrespective:1 represented:5 along:4 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quant-ph9708012.txt |@word ψcs:2 exponentiation:1 magnetic:1 four:1 eds:1 supplement:1 follow:1 issues:1 aug:1 everything:1 shouldn:1 alamos:2 young:1 critical:1 include:1 sent:1 division:1 zhang:1 far:1 wave:14 nieto:7 preuss:1 entire:1 went:1 hamiltonian:1 did:1 michael:1 t0:1 discrete:1 truax:1 eigenvalues:1 optics:2 der:2 version:1 oscillation:1 sin2:3 za:1 cosh:10 surrounding:1 further:3 copenhagen:2 linear:1 even:2 erwin:1 appear:2 section:1 experiment:1 above:1 new:6 learned:1 method:11 fujiwara:1 satisfy:1 full:1 ibid:1 kennard:13 here:1 promoted:1 identifications:1 cos2:1 groups:3 concluding:1 z1:1 change:2 obtained:1 great:1 ahead:1 involved:1 studied:1 controversy:1 p2:1 proof:1 makes:1 followed:2 hermite:1 yuen:2 plenum:1 use:1 feb:1 takes:2 would:5 commun:2 pedagogical:1 next:1 going:1 archives:1 theoretical:2 more:3 operators:2 factor:1 corrected:1 odinger:18 ip0:3 particular:3 known:1 iii:1 me:1 helped:1 springer:2 equation:1 work:5 oscillator:5 iα2:1 reviews:1 minimum:7 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quant-ph9708013.txt |@word concept:1 consider:2 results:2 compact:1 whose:1 aug:1 quasi:1 squeezing:1 γm:1 under:1 hermann:1 ingredient:2 g2:1 g1:1 fan:1 matrices:1 bxy:1 ro:1 hamiltonian:1 ref:1 vector:1 o1:3 oscillator:6 b22:1 b21:1 sign:1 canonical:2 nel:1 appear:1 above:2 new:1 method:2 satisfy:2 respectively:1 here:1 plenio:1 balian:2 obtained:5 remark:2 institute:1 vogt:1 twamley:5 scutaru:4 studies:1 schur:1 hermite:1 app:4 apq:3 f2:1 use:4 positive:1 contains:1 negative:1 axioms:1 choosen:1 few:1 nucl:1 m1:9 m2:9 theoretical:1 more:1 operators:5 known:2 hold:1 negativity:1 must:2 dominicis:2 mg:1 det:2 following:4 root:3 f1:2 compare:1 f3:1 f4:4 states:13 sinθ:1 evidently:1 rippin:1 hampered:1 pure:4 sin:1 barnum:1 may:1 after:1 fd:1 law:2 natural:2 revzen:1 correlation:1 representation:2 order:4 indeed:1 thermal:13 paper:1 statistical:1 fidelity:9 its:1 metaplectic:1 ln:4 mail:1 main:1 lx:3 correspondence:1 closure:1 non:2 integral:4 now:2 nos:1 inversion:1 university:1 l2:4 ifa:1 2b:8 l1:1 found:1 square:3 ed:1 gaussian:4 obtains:1 measure:1 quantum:4 our:2 matrix:4 space:3 rep:1 rev:2 geom:1 gaussians:1 rel:1 denoted:1 formula:2 rea:4 suffices:1 ri:1 math:5 benjamin:1 theory:1 free:1 completely:1 complicated:2 generate:1 r1:3 r2:2 transition:2 m22:1 impure:1 hence:1 first:2 romania:1 reobtained:1 probability:2 entanglement:2 xy:1 directly:1 knight:1 given:6 calculations:2 x2:1 cosh:1 uhlmann:2 cosθ:1 distance:3 princeton:1 fuchs:1 minus:1 magurele:1 ann:1 seem:1 treatement:1 any:3 ax2:4 k2:1 k1:3 concerns:1 trace:2 shall:4 xψ:1 wells:1 simplify:1 phase:1 wilcox:2 calculated:1 exp:12 ky:3 quantifying:1 fact:1 show:1 spaces:1 d2:1 d1:7 q2:1 parameters:1 only:4 factor:1 do:1 de:1 dz:2 dx:1 dy:5 dv:1 gen:1 normalizable:1 qp:1 arxiv:1 factorization:1 ut:4 kernel:1 set:1 computed:1 see:1 result:7 said:1 definite:1 state:10 difficulties:1 between:2 invariant:1 reading:1 recently:2 ker:1 cos:1 symplectic:1 reobtain:1 mann:1 feynmann:1 simple:1 harmonic:1 dimensional:2 m12:1 cs:1 pob:1 kernels:1 combines:1 transformations:1 due:2 been:4 describes:1 basic:2 ph:2 define:2 corresponding:4 itzykson:2 concerning:2 coherent:1 denote:1 case:3 exponential:1 pacs:1 technical:1 squeezed:9 aid:1 property:1 gτ:1 unitary:1 belong:1 same:1 quant:2 arguments:1 finite:1 composition:3 hermiticity:1 independent:1 used:1 folland:1 infinite:1 well:1 analysis:1 concrete:2 howe:1 coordinate:1 obtain:2 atomic:1 candidate:1 density:9 parametrization:3 belongs:1 lett:4 easy:1 theorem:1 semigroup:11 iτ:3 opt:2 rules:1 entangled:1 press:1 using:3 ams:1 mod:2 displaced:10 deta:5 follows:2 bucharest:1 b1:2 b2:3 gα:2 investigations:1 providence:1 does:1 decomposition:2 proc:1 o2:1 minimal:1 symp:1 hilbert:2 bz:1 most:1 column:1 o:3 range:1 oz:1 mixed:2 ot:1 ubner:2 mathematical:1 communication:1 into:1 disentangled:1 hermitian:1 weyl:1 operator:8 vedral:1 long:1 heritage:1 was:2 function:1 aqq:4 form:2 but:2 bures:3 he:1 characteristic:1 us:1 until:1 mechanics:1 ad:1 computing:1 describe:2 general:1 identities:1 inc:1 when:1 valid:1 details:1 department:1 josza:2 elements:1 a1:8 generated:1 a2:13 massachusetts:1 qψ:1 structure:1 dψ:1 physics:2 baker:1 factors:2 rule:3 requires:1
