November  2011, 10(6): 1687-1706. doi: 10.3934/cpaa.2011.10.1687

Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

1. 

Universidade Lusófona de Humanidades e Tecnologias, Av. Campo Grande 376, 1749-024 Lisboa, Portugal, Portugal

2. 

Dipartimento di Scienze Fisiche e Matematiche, Università dell'Insubria, via valleggio 11, I-22100 Como, Italy

Received  March 2010 Revised  February 2011 Published  May 2011

For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(R^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(R^d)$ that dynamically confine the system to an open set $\Omega \subset \RE^d$ while reproducing the action of $ H_0$ on an appropriate operator domain. In the case $H_0=-\Delta +V$ we construct these Hamiltonians explicitly showing that they can be written in the form $H=H_0+ B$, where $B$ is a singular boundary potential and $H$ is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.
Citation: Nuno Costa Dias, Andrea Posilicano, João Nuno Prata. Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1687-1706. doi: 10.3934/cpaa.2011.10.1687
References:
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N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Space," Pitman, Boston, 1981.

[2]

S. Albeverio, F. Gesztesy, R. Högh-Krohn and H. Holden, "Solvable Models in Quantum Mechanics," 2nd edition, Am. Math. Soc., Providence, Rhode Island, 2005.

[3]

G. A. Baker, Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space, Phys. Rev., 109 (1958), 2198-2206.

[4]

F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. F. Sternheimer, Deformation theory and quantization, I and II, Ann. Phys., 111 (1978), 61-110; Ann. Phys., 110 (1978), 111-151.

[5]

F. A. Berezin and L. D. Fadeev, Remark on the Schröinger equation with singular potential, Dokl. Akad. Nauk. SSSR, 137 (1961), 1011-1014.

[6]

J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536-565.

[7]

M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Spaces," Reidel, Dordrecht, Holland, 1987.

[8]

Ph. Blanchard, R. Figari and A. Mantile, Point interaction Hamiltonians in bounded domains, J. Math. Phys., 48 (2007), 082108.

[9]

J. Blank, P. Exner and M. Havlíček, "Hilbert Space Operators in Quantum Physics,'' 2nd edition, Springer-Verlag, Berlin, 2008.

[10]

G. Bonneau, J. Faraut and G. Valent, Self-adjoint extensions of operators and the teaching of quantum mechanics, Am. J. Phys., 69 (2001), 322-331.

[11]

A. Bracken, G. Cassinelli and J. Wood, Quantum symmetries and the Weyl-Wigner product of group representations,, preprint, (). 

[12]

B. M. Brown, M. Marletta, S. Naboko and I. G. Wood, Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, J. Lond. Math. Soc., 77 (2008), 700-718.

[13]

B. M. Brown, G. Grubb and I. G. Wood, M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems, Math. Nachr., 282 (2009), 314-347.

[14]

C. Cacciapuoti, R. Carlone and R. Figari, Spin dependent point potentials in one and three dimensions, J. Phys. A: Math. Gen., 40 (2007), 249-261.

[15]

J. W. Calkin, Abstract symmetric boundary conditions, Trans. Am. Math. Soc., 45 (1939), 369-342.

[16]

A. Connes, "Noncommutative Geometry," Academic Press, New-York, 1994.

[17]

C. R. de Oliveira, "Intermediate Spectral Theory and Quantum Dynamics," Birkhäuser, Basel, 2009.

[18]

N. C. Dias and J. N. Prata, Wigner functions with boundaries, J. Math. Phys., 43 (2002), 4602-4627.

[19]

N. C. Dias, A. Posilicano and J. N. Prata, in, in preparation., (). 

[20]

N. C. Dias and J. N. Prata, Admissible states in quantum phase space, Ann. Phys., 313 (2004) 110-146.

[21]

N. C. Dias and J. N. Prata, Comment on "On infinite walls in deformation quantization", Ann. Phys., 321, (2006) 495-502.

[22]

D. Dubin, M. Hennings and T. Smith, "Mathematical Aspects of Weyl Quantization," World Scientific, Singapore, 2000.

[23]

W. Faris, "Self-Adjoint Operators," Lecture Notes in Mathematics 433, Springer-Verlag, Berlin, 1975.

[24]

D. Fairlie, The formulation of quantum mechanics in terms of phase space functions, Proc. Camb. Phil. Soc., 60 (1964), 581-586.

