Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators

Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates |ϖxβg6ξααij(x, ξ, h)| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + |ξ|2)1/2 for every multi-indices α, β. We also assume that aij(...
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Autor*in:	Rabinovich, V. S. [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2009

Schlagwörter:	Dirac Operator Essential Spectrum Inverse Operator Exponential Estimate Analytic Symbol

Anmerkung:	© Pleiades Publishing, Ltd. 2009

Übergeordnetes Werk:	Enthalten in: Russian journal of mathematical physics - Moscow : MAIK Nauka/Interperiodics Publ., 2006, 16(2009), 2 vom: Juni, Seite 300-308
Übergeordnetes Werk:	volume:16 ; year:2009 ; number:2 ; month:06 ; pages:300-308

Links:	Volltext

DOI / URN:	10.1134/S1061920809020149

Katalog-ID:	SPR020321007

Internformat


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100	1		\|a Rabinovich, V. S. \|e verfasserin \|4 aut
245	1	0	\|a Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators
264		1	\|c 2009
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520			\|a Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates \|ϖxβg6ξααij(x, ξ, h)\| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + \|ξ\|2)1/2 for every multi-indices α, β. We also assume that aij(x, ξ) is analytically continued with respect to ξ to a tube domain $ ℝ^{n} $ + i%$, where %$ is a bounded domain in $ ℝ^{n} $ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let Hhs ($ ℝ^{n} $, $ ℂ^{N} $) be the space of distributions with values in $ ℂ^{N} $ which is equipped with the norm %$, let Ω ⊂ $ ℝ^{n} $ be a bounded open set, let v ∈ C∞($ ℝ^{n} $), let ▿v(x) ∈ %$ for any x ∈ Ω, and let %$. Let uh(∈ Hhs($ ℝ^{n} $,‒N)) be a solution of the equation $ Op_{h} $(α)u = 0. In this case, for every ϕ ∈ C0∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h0 > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h0]: (1)%$ We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schrödinger, Dirac, and square-root Klein-Gordon operators.
650		4	\|a Dirac Operator \|7 (dpeaa)DE-He213
650		4	\|a Essential Spectrum \|7 (dpeaa)DE-He213
650		4	\|a Inverse Operator \|7 (dpeaa)DE-He213
650		4	\|a Exponential Estimate \|7 (dpeaa)DE-He213
650		4	\|a Analytic Symbol \|7 (dpeaa)DE-He213
773	0	8	\|i Enthalten in \|t Russian journal of mathematical physics \|d Moscow : MAIK Nauka/Interperiodics Publ., 2006 \|g 16(2009), 2 vom: Juni, Seite 300-308 \|w (DE-627)510109861 \|w (DE-600)2229520-3 \|x 1555-6638 \|7 nnns
773	1	8	\|g volume:16 \|g year:2009 \|g number:2 \|g month:06 \|g pages:300-308
856	4	0	\|u https://dx.doi.org/10.1134/S1061920809020149 \|z lizenzpflichtig \|3 Volltext
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912			\|a GBV_ILN_63
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912			\|a GBV_ILN_69
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912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
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912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2070
912			\|a GBV_ILN_2086
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2116
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912			\|a GBV_ILN_2548
912			\|a GBV_ILN_4035
912			\|a GBV_ILN_4037
912			\|a GBV_ILN_4046
912			\|a GBV_ILN_4112
912			\|a GBV_ILN_4125
912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
912			\|a GBV_ILN_4305
912			\|a GBV_ILN_4306
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912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
912			\|a GBV_ILN_4324
912			\|a GBV_ILN_4325
912			\|a GBV_ILN_4326
912			\|a GBV_ILN_4333
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912			\|a GBV_ILN_4336
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951			\|a AR
952			\|d 16 \|j 2009 \|e 2 \|c 06 \|h 300-308