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quant-ph9708014.txt |@word limited:1 caused:1 magnetic:10 excellent:1 results:3 leads:2 garching:1 grateful:1 consists:1 expressions:1 causes:2 olshanii:1 depend:1 calculate:2 aug:1 δv:2 quasi:3 preserved:1 straße:1 unchen:2 figs:2 resonant:2 experimentally:1 condition:2 splitter:1 level:2 die:1 solution:3 leave:1 differential:2 small:5 optics:1 ohrl:1 limitations:1 smaller:1 compensated:2 sin2:1 pulse:2 approximation:7 zero:3 further:1 dynamics:1 44e:1 spectral:1 nonlinear:2 rate:11 sum:1 localization:1 transforms:1 uniform:2 current:2 ballagh:1 above:2 consisting:1 funtion:1 refilled:1 edited:1 full:1 chose:1 component:4 here:3 atoms:23 following:4 andrews:3 along:1 strong:2 equations:2 obtained:1 technical:1 study:1 studied:3 leaving:4 prior:1 resulting:1 resonance:9 spatially:1 castin:1 private:1 addressed:1 decrease:3 condensates:2 total:2 planck:1 would:1 until:1 periodically:1 calculation:2 therefore:1 scope:1 theoretical:2 more:1 ejected:1 becomes:2 inguscio:1 correspondingly:1 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quant-ph9708015.txt |@word partial:5 consider:10 remarkable:1 incorporate:1 rev:24 results:8 leads:5 scalar:3 supplement:1 entropy:6 comments:2 computation:2 lahti:1 follow:1 research:2 increase:1 certainly:1 barnett:1 violate:15 integral:1 similar:1 producing:1 subjected:6 technique:1 tr:7 nonlocal:1 environment:1 finally:4 protocols:7 excluded:1 under:6 opt:1 analogous:5 tb:2 worth:1 norm:1 odinger:1 case:15 belonging:1 supported:1 advantage:1 orthogonal:1 very:3 version:2 removing:1 choice:1 mν:1 mattle:2 experimentally:1 trouble:1 little:1 matrices:7 condition:26 generalised:1 preserving:2 level:2 did:2 cause:1 trb:1 solution:2 leave:1 vector:2 science:1 manifestly:1 eigenvalues:4 noted:2 vincenzo:1 employing:1 says:1 disordered:1 foundation:2 haar:1 ψi:3 direct:2 zero:2 equal:4 further:4 10p:2 estimated:1 permutations:1 λcp:7 even:1 what:2 xor:4 section:9 contributed:1 analogously:1 uniform:1 pawel:2 sequential:1 above:12 sup:1 new:6 increasing:1 order:1 method:7 contrast:3 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quant-ph9708021.txt |@word code:45 allowed:1 consider:4 caused:1 results:12 projected:1 four:2 roy:1 muller:1 entropy:4 computation:26 transversally:1 contributions:1 row:1 whose:2 certainly:2 amplification:2 depend:1 aug:1 similar:3 knill:6 involve:5 opportunities:8 environment:3 finally:2 those:4 edmund:1 under:2 stochastic:1 include:1 worth:1 resources:3 p2:3 digit:3 amplitudes:1 advantage:8 evolves:1 very:4 desired:1 end:1 none:1 choice:12 eration:1 every:1 fall:1 little:1 generalised:1 illustrated:1 level:16 turns:1 conc:2 fewer:1 correct:5 large:9 solved:2 unrelated:1 recall:1 small:8 accumulation:1 related:2 limitations:1 smaller:6 slower:1 deutsch:1 dealing:1 consistent:3 estimates:2 sign:16 approximation:1 carefully:1 rate:6 cost:3 further:10 lanl:9 estimated:1 leakage:1 machines:2 belief:1 even:8 ǫs:1 what:3 errors:72 xor:13 appear:1 advantageous:1 giving:2 section:2 mw:1 networks:4 current:2 comparision:1 anywhere:1 manufactured:1 deal:1 new:1 3a:1 method:29 freely:1 full:1 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quant-ph9708022.txt |@word limited:2 saturating:1 similarity:1 magnetic:19 retain:2 reversibility:1 four:7 rapid:1 entropy:14 anticorrelated:1 consists:2 evolving:1 equiprobable:1 relationships:1 whose:7 boghosian:2 calculate:4 speeded:1 contained:2 restriction:1 swap:2 edmund:1 under:7 worth:1 conjecture:1 cial:1 differently:1 every:11 experimentally:3 excited:2 vast:1 copy:2 disturb:1 publicly:4 progression:1 cause:6 conceive:1 solution:10 