[25]

B. Fedosov, A simple geometric construction of deformation quantization, J. Diff. Geom., 40 (1994), 213-238.

[26]

R. Gambini and R. A. Porto, Relational time in generally covariant quantum systems: four models, Phys. Rev., D 63 (2001), 105014.

[27]

P. Garbaczewski and W. Karwowski, Impenetrable barriers and canonical quantization, Am. J. Phys., 72 (2004), 924-933.

[28]

F. Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type 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.), Birkhäuser Verlag, 2009, 81-113.

[29]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Perspectives in Partial Differential Equations, Harmonic Analysis and Applications'' (Proc. Sympos. Pure Math., 79), Amer. Math. Soc., Providence, Rhode Island, 2008, 105-173.

[30]

V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations," Kluver, Dordrecht, 1991.

[31]

M. de Gosson and F. Luef, A new approach to the $\star$-genvalue equation, Lett. Math. Phys., 85 (2008), 173-183.

[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), 425-513.

[33]

G. Grubb, Krein resolvent formulas for elliptic boundary problems in nonsmooth domains, Rend. Sem. Mat. Univ. Pol. Torino, 66 (2008), 271-297.

[34]

C. Isham, Topological and global aspects of quantum theory, in "Relativity, groups and topology II, Les Houches Session XL" (eds. B.S. DeWitt and R. Stora), North-Holland, Amsterdam, 1984, 1059-1290.

[35]

M. Kontsevich, Deformation quantization of Poisson manifolds I, Lett. Math. Phys., 66 (2003), 157-216.

[36]

M. G. Kre\u\i n, The theory of self-adjoint extensions of half-bounded Hermitean operators and their applications I, Mat. Sbornik N.S., 20 (1947), 431-495, (Russian).

[37]

M. G. Kreĭn, The theory of self-adjoint extensions of half-bounded Hermitean operators and their applications II, Mat. Sbornik N.S., 21 (1947), 365-404, (Russian).

[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), 032118-1-9.

[39]

S. Kryukov and M. A. Walton, On infinite walls in deformation quantization, Ann. Phys., 317 (2005), 474-491.

[40]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes II, Ann. Institut Fourier, 11 (1961), 137-178.

[41]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I," Springer-Verlag, Berlin, 1972.

[42]

J. Madore, "An Introduction to Noncommutative Differential Geometry and its Physical Applications," 2nd edition, Cambridge University Press, Cambridge, 2000.

[43]

M. A. Naimark, "Theory of Linear Differential Operators," Frederick Ungar Publishing Co., New York, 1967.

[44]

J. von Neumann, Allgemeine eigenwerttheorie Hermitscher funktionaloperatoren, Math. Ann., 102 (1929), 49-131.

[45]

J. von Neumann, "Mathematische Grundlagen der Quantenmechanik," Springer-Verlag, Berlin, 1932.

[46]

A. Pinzul and A. Stern, Absence of the holographic principle in noncommutative Chern-Simons theory, J. High Energy Phys., 0111 (2001), Paper 23, 14 pp.

[47]

A. Posilicano, A Krein-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal., 183 (2001), 109-147.

[48]

A. Posilicano, Self-adjoint extensions by additive perturbations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 1-20.

[49]

A. Posilicano, Self-adjoint extensions of restrictions, Oper. Matrices, 2 (2008), 483-506.

[50]

A. Posilicano and L. Raimondi, Krein's resolvent formula for self-adjoint extensions of symmetric second order elliptic differential operators, J. Phys. A: Math. Theor., 42 (2009), 015204.

[51]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, Self-Adjointness," Academic Press, London, 1975.

[52]

V. Ryzhov, A general boundary value problem and its Weyl function, Opuscula Math., 27 (2007), 305-331.

[53]

N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys., 9909 (1999), Paper 32, 93 pp.

[54]

M. L. Višik, On general boundary problems for elliptic differential equations, Trudy Mosc. Mat. Obsv., 1 (1952), 186-246 (Russian); translated in Amer. Math. Soc. Trans., 24 (1963), 107-172.

[55]

B. Voronov, D. Gitman and I. Tyutin, Self-adjoint differential operators associated with self-adjoint differential expressions,, preprint, (). 

