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Selfadjoint, globally defined Hamiltonian operators for systems with boundaries
1.  Universidade Lusófona de Humanidades e Tecnologias, Av. Campo Grande 376, 1749024 Lisboa, Portugal, Portugal 
2.  Dipartimento di Scienze Fisiche e Matematiche, Università dell'Insubria, via valleggio 11, I22100 Como, Italy 
References:
[1] 
N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Space,", Pitman, (1981). Google Scholar 
[2] 
S. Albeverio, F. Gesztesy, R. HöghKrohn and H. Holden, "Solvable Models in Quantum Mechanics,", 2$^{nd}$ edition, (2005). Google Scholar 
[3] 
G. A. Baker, Formulation of quantum mechanics based on the quasiprobability distribution induced on phase space,, Phys. Rev., 109 (1958), 2198. Google Scholar 
[4] 
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. F. Sternheimer, Deformation theory and quantization, I and II,, Ann. Phys., 111 (1978), 61. Google Scholar 
[5] 
F. A. Berezin and L. D. Fadeev, Remark on the Schröinger equation with singular potential,, Dokl. Akad. Nauk. SSSR, 137 (1961), 1011. Google Scholar 
[6] 
J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains,, J. Funct. Anal., 243 (2007), 536. Google Scholar 
[7] 
M. S. Birman and M. Z. Solomjak, "Spectral Theory of SelfAdjoint Operators in Hilbert Spaces,", Reidel, (1987). Google Scholar 
[8] 
Ph. Blanchard, R. Figari and A. Mantile, Point interaction Hamiltonians in bounded domains,, J. Math. Phys., 48 (2007). Google Scholar 
[9] 
J. Blank, P. Exner and M. Havlíček, "Hilbert Space Operators in Quantum Physics,'', 2$^{nd}$ edition, (2008). Google Scholar 
[10] 
G. Bonneau, J. Faraut and G. Valent, Selfadjoint extensions of operators and the teaching of quantum mechanics,, Am. J. Phys., 69 (2001), 322. Google Scholar 
[11] 
A. Bracken, G. Cassinelli and J. Wood, Quantum symmetries and the WeylWigner product of group representations,, preprint, (). Google Scholar 
[12] 
B. M. Brown, M. Marletta, S. Naboko and I. G. Wood, Boundary triplets and Mfunctions for nonselfadjoint operators, with applications to elliptic PDEs and block operator matrices,, J. Lond. Math. Soc., 77 (2008), 700. Google Scholar 
[13] 
B. M. Brown, G. Grubb and I. G. Wood, Mfunctions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems,, Math. Nachr., 282 (2009), 314. Google Scholar 
[14] 
C. Cacciapuoti, R. Carlone and R. Figari, Spin dependent point potentials in one and three dimensions,, J. Phys. A: Math. Gen., 40 (2007), 249. Google Scholar 
[15] 
J. W. Calkin, Abstract symmetric boundary conditions,, Trans. Am. Math. Soc., 45 (1939), 369. Google Scholar 
[16] 
A. Connes, "Noncommutative Geometry,", Academic Press, (1994). Google Scholar 
[17] 
C. R. de Oliveira, "Intermediate Spectral Theory and Quantum Dynamics,", Birkh\, (2009). Google Scholar 
[18] 
N. C. Dias and J. N. Prata, Wigner functions with boundaries,, J. Math. Phys., 43 (2002), 4602. Google Scholar 
[19] 
N. C. Dias, A. Posilicano and J. N. Prata, in, in preparation., (). Google Scholar 
[20] 
N. C. Dias and J. N. Prata, Admissible states in quantum phase space,, Ann. Phys., 313 (2004), 110. Google Scholar 
[21] 
N. C. Dias and J. N. Prata, Comment on "On infinite walls in deformation quantization",, Ann. Phys., 321 (2006), 495. Google Scholar 
[22] 
D. Dubin, M. Hennings and T. Smith, "Mathematical Aspects of Weyl Quantization,", World Scientific, (2000). Google Scholar 
[23] 
W. Faris, "SelfAdjoint Operators,", Lecture Notes in Mathematics {\bf 433}, 433 (1975). Google Scholar 
[24] 
D. Fairlie, The formulation of quantum mechanics in terms of phase space functions,, Proc. Camb. Phil. Soc., 60 (1964), 581. Google Scholar 
[25] 
B. Fedosov, A simple geometric construction of deformation quantization,, J. Diff. Geom., 40 (1994), 213. Google Scholar 
[26] 
R. Gambini and R. A. Porto, Relational time in generally covariant quantum systems: four models,, Phys. Rev., D 63 (2001). Google Scholar 
[27] 
P. Garbaczewski and W. Karwowski, Impenetrable barriers and canonical quantization,, Am. J. Phys., 72 (2004), 924. Google Scholar 
[28] 