vector:15 c00:2 c01:2 markov:1 force:3 infancy:1 consistent:3 direct:1 pulse:6 ation:1 feasibility:2 likely:15 qeccs:2 air:1 machines:13 even:27 established:7 errors:37 trapping:1 incompatible:2 correc:1 specification:1 selected:1 supplied:2 decoding:4 above:11 designing:1 resilient:1 new:29 net:1 increasing:3 ner:1 seeks:1 pioneering:1 1997b:4 deemed:4 simultaneously:1 never:6 here:21 atoms:6 met:1 fairly:1 plenio:2 represented:4 digits:5 luther:1 glo:1 items:2 study:1 approaches:1 permit:14 adopting:1 fourier:15 changes:3 mutual:10 secure:8 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quant-ph9708024.txt |@word represent:3 partial:1 consider:2 ceases:2 localizes:1 magnetic:1 global:1 dynamic:1 results:12 leads:1 eds:1 proposition:2 perturbations:1 expressions:5 causes:3 scr:1 certainly:1 depend:1 calculate:1 aug:1 ott:1 technique:2 containing:3 environment:6 compensate:1 tj:39 arisen:1 under:3 hermann:1 stochastic:3 prigogine:1 case:10 amplitudes:28 kτ:3 very:8 rise:2 springer:2 induce:2 wave:12 every:14 cook:2 experimentally:1 difference:3 exact:1 paz:1 randomness:1 facilitate:1 level:7 cosine:1 notes:1 ikeda:1 large:5 solved:1 math:2 small:3 noted:2 related:2 force:3 destroys:1 cosn:6 repetitive:8 sin2:4 direct:1 pulse:1 approximation:5 zero:2 further:4 dynamics:65 gm:2 theory:1 even:5 established:1 what:1 nonlinear:4 rate:1 brackets:2 section:4 expressed:3 find:2 experiment:1 localized:1 above:5 new:2 occupation:1 method:5 multitude:2 coefficient:5 edited:1 full:7 degree:2 respectively:3 equals:2 here:5 understanding:1 adachi:1 cos2:4 relaxation:1 represented:6 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quant-ph9708025.txt |@word vaagen:1 limited:2 sci:1 consider:2 results:2 leads:2 ge:1 looking:1 inverse:1 aug:1 tail:1 compensate:1 those:1 nodes:1 jensen:6 outside:1 rise:2 wave:9 fall:1 experimentally:1 exact:1 condition:2 diverging:1 applicable:1 occurs:1 large:18 small:12 shows:1 danilin:1 dαi:1 smaller:1 α2i:1 b22:1 c49:1 zero:20 ter:1 even:3 established:1 richard:1 sum:1 expressed:2 d3:1 reminiscent:1 above:1 3a:1 increasing:1 method:4 contrast:1 satisfy:2 respectively:1 here:1 q11:3 cos2:1 based:1 equations:9 logarithmic:1 obtained:1 search:1 institute:1 study:1 larger:3 allows:1 reason:1 resulting:1 digamma:1 plotted:1 formulated:1 spatially:3 a21:1 marked:1 esry:1 anomalies:1 total:3 imaginary:2 derivatives:1 use:3 asymptotic:3 remains:1 negative:1 nucl:3 a52:1 therefore:13 survive:1 more:1 formally:1 becomes:2 odinger:4 thompson:1 everywhere:1 particular:1 cases:2 iii:1 must:4 logarithm:1 none:1 equation:15 centrifugal:2 values:1 following:1 hansen:1 root:1 weaker:1 figure:2 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quant-ph9708026.txt |@word limited:1 code:1 forget:2 founder:2 rev:4 forbidden:1 results:5 leads:1 concept:1 generalize:1 expressions:2 denotes:1 transits:1 consider:5 whose:2 calculate:1 by7:4 assumptions:1 deterministic:2 edward:2 present:1 under:3 stochastic:1 nodes:2 must:6 case:2 mainstream:1 ermakov:1 very:1 implicitly:1 mg:2 wave:21 difference:3 exact:2 condition:1 school:2 riemann:5 hamiltonian:2 louis:1 mirror:1 solution:5 occurs:1 large:3 small:9 path:1 noted:1 correct:1 wronskian:4 neighborhood:2 findings:3 construed:1 maclaurin:2 consistent:6 prediction:1 sign:5 tex:2 second:2 traveling:6 copenhagen:46 canonical:4 even:5 what:4 plays:3 stands:1 giving:1 section:4 brief:1 expressed:2 uniform:4 schwarzian:3 above:4 new:6 increasing:1 equivalent:1 satisfy:1 evaluating:1 2kx:2 here:5 separation:3 address:1 based:7 along:2 strong:6 equations:6 hkg:2 change:7 knowledge:3 antisymmetric:1 cts:2 shift:1 study:1 precedent:1 wherever:1 reason:1 mx:2 amount:3 published:1 action:2 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quant-ph9708031.txt |@word limited:1 concept:1 consider:2 caused:4 reversibility:1 results:1 illustrated:1 quadrature:11 contributions:1 causes:1 amplification:2 shall:1 compensate:6 preserved:1 stochastic:2 must:1 indicating:1 case:1 orthogonal:3 