[56]

J. Weidmann, "Linear Operators in Hilbert Spaces," Springer-Verlag, Berlin, 1980.

[57]

M. A. Walton, Wigner functions, contact interactions, and matching, Ann. Phys., 322 (2007), 2233-2248.

[58]

M. W. Wong, "Weyl Transforms," Springer-Verlag, Berlin, 1998.

show all references

References:
[1]

N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Space," Pitman, Boston, 1981.

[2]

S. Albeverio, F. Gesztesy, R. Högh-Krohn and H. Holden, "Solvable Models in Quantum Mechanics," 2nd edition, Am. Math. Soc., Providence, Rhode Island, 2005.

[3]

G. A. Baker, Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space, Phys. Rev., 109 (1958), 2198-2206.

[4]

F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. F. Sternheimer, Deformation theory and quantization, I and II, Ann. Phys., 111 (1978), 61-110; Ann. Phys., 110 (1978), 111-151.

[5]

F. A. Berezin and L. D. Fadeev, Remark on the Schröinger equation with singular potential, Dokl. Akad. Nauk. SSSR, 137 (1961), 1011-1014.

[6]

J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536-565.

[7]

M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Spaces," Reidel, Dordrecht, Holland, 1987.

[8]

Ph. Blanchard, R. Figari and A. Mantile, Point interaction Hamiltonians in bounded domains, J. Math. Phys., 48 (2007), 082108.

[9]

J. Blank, P. Exner and M. Havlíček, "Hilbert Space Operators in Quantum Physics,'' 2nd edition, Springer-Verlag, Berlin, 2008.

[10]

G. Bonneau, J. Faraut and G. Valent, Self-adjoint extensions of operators and the teaching of quantum mechanics, Am. J. Phys., 69 (2001), 322-331.

[11]

A. Bracken, G. Cassinelli and J. Wood, Quantum symmetries and the Weyl-Wigner product of group representations,, preprint, (). 

[12]

B. M. Brown, M. Marletta, S. Naboko and I. G. Wood, Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, J. Lond. Math. Soc., 77 (2008), 700-718.

[13]

B. M. Brown, G. Grubb and I. G. Wood, M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems, Math. Nachr., 282 (2009), 314-347.

[14]

C. Cacciapuoti, R. Carlone and R. Figari, Spin dependent point potentials in one and three dimensions, J. Phys. A: Math. Gen., 40 (2007), 249-261.

[15]

J. W. Calkin, Abstract symmetric boundary conditions, Trans. Am. Math. Soc., 45 (1939), 369-342.

[16]

A. Connes, "Noncommutative Geometry," Academic Press, New-York, 1994.

[17]

C. R. de Oliveira, "Intermediate Spectral Theory and Quantum Dynamics," Birkhäuser, Basel, 2009.

[18]

N. C. Dias and J. N. Prata, Wigner functions with boundaries, J. Math. Phys., 43 (2002), 4602-4627.

[19]

N. C. Dias, A. Posilicano and J. N. Prata, in, in preparation., (). 

[20]

N. C. Dias and J. N. Prata, Admissible states in quantum phase space, Ann. Phys., 313 (2004) 110-146.

[21]

N. C. Dias and J. N. Prata, Comment on "On infinite walls in deformation quantization", Ann. Phys., 321, (2006) 495-502.

[22]

D. Dubin, M. Hennings and T. Smith, "Mathematical Aspects of Weyl Quantization," World Scientific, Singapore, 2000.

[23]

W. Faris, "Self-Adjoint Operators," Lecture Notes in Mathematics 433, Springer-Verlag, Berlin, 1975.

[24]

D. Fairlie, The formulation of quantum mechanics in terms of phase space functions, Proc. Camb. Phil. Soc., 60 (1964), 581-586.

[25]

B. Fedosov, A simple geometric construction of deformation quantization, J. Diff. Geom., 40 (1994), 213-238.

[26]

R. Gambini and R. A. Porto, Relational time in generally covariant quantum systems: four models, Phys. Rev., D 63 (2001), 105014.

[27]

P. Garbaczewski and W. Karwowski, Impenetrable barriers and canonical quantization, Am. J. Phys., 72 (2004), 924-933.

[28]

F. Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type 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.), Birkhäuser Verlag, 2009, 81-113.