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author_variant	v s r vs vsr
matchkey_str	article:15556638:2009----::oaepnnilsiaefrpeddfeetaoeaosntneigoshdnedrc
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publishDate	2009
allfields	10.1134/S1061920809020149 doi (DE-627)SPR020321007 (SPR)S1061920809020149-e DE-627 ger DE-627 rakwb eng Rabinovich, V. S. verfasserin aut Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators 2009 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2009 Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates \|ϖxβg6ξααij(x, ξ, h)\| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + \|ξ\|2)1/2 for every multi-indices α, β. We also assume that aij(x, ξ) is analytically continued with respect to ξ to a tube domain $ ℝ^{n} $ + i%$, where %$ is a bounded domain in $ ℝ^{n} $ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let Hhs ($ ℝ^{n} $, $ ℂ^{N} $) be the space of distributions with values in $ ℂ^{N} $ which is equipped with the norm %$, let Ω ⊂ $ ℝ^{n} $ be a bounded open set, let v ∈ C∞($ ℝ^{n} $), let ▿v(x) ∈ %$ for any x ∈ Ω, and let %$. Let uh(∈ Hhs($ ℝ^{n} $,‒N)) be a solution of the equation $ Op_{h} $(α)u = 0. In this case, for every ϕ ∈ C0∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h0 > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h0]: (1)%$ We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schrödinger, Dirac, and square-root Klein-Gordon operators. Dirac Operator (dpeaa)DE-He213 Essential Spectrum (dpeaa)DE-He213 Inverse Operator (dpeaa)DE-He213 Exponential Estimate (dpeaa)DE-He213 Analytic Symbol (dpeaa)DE-He213 Enthalten in Russian journal of mathematical physics Moscow : MAIK Nauka/Interperiodics Publ., 2006 16(2009), 2 vom: Juni, Seite 300-308 (DE-627)510109861 (DE-600)2229520-3 1555-6638 nnns volume:16 year:2009 number:2 month:06 pages:300-308 https://dx.doi.org/10.1134/S1061920809020149 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2009 2 06 300-308
spelling	10.1134/S1061920809020149 doi (DE-627)SPR020321007 (SPR)S1061920809020149-e DE-627 ger DE-627 rakwb eng Rabinovich, V. S. verfasserin aut Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators 2009 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2009 Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates \|ϖxβg6ξααij(x, ξ, h)\| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + \|ξ\|2)1/2 for every multi-indices α, β. We also assume that aij(x, ξ) is analytically continued with respect to ξ to a tube domain $ ℝ^{n} $ + i%$, where %$ is a bounded domain in $ ℝ^{n} $ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let Hhs ($ ℝ^{n} $, $ ℂ^{N} $) be the space of distributions with values in $ ℂ^{N} $ which is equipped with the norm %$, let Ω ⊂ $ ℝ^{n} $ be a bounded open set, let v ∈ C∞($ ℝ^{n} $), let ▿v(x) ∈ %$ for any x ∈ Ω, and let %$. Let uh(∈ Hhs($ ℝ^{n} $,‒N)) be a solution of the equation $ Op_{h} $(α)u = 0. In this case, for every ϕ ∈ C0∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h0 > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h0]: (1)%$ We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schrödinger, Dirac, and square-root Klein-Gordon operators. Dirac Operator (dpeaa)DE-He213 Essential Spectrum (dpeaa)DE-He213 Inverse Operator (dpeaa)DE-He213 Exponential Estimate (dpeaa)DE-He213 Analytic Symbol (dpeaa)DE-He213 Enthalten in Russian journal of mathematical physics Moscow : MAIK Nauka/Interperiodics Publ., 2006 16(2009), 2 vom: Juni, Seite 300-308 (DE-627)510109861 (DE-600)2229520-3 1555-6638 nnns volume:16 year:2009 number:2 month:06 pages:300-308 https://dx.doi.org/10.1134/S1061920809020149 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2009 2 06 300-308