2re:1 rise:1 wave:1 affect:1 difference:3 level:5 knight:1 large:1 vector:10 small:1 ueda:1 optics:2 mahler:3 smaller:2 compensated:7 dealing:1 sign:3 approximation:1 zero:4 compensates:1 timescales:1 dynamics:3 kitagawa:1 linear:5 even:2 established:1 what:2 giving:1 sum:1 expressed:1 new:2 revealing:1 full:1 respectively:4 unobserved:2 component:19 atoms:1 behaviour:1 plenio:1 represented:1 path:1 along:1 strong:1 observing:2 interpret:2 obtained:2 incoming:1 holger:1 institute:2 shift:2 approaches:1 sx:22 action:1 makes:3 corrects:1 composed:2 useful:1 apr:1 requiring:1 vii:1 plenum:2 logically:2 odinger:1 apply:1 total:2 use:1 takes:1 would:2 remains:1 positive:3 negative:3 next:1 scenarios:1 classically:1 a55:1 opposed:1 conservation:3 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quant-ph9708032.txt |@word limited:1 consider:6 remarkable:1 keep:1 shor:2 lr1:1 lr0:2 commute:1 computation:1 consists:1 gk:2 fmrx:1 whose:1 maximally:1 depend:1 aug:1 transferred:3 iii:1 technique:1 send:2 passage:1 environment:5 finally:2 tj:2 those:2 under:1 nodes:2 include:1 resources:4 ta:4 case:5 supported:1 g2:2 g1:3 far:3 frequencies:2 induce:1 electron:1 lrr:1 summarized:1 experimentally:1 categories:1 four:1 hamiltonian:2 transmission:10 pellizzari:1 proposals:3 t0:6 t1:7 travels:1 optics:1 application:1 occurrence:1 pulse:4 sign:2 approximation:1 likely:1 implemented:2 linear:1 even:4 errors:30 incompatible:1 giving:1 near:1 diagrammatic:1 above:2 new:1 contrast:1 satisfy:1 full:1 simultaneously:1 degree:3 respectively:5 error:7 here:5 atoms:19 desired:1 inverse:3 possibility:1 standard:2 change:3 involved:1 leaves:1 losses:2 allows:1 leaving:1 suggests:1 knill:2 usually:1 resonance:2 implement:2 involves:1 via:1 useful:1 secure:1 extra:1 9a:1 9b:1 austria:1 turchette:1 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quant-ph9708033.txt |@word limited:1 grobe:2 represent:2 rev:13 lack:1 results:5 existing:1 lewenstein:1 aug:1 including:1 passage:1 tail:2 present:1 analogous:1 norm:1 01a:2 amplitudes:1 very:4 frequencies:2 peaks:1 supercomputer:1 harmonics:8 wave:6 electron:5 matrices:1 tri:1 presented:6 solution:1 occurs:1 swirling:1 dealt:1 optics:1 smaller:1 compensated:1 ten:2 pulse:8 approximation:2 second:1 equal:3 dynamics:4 implemented:1 theory:2 even:1 established:1 section:1 current:1 experiment:1 above:1 new:2 net:1 method:5 pictorially:1 evolving:1 degree:2 eksp:1 gif:4 residual:1 here:4 atoms:6 ionization:25 possibility:2 others:1 directions:1 based:1 along:1 strong:9 box:3 great:1 stark:1 institute:2 study:3 funded:1 approaches:5 through:3 eberly:4 amount:4 products:1 action:2 fourier:1 makes:1 plotted:1 followed:1 mobility:1 plenum:1 private:1 scaled:1 addressed:1 org:2 total:1 circumvent:1 linearly:1 would:1 finally:1 properties:1 explicitly:1 taken:1 theoretical:3 reflects:1 more:3 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quant-ph9708034.txt |@word partial:1 consider:4 magnetic:1 particular:1 results:1 leads:3 consists:2 whose:1 depend:1 calculate:6 aug:1 concerned:1 tn:5 nπ:1 under:3 stochastic:1 analogous:2 forbidden:1 hill:1 hibbs:1 springer:1 wave:23 valued:2 putting:1 bounding:1 unlike:1 difference:1 exact:1 condition:2 school:1 exponential:2 level:4 t2:2 solution:1 large:1 discounting:3 optics:2 setup:1 tel:4 measurable:1 maintain:1 zero:4 traveling:2 further:1 scientific:1 plays:1 itˆ:1 experiment:1 above:3 kleinert:2 new:1 method:3 contrast:1 coefficient:3 2ma2:3 never:1 consideration:1 along:2 equations:3 obtained:4 antisymmetric:1 schuss:4 resulting:1 action:1 makes:1 useful:1 polymer:1 explained:1 limn:1 remains:1 positive:1 contains:3 next:3 ψj:2 classically:1 ψn:1 therefore:4 survive:4 mckean:1 type:2 more:1 introducing:1 israel:2 initial:7 odinger:4 separates:1 k02:1 endpoint:1 defines:1 weighting:1 equation:5 remain:1 obvious:1 following:2 example:5 numbers:1 tau:2 do:3 calculation:2 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quant-ph9708035.txt |@word results:5 ge:1 