F. Gesztesy and M. Mitrea, RobintoRobin maps and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Modern Analysis and Applications. The Mark Krein Centenary Conference. Vol. 2: Differential Operators and Mechanics'', (eds. V. Adamyan et al.), (2009), 81. Google Scholar 
[29] 
F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, RobintoDirichlet maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains,, in, 79 (2008), 105. Google Scholar 
[30] 
V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations,", Kluver, (1991). Google Scholar 
[31] 
M. de Gosson and F. Luef, A new approach to the $\star$genvalue equation,, Lett. Math. Phys., 85 (2008), 173. Google Scholar 
[32] 
G. Grubb, A characterization of the non local boundary value problems associated with an elliptic operator,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 425. Google Scholar 
[33] 
G. Grubb, Krein resolvent formulas for elliptic boundary problems in nonsmooth domains,, Rend. Sem. Mat. Univ. Pol. Torino, 66 (2008), 271. Google Scholar 
[34] 
C. Isham, Topological and global aspects of quantum theory,, in, (1984), 1059. Google Scholar 
[35] 
M. Kontsevich, Deformation quantization of Poisson manifolds I,, Lett. Math. Phys., 66 (2003), 157. Google Scholar 
[36] 
M. G. Kre\u\i n, The theory of selfadjoint extensions of halfbounded Hermitean operators and their applications I,, Mat. Sbornik N.S., 20 (1947), 431. Google Scholar 
[37] 
M. G. Kreĭn, The theory of selfadjoint extensions of halfbounded Hermitean operators and their applications II,, Mat. Sbornik N.S., 21 (1947), 365. Google Scholar 
[38] 
K. Kowalski, K. Podlaski and J. Rembieliński, Quantum mechanics of a free particle on a plane with an extracted point,, Phys. Rev. A, 66 (2002), 032118. Google Scholar 
[39] 
S. Kryukov and M. A. Walton, On infinite walls in deformation quantization,, Ann. Phys., 317 (2005), 474. Google Scholar 
[40] 
J. L. Lions and E. Magenes, Problèmes aux limites non homogènes II,, Ann. Institut Fourier, 11 (1961), 137. Google Scholar 
[41] 
J. L. Lions and E. Magenes, "NonHomogeneous Boundary Value Problems and Applications I,", SpringerVerlag, (1972). Google Scholar 
[42] 
J. Madore, "An Introduction to Noncommutative Differential Geometry and its Physical Applications,", 2$^{nd}$ edition, (2000). Google Scholar 
[43] 
M. A. Naimark, "Theory of Linear Differential Operators,", Frederick Ungar Publishing Co., (1967). Google Scholar 
[44] 
J. von Neumann, Allgemeine eigenwerttheorie Hermitscher funktionaloperatoren,, Math. Ann., 102 (1929), 49. Google Scholar 
[45] 
J. von Neumann, "Mathematische Grundlagen der Quantenmechanik,", SpringerVerlag, (1932). Google Scholar 
[46] 
A. Pinzul and A. Stern, Absence of the holographic principle in noncommutative ChernSimons theory,, J. High Energy Phys., 0111 (2001). Google Scholar 
[47] 
A. Posilicano, A Kreinlike formula for singular perturbations of selfadjoint operators and applications,, J. Funct. Anal., 183 (2001), 109. Google Scholar 
[48] 
A. Posilicano, Selfadjoint extensions by additive perturbations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 1. Google Scholar 
[49] 
A. Posilicano, Selfadjoint extensions of restrictions,, Oper. Matrices, 2 (2008), 483. Google Scholar 
[50] 
A. Posilicano and L. Raimondi, Krein's resolvent formula for selfadjoint extensions of symmetric second order elliptic differential operators,, J. Phys. A: Math. Theor., 42 (2009). Google Scholar 
[51] 
M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, SelfAdjointness,", Academic Press, (1975). Google Scholar 
[52] 
V. Ryzhov, A general boundary value problem and its Weyl function,, Opuscula Math., 27 (2007), 305. Google Scholar 
[53] 
N. Seiberg and E. Witten, String theory and noncommutative geometry,, J. High Energy Phys., 9909 (1999). Google Scholar 
[54] 
M. L. Višik, On general boundary problems for elliptic differential equations,, Trudy Mosc. Mat. Obsv., 1 (1952), 186. Google Scholar 
[55] 
B. Voronov, D. Gitman and I. Tyutin, Selfadjoint differential operators associated with selfadjoint differential expressions,, preprint, (). Google Scholar 
[56] 
J. Weidmann, "Linear Operators in Hilbert Spaces,", SpringerVerlag, (1980). Google Scholar 
[57] 