allfields_unstemmed	10.1134/S1061920809020149 doi (DE-627)SPR020321007 (SPR)S1061920809020149-e DE-627 ger DE-627 rakwb eng Rabinovich, V. S. verfasserin aut Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators 2009 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2009 Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates \|ϖxβg6ξααij(x, ξ, h)\| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + \|ξ\|2)1/2 for every multi-indices α, β. We also assume that aij(x, ξ) is analytically continued with respect to ξ to a tube domain $ ℝ^{n} $ + i%$, where %$ is a bounded domain in $ ℝ^{n} $ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let Hhs ($ ℝ^{n} $, $ ℂ^{N} $) be the space of distributions with values in $ ℂ^{N} $ which is equipped with the norm %$, let Ω ⊂ $ ℝ^{n} $ be a bounded open set, let v ∈ C∞($ ℝ^{n} $), let ▿v(x) ∈ %$ for any x ∈ Ω, and let %$. Let uh(∈ Hhs($ ℝ^{n} $,‒N)) be a solution of the equation $ Op_{h} $(α)u = 0. In this case, for every ϕ ∈ C0∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h0 > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h0]: (1)%$ We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schrödinger, Dirac, and square-root Klein-Gordon operators. Dirac Operator (dpeaa)DE-He213 Essential Spectrum (dpeaa)DE-He213 Inverse Operator (dpeaa)DE-He213 Exponential Estimate (dpeaa)DE-He213 Analytic Symbol (dpeaa)DE-He213 Enthalten in Russian journal of mathematical physics Moscow : MAIK Nauka/Interperiodics Publ., 2006 16(2009), 2 vom: Juni, Seite 300-308 (DE-627)510109861 (DE-600)2229520-3 1555-6638 nnns volume:16 year:2009 number:2 month:06 pages:300-308 https://dx.doi.org/10.1134/S1061920809020149 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2009 2 06 300-308
allfieldsGer	10.1134/S1061920809020149 doi (DE-627)SPR020321007 (SPR)S1061920809020149-e DE-627 ger DE-627 rakwb eng Rabinovich, V. S. verfasserin aut Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators 2009 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2009 Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates \|ϖxβg6ξααij(x, ξ, h)\| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + \|ξ\|2)1/2 for every multi-indices α, β. We also assume that aij(x, ξ) is analytically continued with respect to ξ to a tube domain $ ℝ^{n} $ + i%$, where %$ is a bounded domain in $ ℝ^{n} $ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let Hhs ($ ℝ^{n} $, $ ℂ^{N} $) be the space of distributions with values in $ ℂ^{N} $ which is equipped with the norm %$, let Ω ⊂ $ ℝ^{n} $ be a bounded open set, let v ∈ C∞($ ℝ^{n} $), let ▿v(x) ∈ %$ for any x ∈ Ω, and let %$. Let uh(∈ Hhs($ ℝ^{n} $,‒N)) be a solution of the equation $ Op_{h} $(α)u = 0. In this case, for every ϕ ∈ C0∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h0 > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h0]: (1)%$ We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schrödinger, Dirac, and square-root Klein-Gordon operators. Dirac Operator (dpeaa)DE-He213 Essential Spectrum (dpeaa)DE-He213 Inverse Operator (dpeaa)DE-He213 Exponential Estimate (dpeaa)DE-He213 Analytic Symbol (dpeaa)DE-He213 Enthalten in Russian journal of mathematical physics Moscow : MAIK Nauka/Interperiodics Publ., 2006 16(2009), 2 vom: Juni, Seite 300-308 (DE-627)510109861 (DE-600)2229520-3 1555-6638 nnns volume:16 year:2009 number:2 month:06 pages:300-308 https://dx.doi.org/10.1134/S1061920809020149 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2009 2 06 300-308