go:1 follow:1 expressions:1 depend:1 calculate:1 aug:1 uniqueness:2 suffered:1 case:1 rise:1 wave:3 exact:7 condition:1 t0:2 solved:1 small:1 eigenvalues:3 sukhatme:1 sin2:5 approximation:3 notwithstanding:1 further:4 even:1 aim:1 what:1 appear:2 sum:1 expressed:1 lifshitz:1 new:1 method:3 wavefunctions:1 degree:2 here:1 sinm:1 address:1 appears:1 obtained:4 remark:1 rigorously:1 allows:1 usually:1 via:4 useful:1 requiring:1 apply:1 unit:2 interest:1 therefore:1 m2:5 until:1 more:1 formally:1 company:1 odinger:2 attained:1 known:1 hold:1 iii:1 springer:1 equation:6 work:5 modified:1 del:1 following:1 making:1 barclay:2 f0:2 theorems:1 give:1 calculation:3 topic:1 junker:1 sought:1 axis:1 needs:1 end:1 holds:1 pura:1 instead:1 parameter:5 sin:4 fs:1 may:3 after:2 derived:1 celoria:1 consequence:1 wrong:1 λ2h:1 outline:1 sq:1 lines:1 sk:1 sn:5 enter:1 order:14 s0:1 over:1 paper:1 statistical:1 its:7 roots:1 ln:1 solving:1 gen:3 systems:2 maxwell:1 mail:2 main:2 guess:1 landau:2 framework:1 bandrauk:3 identically:1 now:3 eigenvalue:5 term:2 always:1 l2:5 luca:1 reasonable:1 each:1 2m:2 universit:2 side:2 referred:1 wentzel:1 ed:1 energy:2 applicata:1 literature:1 quantum:9 our:1 eq:5 conjectures:1 shown:3 3rd:1 since:3 rev:1 dsk:1 istituto:4 formula:4 get:1 ref:2 correct:3 math:4 shows:1 turning:1 theory:3 associate:1 fisica:3 standard:3 difficulty:1 khare:2 besides:1 put:1 derivation:1 recovering:1 could:1 ds0:1 leading:7 corrections:1 think:1 first:5 already:1 number:1 done:1 angular:7 differ:1 ordering:1 unknown:1 dipartimento:2 their:1 gives:3 exactly:2 part:2 than:1 approximations:1 ingegneria:1 powers:1 lowest:1 explicit:1 correction:2 dθ2:1 kepler:1 null:1 orders:1 complete:1 depending:1 latter:1 converges:1 potential:2 coincide:1 subject:1 unless:1 shall:1 most:1 class:1 calculated:2 milano:2 points:1 exp:1 left:1 physics:2 fact:2 show:2 comtet:3 text:1 supported:1 relation:1 substituting:1 d2:1 find:1 ground:3 based:1 parameters:2 analytic:1 summarizing:1 override:1 hamiltonians:1 preliminaries:1 do:1 legendre:1 di:10 brillouin:1 dx:2 during:1 ds:1 arxiv:1 nθ:9 analytical:1 requirement:1 set:4 spherically:2 observed:1 edition:1 see:5 full:1 result:4 recently:2 contour:1 techniques:1 eigenfunctions:1 symmetric:2 state:3 invariant:2 conditions:1 stati:2 notice:2 terms:6 importance:1 were:2 however:1 robnik:7 cot:5 taking:1 quantization:9 expression:1 finds:1 otherwise:1 detailed:1 ch:1 dutt:2 dimensional:3 publishing:1 three:2 been:6 azimuthal:1 ph:1 define:1 supersymmetric:5 gas:1 spectrum:1 kramers:1 careful:1 former:1 present:2 applied:1 encircles:1 replaces:1 pacs:1 ǫn:1 value:3 will:2 della:2 wkb:6 ii:3 susy:6 partner:2 recover:1 italy:6 grant:1 perform:1 recursion:1 make:1 same:1 any:2 complex:2 quant:1 gets:1 murst:1 higher:4 independent:2 used:1 upon:2 hand:2 devoid:1 potentials:3 infinite:1 schr:2 nucleare:1 well:1 analysis:2 obviously:1 solve:1 position:1 salasnich:4 eimφ:1 they:2 just:1 generally:1 obtain:4 torus:1 shape:2 summed:1 campbell:1 lett:3 solvable:1 momentum:6 real:1 lacking:1 easily:2 rules:1 couple:1 possible:1 using:5 semiclassical:4 modifications:1 necessary:1 marzolo:1 semi:1 yields:2 sezione:1 fully:1 replacement:1 therein:1 reidel:1 right:1 methods:1 classical:1 some:1 proofs:1 cnr:1 though:1 stops:1 does:2 unipd:1 demonstrated:2 equivalent:1 radial:1 expansion:4 substitution:2 about:1 freedom:1 eqns:1 maslov:2 langer:5 italian:1 statements:1 pergamon:1 into:1 within:1 integral:3 included:1 references:2 operator:3 there:1 approximate:1 long:1 series:3 function:4 fabio:1 pd:1 form:3 but:3 corso:2 matematica:1 padova:9 compute:1 2πi:2 us:1 limit:1 problem:3 called:1 mechanics:6 cbc:1 constant:1 defined:1 abstract:1 pˆ:1 dθ:2 nazionale:3 formalism:1 materia:2 when:2 vanishes:1 arctan:1 other:1 role:1 details:1 igi:1 conclusion:1 briefly:1 repeat:1 problems:2 polynomials:1 a1:4 a0:2 important:2 variable:2 longer:1 time:1