M. A. Walton, Wigner functions, contact interactions, and matching,, Ann. Phys., 322 (2007), 2233. Google Scholar 
[58] 
M. W. Wong, "Weyl Transforms,", SpringerVerlag, (1998). Google Scholar 
show all references
References:
[1] 
N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Space,", Pitman, (1981). Google Scholar 
[2] 
S. Albeverio, F. Gesztesy, R. HöghKrohn and H. Holden, "Solvable Models in Quantum Mechanics,", 2$^{nd}$ edition, (2005). Google Scholar 
[3] 
G. A. Baker, Formulation of quantum mechanics based on the quasiprobability distribution induced on phase space,, Phys. Rev., 109 (1958), 2198. Google Scholar 
[4] 
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. F. Sternheimer, Deformation theory and quantization, I and II,, Ann. Phys., 111 (1978), 61. Google Scholar 
[5] 
F. A. Berezin and L. D. Fadeev, Remark on the Schröinger equation with singular potential,, Dokl. Akad. Nauk. SSSR, 137 (1961), 1011. Google Scholar 
[6] 
J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains,, J. Funct. Anal., 243 (2007), 536. Google Scholar 
[7] 
M. S. Birman and M. Z. Solomjak, "Spectral Theory of SelfAdjoint Operators in Hilbert Spaces,", Reidel, (1987). Google Scholar 
[8] 
Ph. Blanchard, R. Figari and A. Mantile, Point interaction Hamiltonians in bounded domains,, J. Math. Phys., 48 (2007). Google Scholar 
[9] 
J. Blank, P. Exner and M. Havlíček, "Hilbert Space Operators in Quantum Physics,'', 2$^{nd}$ edition, (2008). Google Scholar 
[10] 
G. Bonneau, J. Faraut and G. Valent, Selfadjoint extensions of operators and the teaching of quantum mechanics,, Am. J. Phys., 69 (2001), 322. Google Scholar 
[11] 
A. Bracken, G. Cassinelli and J. Wood, Quantum symmetries and the WeylWigner product of group representations,, preprint, (). Google Scholar 
[12] 
B. M. Brown, M. Marletta, S. Naboko and I. G. Wood, Boundary triplets and Mfunctions for nonselfadjoint operators, with applications to elliptic PDEs and block operator matrices,, J. Lond. Math. Soc., 77 (2008), 700. Google Scholar 
[13] 
B. M. Brown, G. Grubb and I. G. Wood, Mfunctions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems,, Math. Nachr., 282 (2009), 314. Google Scholar 
[14] 
C. Cacciapuoti, R. Carlone and R. Figari, Spin dependent point potentials in one and three dimensions,, J. Phys. A: Math. Gen., 40 (2007), 249. Google Scholar 
[15] 
J. W. Calkin, Abstract symmetric boundary conditions,, Trans. Am. Math. Soc., 45 (1939), 369. Google Scholar 
[16] 
A. Connes, "Noncommutative Geometry,", Academic Press, (1994). Google Scholar 
[17] 
C. R. de Oliveira, "Intermediate Spectral Theory and Quantum Dynamics,", Birkh\, (2009). Google Scholar 
[18] 
N. C. Dias and J. N. Prata, Wigner functions with boundaries,, J. Math. Phys., 43 (2002), 4602. Google Scholar 
[19] 
N. C. Dias, A. Posilicano and J. N. Prata, in, in preparation., (). Google Scholar 
[20] 
N. C. Dias and J. N. Prata, Admissible states in quantum phase space,, Ann. Phys., 313 (2004), 110. Google Scholar 
[21] 
N. C. Dias and J. N. Prata, Comment on "On infinite walls in deformation quantization",, Ann. Phys., 321 (2006), 495. Google Scholar 
[22] 
D. Dubin, M. Hennings and T. Smith, "Mathematical Aspects of Weyl Quantization,", World Scientific, (2000). Google Scholar 
[23] 
W. Faris, "SelfAdjoint Operators,", Lecture Notes in Mathematics {\bf 433}, 433 (1975). Google Scholar 
[24] 
D. Fairlie, The formulation of quantum mechanics in terms of phase space functions,, Proc. Camb. Phil. Soc., 60 (1964), 581. Google Scholar 
[25] 
B. Fedosov, A simple geometric construction of deformation quantization,, J. Diff. Geom., 40 (1994), 213. Google Scholar 
[26] 
R. Gambini and R. A. Porto, Relational time in generally covariant quantum systems: four models,, Phys. Rev., D 63 (2001). Google Scholar 
[27] 
P. Garbaczewski and W. Karwowski, Impenetrable barriers and canonical quantization,, Am. J. Phys., 72 (2004), 924. Google Scholar 
[28] 
F. Gesztesy and M. Mitrea, RobintoRobin maps and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Modern Analysis and Applications. The Mark Krein Centenary Conference. Vol. 2: Differential Operators and Mechanics'', (eds. V. Adamyan et al.), (2009), 81. Google Scholar 
[29] 