[29]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Perspectives in Partial Differential Equations, Harmonic Analysis and Applications'' (Proc. Sympos. Pure Math., 79), Amer. Math. Soc., Providence, Rhode Island, 2008, 105-173.

[30]

V. I. Gorbachuk and M. L. Gorbachuk, "Boundary Value Problems for Operator Differential Equations," Kluver, Dordrecht, 1991.

[31]

M. de Gosson and F. Luef, A new approach to the $\star$-genvalue equation, Lett. Math. Phys., 85 (2008), 173-183.

[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), 425-513.

[33]

G. Grubb, Krein resolvent formulas for elliptic boundary problems in nonsmooth domains, Rend. Sem. Mat. Univ. Pol. Torino, 66 (2008), 271-297.

[34]

C. Isham, Topological and global aspects of quantum theory, in "Relativity, groups and topology II, Les Houches Session XL" (eds. B.S. DeWitt and R. Stora), North-Holland, Amsterdam, 1984, 1059-1290.

[35]

M. Kontsevich, Deformation quantization of Poisson manifolds I, Lett. Math. Phys., 66 (2003), 157-216.

[36]

M. G. Kre\u\i n, The theory of self-adjoint extensions of half-bounded Hermitean operators and their applications I, Mat. Sbornik N.S., 20 (1947), 431-495, (Russian).

[37]

M. G. Kreĭn, The theory of self-adjoint extensions of half-bounded Hermitean operators and their applications II, Mat. Sbornik N.S., 21 (1947), 365-404, (Russian).

[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), 032118-1-9.

[39]

S. Kryukov and M. A. Walton, On infinite walls in deformation quantization, Ann. Phys., 317 (2005), 474-491.

[40]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes II, Ann. Institut Fourier, 11 (1961), 137-178.

[41]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications I," Springer-Verlag, Berlin, 1972.

[42]

J. Madore, "An Introduction to Noncommutative Differential Geometry and its Physical Applications," 2nd edition, Cambridge University Press, Cambridge, 2000.

[43]

M. A. Naimark, "Theory of Linear Differential Operators," Frederick Ungar Publishing Co., New York, 1967.

[44]

J. von Neumann, Allgemeine eigenwerttheorie Hermitscher funktionaloperatoren, Math. Ann., 102 (1929), 49-131.

[45]

J. von Neumann, "Mathematische Grundlagen der Quantenmechanik," Springer-Verlag, Berlin, 1932.

[46]

A. Pinzul and A. Stern, Absence of the holographic principle in noncommutative Chern-Simons theory, J. High Energy Phys., 0111 (2001), Paper 23, 14 pp.

[47]

A. Posilicano, A Krein-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal., 183 (2001), 109-147.

[48]

A. Posilicano, Self-adjoint extensions by additive perturbations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 1-20.

[49]

A. Posilicano, Self-adjoint extensions of restrictions, Oper. Matrices, 2 (2008), 483-506.

[50]

A. Posilicano and L. Raimondi, Krein's resolvent formula for self-adjoint extensions of symmetric second order elliptic differential operators, J. Phys. A: Math. Theor., 42 (2009), 015204.

[51]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, Self-Adjointness," Academic Press, London, 1975.

[52]

V. Ryzhov, A general boundary value problem and its Weyl function, Opuscula Math., 27 (2007), 305-331.

[53]

N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys., 9909 (1999), Paper 32, 93 pp.

[54]

M. L. Višik, On general boundary problems for elliptic differential equations, Trudy Mosc. Mat. Obsv., 1 (1952), 186-246 (Russian); translated in Amer. Math. Soc. Trans., 24 (1963), 107-172.

[55]

B. Voronov, D. Gitman and I. Tyutin, Self-adjoint differential operators associated with self-adjoint differential expressions,, preprint, (). 

[56]

J. Weidmann, "Linear Operators in Hilbert Spaces," Springer-Verlag, Berlin, 1980.

[57]

M. A. Walton, Wigner functions, contact interactions, and matching, Ann. Phys., 322 (2007), 2233-2248.

[58]

M. W. Wong, "Weyl Transforms," Springer-Verlag, Berlin, 1998.

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