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quant-ph9708036.txt |@word consider:1 global:1 ge:1 expressions:1 manfredi:2 worked:1 aug:1 marko:1 finally:1 compensate:1 conjecture:3 very:1 wave:1 exact:11 condition:1 entire:1 level:1 positions:1 solution:2 t0:2 correct:3 solved:2 dealt:2 small:1 eigenvalues:4 referred:3 sukhatme:1 sin2:2 approximation:3 rigorous:1 odd:3 even:4 appear:1 section:2 3n:4 lifshitz:3 new:1 method:1 1997b:3 1997a:3 full:2 degree:1 spacing:1 behaviour:1 following:1 sinm:1 cos2:3 address:1 sinf:1 along:1 appears:1 extreme:1 obtained:4 rigorously:2 study:2 studied:1 graffi:2 barclay:1 via:4 requiring:1 ls:1 apply:1 unit:2 use:1 prove:3 few:1 therefore:3 bender:3 m2:3 until:1 more:2 formally:1 becomes:2 company:1 odinger:2 particular:1 known:2 cases:1 iii:2 springer:2 valued:1 equation:3 work:5 modified:1 del:1 believed:1 proof:1 f0:2 wang:3 predict:1 junker:2 celoria:1 end:1 recursive:1 means:1 pura:1 instead:2 parameter:4 sin:4 dσ0:1 dσ1:1 fs:1 may:5 after:3 goal:1 λ2h:1 dσk:1 sq:1 spectra:1 murst:1 complexity:1 allow:1 order:8 paper:5 including:1 statistical:1 its:3 roots:1 ln:1 systems:3 pergamon:1 mail:2 main:1 them:1 landau:3 framework:1 maslov:2 half:1 now:2 discuss:1 eigenvalue:5 term:4 l37:1 university:1 l2:1 luca:1 reasonable:1 radial:2 2m:1 universit:2 higher:5 side:1 prosen:2 series:6 energy:4 ef:2 applicata:1 quantum:11 our:3 eq:12 out:1 shown:1 3rd:1 nuovo:1 since:4 rev:4 chaos:3 increase:1 istituto:4 formula:4 get:2 induction:1 integrated:1 math:5 voros:1 associate:1 hull:2 fisica:3 standard:1 difficulty:1 khare:1 york:1 ordering:1 corrections:1 first:4 already:2 number:2 clear:1 done:1 reached:1 angular:13 differ:1 introduction:1 leading:5 dipartimento:2 perfectly:1 gives:4 exactly:4 part:2 x2:1 x1:1 than:3 coefficients:1 approximations:1 cimento:1 ingegneria:1 powers:1 lowest:1 dθ2:1 kepler:3 orders:2 chirikov:2 general:2 latter:1 note:1 potential:2 coincide:1 accuracy:1 formulas:1 fundamental:1 shall:3 phases:1 analytically:1 especially:1 torus:6 milano:2 exp:1 left:1 physics:2 sum:1 fact:1 pd:1 show:3 comtet:4 text:1 supported:1 relation:1 solutions:1 d2:1 find:1 ground:3 completes:1 analytic:1 override:1 integrable:1 constant:1 hamiltonians:1 do:1 legendre:1 di:10 ones:1 dx:1 during:1 gen:4 fields:1 arxiv:1 gutzwiller:2 factorization:1 nθ:12 analytical:1 fixed:1 set:1 computed:1 differentials:1 observed:1 edition:1 see:4 individual:2 result:7 discussions:1 recently:1 contour:1 fails:1 eigenfunctions:2 enlightening:1 please:1 state:3 closed:1 c2k:2 stati:2 terms:8 importance:1 robnik:11 vanishing:1 σk:1 both:1 cos:2 last:2 σn:4 many:2 quantization:18 context:1 mean:1 expression:1 finds:2 detailed:1 cn:2 simple:1 dutt:1 dimensional:3 proved:3 publishing:1 three:1 been:6 quickly:1 azimuthal:1 expected:1 ph:1 pn:2 define:1 supersymmetric:5 gas:1 spectrum:1 observe:3 understand:1 systematic:1 c1:1 c0:1 former:1 present:2 cot:3 pacs:1 ǫn:1 value:2 will:3 2k:4 procedure:1 della:2 chaotic:1 wkb:16 thus:1 ii:7 reads:2 casati:2 susy:3 partner:1 vanish:1 disappear:1 italy:6 grant:1 perform:1 recursion:1 make:1 lett:3 same:3 any:1 quant:1 gets:1 difficult:1 calculation:1 used:2 hand:1 levels:1 potentials:3 infinite:1 recent:1 task:1 schr:2 nucleare:1 nevertheless:1 well:1 solve:1 position:2 salasnich:12 eimφ:1 they:2 just:1 generally:1 obtain:5 infeld:2 thanks:1 summed:3 campbell:4 eigenstates:1 previous:2 integer:1 conclusions:1 had:2 treatment:1 easy:1 dunham:2 mod:1 solvable:1 momentum:12 easily:1 possible:2 know:2 press:1 using:3 formal:1 semiclassical:11 necessary:2 marzolo:1 semi:1 separately:1 yields:1 sezione:1 italian:1 therein:1 reidel:1 because:4 methods:2 deal:1 moreover:1 cnr:1 bandrauk:4 integration:1 does:1 unipd:1 demonstrated:1 equivalent:1 expansion:12 substitution:2 properties:2 about:2 central:1 carried:1 freedom:1 eqns:1 langer:2 mutually:1 introduced:1 slightly:1 cambridge:2 maxwell:1 within:2 integral:1 implications:1 references:2 inverting:1 operator:4 proportional:1 classical:2 fraction:1 ed:1 was:3 function:3 fabio:1 complete:1 form:1 attempted:1 but:2 corso:2 matematica:1 convergence:3 padova:9 limit:1 demonstrate:1 problem:9 called:1 mechanics:6 cbc:1 associated:1 defined:1 certain:1 abstract:1 convergent:2 single:1 proves:1 pˆ:1 dθ:1 nazionale:3 formalism:1 materia:2 no:1 wavefunction:1 when:1 vanishes:1 arctan:1 other:2 role:1 details:2 igi:1 conclusion:1 polynomials:1 important:1 variable:1 structure:1 algorithm:1 4u:3 dσ:2 4m:1 original:1 once:1