allfieldsSound	10.1134/S1061920809020149 doi (DE-627)SPR020321007 (SPR)S1061920809020149-e DE-627 ger DE-627 rakwb eng Rabinovich, V. S. verfasserin aut Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators 2009 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2009 Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates \|ϖxβg6ξααij(x, ξ, h)\| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + \|ξ\|2)1/2 for every multi-indices α, β. We also assume that aij(x, ξ) is analytically continued with respect to ξ to a tube domain $ ℝ^{n} $ + i%$, where %$ is a bounded domain in $ ℝ^{n} $ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let Hhs ($ ℝ^{n} $, $ ℂ^{N} $) be the space of distributions with values in $ ℂ^{N} $ which is equipped with the norm %$, let Ω ⊂ $ ℝ^{n} $ be a bounded open set, let v ∈ C∞($ ℝ^{n} $), let ▿v(x) ∈ %$ for any x ∈ Ω, and let %$. Let uh(∈ Hhs($ ℝ^{n} $,‒N)) be a solution of the equation $ Op_{h} $(α)u = 0. In this case, for every ϕ ∈ C0∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h0 > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h0]: (1)%$ We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schrödinger, Dirac, and square-root Klein-Gordon operators. Dirac Operator (dpeaa)DE-He213 Essential Spectrum (dpeaa)DE-He213 Inverse Operator (dpeaa)DE-He213 Exponential Estimate (dpeaa)DE-He213 Analytic Symbol (dpeaa)DE-He213 Enthalten in Russian journal of mathematical physics Moscow : MAIK Nauka/Interperiodics Publ., 2006 16(2009), 2 vom: Juni, Seite 300-308 (DE-627)510109861 (DE-600)2229520-3 1555-6638 nnns volume:16 year:2009 number:2 month:06 pages:300-308 https://dx.doi.org/10.1134/S1061920809020149 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2009 2 06 300-308
language	English
source	Enthalten in Russian journal of mathematical physics 16(2009), 2 vom: Juni, Seite 300-308 volume:16 year:2009 number:2 month:06 pages:300-308
sourceStr	Enthalten in Russian journal of mathematical physics 16(2009), 2 vom: Juni, Seite 300-308 volume:16 year:2009 number:2 month:06 pages:300-308
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Dirac Operator Essential Spectrum Inverse Operator Exponential Estimate Analytic Symbol
isfreeaccess_bool	false
container_title	Russian journal of mathematical physics
authorswithroles_txt_mv	Rabinovich, V. S. @@aut@@
publishDateDaySort_date	2009-06-01T00:00:00Z
hierarchy_top_id	510109861
id	SPR020321007
language_de	englisch
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author	Rabinovich, V. S.
spellingShingle	Rabinovich, V. S. misc Dirac Operator misc Essential Spectrum misc Inverse Operator misc Exponential Estimate misc Analytic Symbol Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators
authorStr	Rabinovich, V. S.
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issn	1555-6638
topic_title	Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators Dirac Operator (dpeaa)DE-He213 Essential Spectrum (dpeaa)DE-He213 Inverse Operator (dpeaa)DE-He213 Exponential Estimate (dpeaa)DE-He213 Analytic Symbol (dpeaa)DE-He213
topic	misc Dirac Operator misc Essential Spectrum misc Inverse Operator misc Exponential Estimate misc Analytic Symbol
topic_unstemmed	misc Dirac Operator misc Essential Spectrum misc Inverse Operator misc Exponential Estimate misc Analytic Symbol
topic_browse	misc Dirac Operator misc Essential Spectrum misc Inverse Operator misc Exponential Estimate misc Analytic Symbol
format_facet	Elektronische Aufsätze Aufsätze Elektronische Ressource
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hierarchy_parent_title	Russian journal of mathematical physics
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title	Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators
ctrlnum	(DE-627)SPR020321007 (SPR)S1061920809020149-e
title_full	Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators
author_sort	Rabinovich, V. S.
journal	Russian journal of mathematical physics
journalStr	Russian journal of mathematical physics
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container_start_page	300
author_browse	Rabinovich, V. S.
container_volume	16
format_se	Elektronische Aufsätze
author-letter	Rabinovich, V. S.