F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, RobintoDirichlet maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains,, in, 79 (2008), 105. Google Scholar 
[30] 
V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations,", Kluver, (1991). Google Scholar 
[31] 
M. de Gosson and F. Luef, A new approach to the $\star$genvalue equation,, Lett. Math. Phys., 85 (2008), 173. Google Scholar 
[32] 
G. Grubb, A characterization of the non local boundary value problems associated with an elliptic operator,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 425. Google Scholar 
[33] 
G. Grubb, Krein resolvent formulas for elliptic boundary problems in nonsmooth domains,, Rend. Sem. Mat. Univ. Pol. Torino, 66 (2008), 271. Google Scholar 
[34] 
C. Isham, Topological and global aspects of quantum theory,, in, (1984), 1059. Google Scholar 
[35] 
M. Kontsevich, Deformation quantization of Poisson manifolds I,, Lett. Math. Phys., 66 (2003), 157. Google Scholar 
[36] 
M. G. Kre\u\i n, The theory of selfadjoint extensions of halfbounded Hermitean operators and their applications I,, Mat. Sbornik N.S., 20 (1947), 431. Google Scholar 
[37] 
M. G. Kreĭn, The theory of selfadjoint extensions of halfbounded Hermitean operators and their applications II,, Mat. Sbornik N.S., 21 (1947), 365. Google Scholar 
[38] 
K. Kowalski, K. Podlaski and J. Rembieliński, Quantum mechanics of a free particle on a plane with an extracted point,, Phys. Rev. A, 66 (2002), 032118. Google Scholar 
[39] 
S. Kryukov and M. A. Walton, On infinite walls in deformation quantization,, Ann. Phys., 317 (2005), 474. Google Scholar 
[40] 
J. L. Lions and E. Magenes, Problèmes aux limites non homogènes II,, Ann. Institut Fourier, 11 (1961), 137. Google Scholar 
[41] 
J. L. Lions and E. Magenes, "NonHomogeneous Boundary Value Problems and Applications I,", SpringerVerlag, (1972). Google Scholar 
[42] 
J. Madore, "An Introduction to Noncommutative Differential Geometry and its Physical Applications,", 2$^{nd}$ edition, (2000). Google Scholar 
[43] 
M. A. Naimark, "Theory of Linear Differential Operators,", Frederick Ungar Publishing Co., (1967). Google Scholar 
[44] 
J. von Neumann, Allgemeine eigenwerttheorie Hermitscher funktionaloperatoren,, Math. Ann., 102 (1929), 49. Google Scholar 
[45] 
J. von Neumann, "Mathematische Grundlagen der Quantenmechanik,", SpringerVerlag, (1932). Google Scholar 
[46] 
A. Pinzul and A. Stern, Absence of the holographic principle in noncommutative ChernSimons theory,, J. High Energy Phys., 0111 (2001). Google Scholar 
[47] 
A. Posilicano, A Kreinlike formula for singular perturbations of selfadjoint operators and applications,, J. Funct. Anal., 183 (2001), 109. Google Scholar 
[48] 
A. Posilicano, Selfadjoint extensions by additive perturbations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 1. Google Scholar 
[49] 
A. Posilicano, Selfadjoint extensions of restrictions,, Oper. Matrices, 2 (2008), 483. Google Scholar 
[50] 
A. Posilicano and L. Raimondi, Krein's resolvent formula for selfadjoint extensions of symmetric second order elliptic differential operators,, J. Phys. A: Math. Theor., 42 (2009). Google Scholar 
[51] 
M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, SelfAdjointness,", Academic Press, (1975). Google Scholar 
[52] 
V. Ryzhov, A general boundary value problem and its Weyl function,, Opuscula Math., 27 (2007), 305. Google Scholar 
[53] 
N. Seiberg and E. Witten, String theory and noncommutative geometry,, J. High Energy Phys., 9909 (1999). Google Scholar 
[54] 
M. L. Višik, On general boundary problems for elliptic differential equations,, Trudy Mosc. Mat. Obsv., 1 (1952), 186. Google Scholar 
[55] 
B. Voronov, D. Gitman and I. Tyutin, Selfadjoint differential operators associated with selfadjoint differential expressions,, preprint, (). Google Scholar 
[56] 
J. Weidmann, "Linear Operators in Hilbert Spaces,", SpringerVerlag, (1980). Google Scholar 
[57] 
M. A. Walton, Wigner functions, contact interactions, and matching,, Ann. Phys., 322 (2007), 2233. Google Scholar 
[58] 
M. W. Wong, "Weyl Transforms,", SpringerVerlag, (1998). Google Scholar 
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