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quant-ph9708037.txt |@word consider:2 rev:14 results:1 leads:1 jawaharlal:1 consists:1 follow:1 row:2 whose:1 careful:1 aug:1 tr:7 noncompact:1 finally:2 γn:2 th:1 under:12 p2:6 advantage:1 very:1 implicitly:1 choice:2 valued:1 every:7 entries:3 overcomes:1 little:1 matrices:1 condition:5 arrange:2 generalised:2 entire:1 smithey:2 presented:2 vector:7 pˆ4:1 science:1 manifestly:1 referred:1 noted:1 possibility:1 concise:3 dealing:1 direct:1 second:2 rigorous:1 further:5 canonical:10 even:1 will:9 what:1 errors:1 stands:1 tamarkin:1 sum:1 current:1 dunn:1 indian:1 subspaces:1 ˆj:2 preserved:1 coefficient:1 full:1 lee:1 respectively:1 tomographically:1 here:2 following:2 bech:2 objects:1 illustration:2 groups:1 meet:1 obtained:1 great:1 suitably:1 institute:2 vectors:2 approaches:1 resulting:1 action:2 fourier:1 studies:1 useful:2 analyzing:1 unit:2 conventional:2 nehru:1 always:1 positive:2 poizat:1 next:1 few:2 raymer:3 bender:1 themselves:1 type:1 theoretical:1 more:3 naive:2 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quant-ph9708038.txt |@word limited:1 concept:1 sci:1 consider:2 remarkable:1 squared:1 global:1 desirable:1 results:2 leads:3 consists:1 reˆ:1 whose:1 inverse:1 depend:1 aug:1 tr:1 γk:3 γj:2 quasi:2 does:2 squeezing:3 γn:12 analogous:1 dutta:1 conjecture:2 advantage:1 orthogonal:1 choice:2 every:4 matrices:1 condition:17 entire:2 incoherent:2 presented:2 turns:2 solution:4 discrete:1 recall:1 small:2 dip:1 noted:4 averaged:2 oscillation:12 direct:1 hillery:3 second:3 notwithstanding:3 qualitatively:1 odd:1 even:4 established:1 what:2 sub:2 invertible:1 tamarkin:1 expressed:1 indian:3 reconstruction:1 new:1 ˆm:1 ˆn:1 lee:1 continuing:2 q3:3 here:1 desired:1 simon:3 decay:1 consideration:1 along:1 z0:2 deduced:1 obtained:1 suitably:2 institute:4 involved:1 reason:1 amount:1 studies:1 involves:2 inference:1 marginal:1 highly:1 apply:1 sanjay:1 commun:1 distinct:1 finally:2 next:1 call:2 totality:2 criteria:3 taken:1 type:3 theoretical:1 more:1 initiated:1 operators:3 everywhere:2 virtue:2 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quant-ph9708040.txt |@word consider:2 crucial:1 desirable:1 results:5 leads:3 generalize:1 comments:1 ψ1n:1 consists:3 preserves:1 aug:1 tr:2 nonlocal:4 environment:1 quasi:3 preserved:3 case:4 straightforward:1 orthogonal:26 far:1 rise:2 wishes:1 experimentally:1 difference:1 matrices:4 presented:4 turns:3 solution:4 vector:3 phase:1 upper:1 deutsch:3 sin2:3 sign:1 imply:1 theory:1 even:2 what:2 nonlinear:5 xor:7 section:2 expressed:1 above:1 new:4 satisfy:1 ψii:1 outgoing:3 respectively:2 component:8 here:5 iσy:2 operating:1 represented:2 along:1 standard:3 change:1 obtained:10 leaves:1 vectors:6 ψ2n:1 stressed:1 bob:6 makes:3 involves:2 followed:1 projection:4 merely:1 outcomes:1 decrease:1 apply:1 2φi:2 total:1 select:1 use:2 figure:4 would:1 positive:5 finally:1 next:2 few:1 ψi:1 ψn:2 therefore:4 selects:1 frot:2 reflects:1 more:1 separated:2 successful:2 flying:2 operators:5 hereby:1 normalized:1 purification:6 known:5 cases:1 must:3 states:45 science:1 originally:1 work:1 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quant-ph9708041.txt |@word consider:1 remarkable:2 devoted:1 whose:1 calculate:1 aug:1 effort:1 tr:2 λn1:1 finally:1 th:1 kν:1 straightforward:1 far:1 putting:2 condition:1 pullback:1 abbreviated:2 2k3:1 cosh:6 second:1 further:1 canonical:2 diag:1 section:2 while:1 expressed:1 new:2 method:2 contrast:1 satisfy:3 groups:1 address:2 path:3 z1:6 obtained:2 published:1 usually:1 