doi_str_mv	10.1134/S1061920809020149
title_sort	local exponential estimates for h-pseudodifferential operators and tunneling for schrödinger, dirac, and square root klein-gordon operators
title_auth	Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators
abstract	Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates \|ϖxβg6ξααij(x, ξ, h)\| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + \|ξ\|2)1/2 for every multi-indices α, β. We also assume that aij(x, ξ) is analytically continued with respect to ξ to a tube domain $ ℝ^{n} $ + i%$, where %$ is a bounded domain in $ ℝ^{n} $ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let Hhs ($ ℝ^{n} $, $ ℂ^{N} $) be the space of distributions with values in $ ℂ^{N} $ which is equipped with the norm %$, let Ω ⊂ $ ℝ^{n} $ be a bounded open set, let v ∈ C∞($ ℝ^{n} $), let ▿v(x) ∈ %$ for any x ∈ Ω, and let %$. Let uh(∈ Hhs($ ℝ^{n} $,‒N)) be a solution of the equation $ Op_{h} $(α)u = 0. In this case, for every ϕ ∈ C0∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h0 > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h0]: (1)%$ We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schrödinger, Dirac, and square-root Klein-Gordon operators. © Pleiades Publishing, Ltd. 2009
abstractGer	Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates \|ϖxβg6ξααij(x, ξ, h)\| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + \|ξ\|2)1/2 for every multi-indices α, β. We also assume that aij(x, ξ) is analytically continued with respect to ξ to a tube domain $ ℝ^{n} $ + i%$, where %$ is a bounded domain in $ ℝ^{n} $ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let Hhs ($ ℝ^{n} $, $ ℂ^{N} $) be the space of distributions with values in $ ℂ^{N} $ which is equipped with the norm %$, let Ω ⊂ $ ℝ^{n} $ be a bounded open set, let v ∈ C∞($ ℝ^{n} $), let ▿v(x) ∈ %$ for any x ∈ Ω, and let %$. Let uh(∈ Hhs($ ℝ^{n} $,‒N)) be a solution of the equation $ Op_{h} $(α)u = 0. In this case, for every ϕ ∈ C0∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h0 > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h0]: (1)%$ We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schrödinger, Dirac, and square-root Klein-Gordon operators. © Pleiades Publishing, Ltd. 2009
abstract_unstemmed	Abstract We consider the class of matrix h-pseudodifferential operators $ Op_{h} $(a) with symbols a = (aij)i,j=1N, where the coefficients aij ∈ C∞(ℝxn × ℝξn ⊗ C(0, 1] satisfy the estimates \|ϖxβg6ξααij(x, ξ, h)\| ⩽ Cαβ〈ξ〉m and 〈ξ〉 = (1 + \|ξ\|2)1/2 for every multi-indices α, β. We also assume that aij(x, ξ) is analytically continued with respect to ξ to a tube domain $ ℝ^{n} $ + i%$, where %$ is a bounded domain in $ ℝ^{n} $ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let Hhs ($ ℝ^{n} $, $ ℂ^{N} $) be the space of distributions with values in $ ℂ^{N} $ which is equipped with the norm %$, let Ω ⊂ $ ℝ^{n} $ be a bounded open set, let v ∈ C∞($ ℝ^{n} $), let ▿v(x) ∈ %$ for any x ∈ Ω, and let %$. Let uh(∈ Hhs($ ℝ^{n} $,‒N)) be a solution of the equation $ Op_{h} $(α)u = 0. In this case, for every ϕ ∈ C0∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h0 > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h0]: (1)%$ We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schrödinger, Dirac, and square-root Klein-Gordon operators. © Pleiades Publishing, Ltd. 2009
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title_short	Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators
url	https://dx.doi.org/10.1134/S1061920809020149
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doi_str	10.1134/S1061920809020149
up_date	2024-07-03T15:18:50.406Z
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Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators

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