total:1 commun:2 next:2 comparing:2 noting:1 lowering:2 becomes:4 operators:3 berlin:1 appendix:2 normalized:1 projective:2 hold:1 eα:2 must:2 springer:1 work:1 oscillator:4 modified:4 fock:1 example:2 f1:38 compare:3 calculation:1 states:2 verlag:1 discussion:2 holds:1 write:3 very:1 ordinary:1 sin:6 map:2 product:5 algebra:2 may:2 oscillators:1 applications:1 natural:4 explicit:9 su:6 element:1 ones:1 so:1 representation:16 restriction:5 course:2 paper:4 looks:1 japan:1 satisfies:5 still:1 its:4 group:1 boson:2 forms:2 mail:2 overcome:4 them:3 number:4 they:3 eξk:1 eigenvalue:4 equality:1 hereafter:2 2r:25 university:1 utilized:1 2z:2 funahashi:4 l2:1 2n:1 2k:60 en:7 hypergeometric:4 extended:6 measure:41 quantum:1 our:3 eq:4 really:1 shown:2 space:5 restricted:1 hill:1 formula:3 induced:17 math:7 turning:1 theory:1 quite:1 reason:1 york:2 put:1 definition:4 skagerstam:1 kazuyuki:1 r2:7 first:3 origin:3 zn:10 xx:1 another:1 xs:1 angular:2 city:1 given:4 sakoda:3 introduction:1 construct:4 πr:1 their:2 identity:1 πz:1 1k:5 zα:7 zβ:3 kind:1 z1n1:2 annihilation:2 behaviors:1 iii:3 apparently:1 contents:1 barut:4 k3:3 k2:1 k1:1 generalized:1 coincide:1 measures:4 why:1 kk:3 clear:1 adopt:2 bateman:1 dot:1 show:3 attempts:1 d2:1 based:1 spite:2 eαα:1 explain:1 eαβ:3 written:3 analytic:11 bases:2 generators:1 notation:1 means:5 df:2 express:1 aα:4 aβ:2 dz:39 words:1 dt:1 dr:2 dp:2 reveal:1 covering:1 arxiv:1 investigate:2 nα:6 nβ:1 ik2:1 exists:1 sec:3 dµ:14 un1:2 review:2 state:41 un:3 august:1 jp:2 however:5 last:1 many:1 according:1 symplectic:21 casimir:4 expression:1 sinh:7 conclude:1 harmonic:5 dimensional:3 cp:5 cu:2 transformations:1 been:3 ph:1 define:2 kashiwa:3 coherent:43 derivative:1 case:3 fujii:6 near:3 behavior:4 property:1 concurrently:1 quesne:1 vol:1 thus:5 iv:1 ii:7 exterior:1 different:6 same:1 unity:5 complex:2 quant:1 physical:1 difficult:1 used:4 singapore:1 hand:4 infinite:3 normalization:1 well:1 academic:1 i2k:1 coordinate:2 dimension:1 manifold:1 being:1 λz:2 obtain:3 abbreviation:1 identify:1 klauder:1 λn:2 previous:2 now:3 preprint:1 lebesgue:1 except:1 theorem:1 trifonov:1 eigenstate:3 momentum:2 iν:3 possible:1 press:1 world:1 apart:1 dµk:2 follows:1 alternatively:1 essential:1 because:1 creation:1 some:3 examples:1 when:1 inrne:1 bessel:4 bg:38 scientific:1 bo:1 although:1 properties:2 comparison:1 extension:6 o:3 range:1 extensions:1 2x1:1 perelomov:5 into:3 integral:6 references:1 operator:7 log:4 vallee:1 assumed:1 there:4 restricts:2 inner:5 function:5 project:1 form:11 conclusively:1 construction:1 subsidiary:1 utilizing:1 whether:2 girardello:4 us:1 ωk:2 convince:1 ωm:2 ac:2 defined:13 abstract:1 dξ:6 dirac:1 provost:1 yokohama:4 constructed:3 functions:5 completeness:2 nn:22 no:1 reiθ:2 schwinger:3 other:5 department:1 mathematics:1 polar:2 briefly:1 june:1 meaning:2 important:1 longer:1 fact:1 n1:17 n2:2 resolution:5
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quant-ph9708042.txt |@word partial:1 consider:1 rev:12 magnetic:1 global:2 results:11 illustrated:1 eds:1 entropy:6 bohm:1 computation:1 consists:2 expressions:1 faster:1 gk:15 c0i:1 dimensionality:1 depend:1 aug:1 concerned:1 simplify:1 tr:4 essentialy:1 environment:16 does:3 fischer:1 those:1 stable:1 finitetemperature:1 stabilizing:1 analogous:3 worth:2 sound:1 belonging:2 eigenstate:1 orthogonal:1 very:5 multi:1 focus:1 worst:1 trough:1 mixing:1 emphasized:1 exact:5 condition:2 paz:1 dim:1 annihilated:2 presented:3 irreducible:3 trb:5 t2:1 solution:1 noiseless:4 integro:1 vector:5 math:2 small:2 eigenvalues:3 trs:1 optics:2 kozak:1 still:1 smaller:3 villa:1 direct:1 approximation:1 zero:7 design:1 further:3 dynamics:14 eigenspaces:1 theory:1 even:3 index:4 length:2 appear:2 section:4 analogously:1 access:1 subspaces:1 obvious:1 increasing:1 consisting:1 submitted:1 exchange:8 d1:3 respectively:3 never:1 here:4 atoms:1 behaviour:5 following:6 objects:1 possibility:1 groups:1 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